2000 character limit reached
Nav-Q: Quantum Deep Reinforcement Learning for Collision-Free Navigation of Self-Driving Cars (2311.12875v2)
Published 20 Nov 2023 in quant-ph, cs.AI, and cs.LG
Abstract: The task of collision-free navigation (CFN) of self-driving cars is an NP-hard problem usually tackled using Deep Reinforcement Learning (DRL). While DRL methods have proven to be effective, their implementation requires substantial computing resources and extended training periods to develop a robust agent. On the other hand, quantum reinforcement learning has recently demonstrated faster convergence and improved stability in simple, non-real-world environments. In this work, we propose Nav-Q, the first quantum-supported DRL algorithm for CFN of self-driving cars, that leverages quantum computation for improving the training performance without the requirement for onboard quantum hardware. Nav-Q is based on the actor-critic approach, where the critic is implemented using a hybrid quantum-classical algorithm suitable for near-term quantum devices. We assess the performance of Nav-Q using the CARLA driving simulator, a de facto standard benchmark for evaluating state-of-the-art DRL methods. Our empirical evaluations showcase that Nav-Q surpasses its classical counterpart in terms of training stability and, in certain instances, with respect to the convergence rate. Furthermore, we assess Nav-Q in relation to effective dimension, unveiling that the incorporation of a quantum component results in a model with greater descriptive power compared to classical baselines. Finally, we evaluate the performance of Nav-Q using noisy quantum simulation, observing that the quantum noise deteriorates the training performances but enhances the exploratory tendencies of the agent during training.
- Chib, P.S., Singh, P.: Recent advancements in end-to-end autonomous driving using deep learning: A survey. IEEE Transactions on Intelligent Vehicles, 1–18 (2023) https://doi.org/10.1109/TIV.2023.3318070 Kiran et al. [2021] Kiran, B.R., Sobh, I., Talpaert, V., Mannion, P., Al Sallab, A.A., Yogamani, S., Pérez, P.: Deep reinforcement learning for autonomous driving: A survey. IEEE Transactions on Intelligent Transportation Systems 23(6), 4909–4926 (2021) Pusse and Klusch [2019] Pusse, F., Klusch, M.: Hybrid online pomdp planning and deep reinforcement learning for safer self-driving cars. In: 2019 IEEE Intelligent Vehicles Symposium (IV), pp. 1013–1020 (2019). https://doi.org/10.1109/IVS.2019.8814125 Gupta and Klusch [2023] Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kiran, B.R., Sobh, I., Talpaert, V., Mannion, P., Al Sallab, A.A., Yogamani, S., Pérez, P.: Deep reinforcement learning for autonomous driving: A survey. IEEE Transactions on Intelligent Transportation Systems 23(6), 4909–4926 (2021) Pusse and Klusch [2019] Pusse, F., Klusch, M.: Hybrid online pomdp planning and deep reinforcement learning for safer self-driving cars. In: 2019 IEEE Intelligent Vehicles Symposium (IV), pp. 1013–1020 (2019). https://doi.org/10.1109/IVS.2019.8814125 Gupta and Klusch [2023] Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pusse, F., Klusch, M.: Hybrid online pomdp planning and deep reinforcement learning for safer self-driving cars. In: 2019 IEEE Intelligent Vehicles Symposium (IV), pp. 1013–1020 (2019). https://doi.org/10.1109/IVS.2019.8814125 Gupta and Klusch [2023] Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Kiran, B.R., Sobh, I., Talpaert, V., Mannion, P., Al Sallab, A.A., Yogamani, S., Pérez, P.: Deep reinforcement learning for autonomous driving: A survey. IEEE Transactions on Intelligent Transportation Systems 23(6), 4909–4926 (2021) Pusse and Klusch [2019] Pusse, F., Klusch, M.: Hybrid online pomdp planning and deep reinforcement learning for safer self-driving cars. In: 2019 IEEE Intelligent Vehicles Symposium (IV), pp. 1013–1020 (2019). https://doi.org/10.1109/IVS.2019.8814125 Gupta and Klusch [2023] Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pusse, F., Klusch, M.: Hybrid online pomdp planning and deep