On fundamental aspects of quantum extreme learning machines (2312.15124v2)
Abstract: Quantum Extreme Learning Machines (QELMs) have emerged as a promising framework for quantum machine learning. Their appeal lies in the rich feature map induced by the dynamics of a quantum substrate - the quantum reservoir - and the efficient post-measurement training via linear regression. Here we study the expressivity of QELMs by decomposing the prediction of QELMs into a Fourier series. We show that the achievable Fourier frequencies are determined by the data encoding scheme, while Fourier coefficients depend on both the reservoir and the measurement. Notably, the expressivity of QELMs is fundamentally limited by the number of Fourier frequencies and the number of observables, while the complexity of the prediction hinges on the reservoir. As a cautionary note on scalability, we identify four sources that can lead to the exponential concentration of the observables as the system size grows (randomness, hardware noise, entanglement, and global measurements) and show how this can turn QELMs into useless input-agnostic oracles. In particular, our result on the reservoir-induced concentration strongly indicates that quantum reservoirs drawn from a highly random ensemble make QELM models unscalable. Our analysis elucidates the potential and fundamental limitations of QELMs, and lays the groundwork for systematically exploring quantum reservoir systems for other machine learning tasks.
- G.-B. Huang, Q.-Y. Zhu, and C.-K. Siew, Extreme learning machine: a new learning scheme of feedforward neural networks, in 2004 IEEE international joint conference on neural networks (IEEE Cat. No. 04CH37541), Vol. 2 (Ieee, 2004) pp. 985–990.
- I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, 2016).
- G.-B. Huang, Q.-Y. Zhu, and C.-K. Siew, Extreme learning machine: theory and applications, Neurocomputing 70, 489 (2006).
- S. Ghosh, T. Paterek, and T. C. H. Liew, Quantum neuromorphic platform for quantum state preparation, Phys. Rev. Lett. 123, 260404 (2019a).
- K. Fujii and K. Nakajima, Harnessing disordered-ensemble quantum dynamics for machine learning, Physical Review Applied 8, 024030 (2017).
- M. Schuld, R. Sweke, and J. J. Meyer, Effect of data encoding on the expressive power of variational quantum-machine-learning models, Physical Review A 103, 032430 (2021).
- F. J. Schreiber, J. Eisert, and J. J. Meyer, Classical surrogates for quantum learning models, Physical Review Letters 131, 100803 (2023).
- M. Cerezo and P. J. Coles, Higher order derivatives of quantum neural networks with barren plateaus, Quantum Science and Technology 6, 035006 (2021).
- C. O. Marrero, M. Kieferová, and N. Wiebe, Entanglement-induced barren plateaus, PRX Quantum 2, 040316 (2021).
- A. Uvarov and J. D. Biamonte, On barren plateaus and cost function locality in variational quantum algorithms, Journal of Physics A: Mathematical and Theoretical 54, 245301 (2021).
- D. García-Martín, M. Larocca, and M. Cerezo, Deep quantum neural networks form gaussian processes, arXiv preprint arXiv:2305.09957 (2023).
- B. Y. Gan, D. Leykam, and S. Thanasilp, A unified framework for trace-induced quantum kernels, arXiv preprint arXiv:2311.13552 https://doi.org/10.48550/arXiv.2311.13552 (2023).
- T. Goto, Q. H. Tran, and K. Nakajima, Universal approximation property of quantum machine learning models in quantum-enhanced feature spaces, Physical Review Letters 127, 090506 (2021).
- L. Gonon and A. Jacquier, Universal approximation theorem and error bounds for quantum neural networks and quantum reservoirs, arXiv preprint arXiv:2307.12904 10.48550/arXiv.2307.12904 (2023).
- U. Sainz de la Maza Gamboa, Quantum extreme learning machine for classification tasks, Ph.D. thesis, Universidad del Pais Vasco (2022).
- S. Shin, Y. Teo, and H. Jeong, Exponential data encoding for quantum supervised learning, Physical Review A 107, 012422 (2023).
- D. A. Roberts and B. Yoshida, Chaos and complexity by design, Journal of High Energy Physics 2017, 121 (2017).
- Y. Liu, S. Arunachalam, and K. Temme, A rigorous and robust quantum speed-up in supervised machine learning, Nature Physics , 1 (2021).
- A. Kutvonen, K. Fujii, and T. Sagawa, Optimizing a quantum reservoir computer for time series prediction, Scientific Reports 10, 14687 (2020).
- A. Sornsaeng, N. Dangniam, and T. Chotibut, Quantum next generation reservoir computing: An efficient quantum algorithm for forecasting quantum dynamics, arXiv preprint arXiv:2308.14239 (2023).
- A. B. Tsybakov, Introduction to Nonparametric Estimation (Springer, 2009).