reinforcement learning for safer self-driving cars. In: 2019 IEEE Intelligent Vehicles Symposium (IV), pp. 1013–1020 (2019). https://doi.org/10.1109/IVS.2019.8814125 Gupta and Klusch [2023] Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Pusse, F., Klusch, M.: Hybrid online pomdp planning and deep reinforcement learning for safer self-driving cars. In: 2019 IEEE Intelligent Vehicles Symposium (IV), pp. 1013–1020 (2019). https://doi.org/10.1109/IVS.2019.8814125 Gupta and Klusch [2023] Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Gupta, D., Klusch, M.: Hylear: Hybrid deep reinforcement learning and planning for safe and comfortable automated driving. In: 2023 IEEE Intelligent Vehicles Symposium (IV), pp. 1–8 (2023). https://doi.org/10.1109/IV55152.2023.10186781 Jerbi et al. [2021] Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Jerbi, S., Gyurik, C., Marshall, S., Briegel, H., Dunjko, V.: Parametrized quantum policies for reinforcement learning. Advances in Neural Information Processing Systems 34, 28362–28375 (2021) Lan [2021] Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Lan, Q.: Variational quantum soft actor-critic. arXiv preprint:2112.11921 (2021) Chen et al. [2020] Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Chen, S.Y.-C., Yang, C.-H.H., Qi, J., Chen, P.-Y., Ma, X., Goan, H.-S.: Variational quantum circuits for deep reinforcement learning. IEEE Access 8, 141007–141024 (2020) Skolik et al. [2022] Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Skolik, A., Jerbi, S., Dunjko, V.: Quantum agents in the gym: a variational quantum algorithm for deep q-learning. Quantum 6, 720 (2022) Brockman et al. [2016] Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Brockman, G., Cheung, V., Pettersson, L., Schneider, J., Schulman, J., Tang, J., Zaremba, W.: Openai gym. arXiv preprint:1606.01540 (2016) Pérez-Salinas et al. [2020] Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Pérez-Salinas, A., Cervera-Lierta, A., Gil-Fuster, E., Latorre, J.I.: Data re-uploading for a universal quantum classifier. Quantum 4, 226 (2020) Mnih et al. [2016] Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Mnih, V., Badia, A.P., Mirza, M., Graves, A., Lillicrap, T., Harley, T., Silver, D., Kavukcuoglu, K.: Asynchronous methods for deep reinforcement learning. In: International Conference on Machine Learning, pp. 1928–1937 (2016). PMLR Mirowski et al. [2016] Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Mirowski, P., Pascanu, R., Viola, F., Soyer, H., Ballard, A.J., Banino, A., Denil, M., Goroshin, R., Sifre, L., Kavukcuoglu, K., et al.: Learning to navigate in complex environments. arXiv preprint:1611.03673 (2016) Everett et al. [2021] Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Everett, M., Chen, Y.F., How, J.P.: Collision avoidance in pedestrian-rich environments with deep reinforcement learning. IEEE Access 9, 10357–10377 (2021) Hochreiter and Schmidhuber [1997] Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Hochreiter, S., Schmidhuber, J.: Long Short-Term Memory. Neural Computation 9(8), 1735–1780 (1997) https://doi.org/10.1162/neco.1997.9.8.1735 Schulman et al. [2017] Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Schulman, J., Wolski, F., Dhariwal, P., Radford, A., Klimov, O.: Proximal policy optimization algorithms. arXiv preprint:1707.06347 (2017) Tang [2019] Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Tang, Y.: Towards learning multi-agent negotiations via self-play. In: Proceedings of the IEEE/CVF International Conference on Computer Vision Workshops (2019) Bergholm et al. [2018] Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Bergholm, V., Izaac, J., Schuld, M., Gogolin, C., Ahmed, S., Ajith, V., Alam, M.S., Alonso-Linaje, G., AkashNarayanan, B., Asadi, A., et al.: Pennylane: Automatic differentiation of hybrid quantum-classical computations. arXiv preprint arXiv:1811.04968 (2018) Lockwood and Si [2020] Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Lockwood, O., Si, M.: Reinforcement learning with quantum variational circuit. In: Proceedings of the AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, vol. 16, pp. 245–251 (2020) Sutton et al. [1999] Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Sutton, R.S., McAllester, D., Singh, S., Mansour, Y.: Policy gradient methods for reinforcement learning with function approximation. Advances in neural information processing systems 12 (1999) Geva and Sitte [1993] Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Geva, S., Sitte, J.: A cartpole experiment benchmark for trainable controllers. Control Systems, IEEE 13, 40–51 (1993) https://doi.org/10.1109/37.236324 Kwak et al. [2021] Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Kwak, Y., Yun, W.J., Jung, S., Kim, J.-K., Kim, J.: Introduction to quantum reinforcement learning: Theory and pennylane-based implementation. In: 2021 International Conference on Information and Communication Technology Convergence (ICTC), pp. 416–420 (2021). IEEE Acuto et al. [2022] Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Acuto, A., Barillà, P., Bozzolo, L., Conterno, M., Pavese, M., Policicchio, A.: Variational quantum soft actor-critic for robotic arm control. arXiv:2212.11681 (2022) Kandala et al. [2017] Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Kandala, A., Mezzacapo, A., Temme, K., Takita, M., Brink, M., Chow, J.M., Gambetta, J.M.: Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature 549(7671), 242–246 (2017) Dosovitskiy et al. [2017] Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Dosovitskiy, A., Ros, G., Codevilla, F., Lopez, A., Koltun, V.: Carla: An open urban driving simulator. In: Conference on Robot Learning, pp. 1–16 (2017). PMLR Bartels and Erbsmehl [2014] Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Bartels, B., Erbsmehl, C.: Bewegungsverhalten von fußgängern im straßenverkehr, teil 1. FAT-Schriftenreihe (267) (2014) Ba et al. [2016] Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Ba, J.L., Kiros, J.R., Hinton, G.E.: Layer normalization. arXiv preprint arXiv:1607.06450 (2016) Mitarai et al. [2018] Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Mitarai, K., Negoro, M., Kitagawa, M., Fujii, K.: Quantum circuit learning. Physical Review A 98(3), 032309 (2018) Schuld et al. [2019] Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Schuld, M., Bergholm, V., Gogolin, C., Izaac, J., Killoran, N.: Evaluating analytic gradients on quantum hardware. Physical Review A 99(3), 032331 (2019) Rigetti Computing [Accessed: 2023-09-19] Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Rigetti Computing: Quantum Gate Errors - PyQuil Documentation. Website. https://pyquil-docs.rigetti.com/en/stable/noise.html#quantum-gate-errors King [2003] King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- King, C.: The capacity of the quantum depolarizing channel. IEEE Transactions on Information Theory 49(1), 221–229 (2003) Berezniuk et al. [2020] Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Berezniuk, O., Figalli, A., Ghigliazza, R., Musaelian, K.: A scale-dependent notion of effective dimension. arXiv preprint arXiv:2001.10872 (2020) Abbas et al. [2021] Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Abbas, A., Sutter, D., Zoufal, C., Lucchi, A., Figalli, A., Woerner, S.: The power of quantum neural networks. Nature Computational Science 1(6), 403–409 (2021) Grant et al. [2019] Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Grant, E., Wossnig, L., Ostaszewski, M., Benedetti, M.: An initialization strategy for addressing barren plateaus in parametrized quantum circuits. Quantum 3, 214 (2019) Skolik et al. [2021] Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Skolik, A., McClean, J.R., Mohseni, M., Smagt, P., Leib, M.: Layerwise learning for quantum neural networks. Quantum Machine Intelligence 3, 1–11 (2021) Holmes et al. [2022] Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022) Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)
- Holmes, Z., Sharma, K., Cerezo, M., Coles, P.J.: Connecting ansatz expressibility to gradient magnitudes and barren plateaus. PRX Quantum 3(1), 010313 (2022)