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When can classical neural networks represent quantum states? (2410.23152v1)

Published 30 Oct 2024 in quant-ph, cond-mat.str-el, and cs.LG

Abstract: A naive classical representation of an n-qubit state requires specifying exponentially many amplitudes in the computational basis. Past works have demonstrated that classical neural networks can succinctly express these amplitudes for many physically relevant states, leading to computationally powerful representations known as neural quantum states. What underpins the efficacy of such representations? We show that conditional correlations present in the measurement distribution of quantum states control the performance of their neural representations. Such conditional correlations are basis dependent, arise due to measurement-induced entanglement, and reveal features not accessible through conventional few-body correlations often examined in studies of phases of matter. By combining theoretical and numerical analysis, we demonstrate how the state's entanglement and sign structure, along with the choice of measurement basis, give rise to distinct patterns of short- or long-range conditional correlations. Our findings provide a rigorous framework for exploring the expressive power of neural quantum states.

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References (76)
  1. Google Quantum AI and Collaborators. Measurement-induced entanglement and teleportation on a noisy quantum processor. Nature, 622(7983):481–486, 2023.
  2. An area law for 2d frustration-free spin systems. arXiv preprint arXiv:2103.02492, 2021.
  3. An area law and sub-exponential algorithm for 1D systems. arXiv preprint arXiv:1301.1162, 2013.
  4. Improved one-dimensional area law for frustration-free systems. Phys. Rev. B, 85:195145, May 2012.
  5. Rigorous RG algorithms and area laws for low energy eigenstates in 1D. Comm. Math. Phys., 356(1):65–105, 2017.
  6. Finite-time teleportation phase transition in random quantum circuits. Phys. Rev. Lett., 132:030401, Jan 2024.
  7. Quantum advantage from measurement-induced entanglement in random shallow circuits. arXiv preprint arXiv:2407.21203, 2024.
  8. The problem of learning long-term dependencies in recurrent networks. In IEEE international conference on neural networks, pages 1183–1188. IEEE, 1993.
  9. Noise and the frontier of quantum supremacy. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science—FOCS 2021, pages 1308–1317. IEEE Computer Soc., Los Alamitos, CA, [2022] ©2022.
  10. (pseudo) random quantum states with binary phase. In Theory of Cryptography Conference, pages 229–250. Springer, 2019.
  11. Finite correlation length implies efficient preparation of quantum thermal states. Communications in Mathematical Physics, pages 1–16, 2016.
  12. A rapidly mixing Markov chain from any gapped quantum many-body system. Quantum, 7:1173, November 2023.
  13. How to simulate quantum measurement without computing marginals. Phys. Rev. Lett., 128:220503, Jun 2022.
  14. Complexity of stoquastic frustration-free Hamiltonians. SIAM J. Comput., 39(4):1462–1485, 2009/10.
  15. Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape. SciPost Phys., 10:147, 2021.
  16. Netket: A machine learning toolkit for many-body quantum systems. SoftwareX, page 100311, 2019.
  17. Solving the quantum many-body problem with artificial neural networks. Science, 355(6325):602–606, 2017.
  18. Reconstructing quantum states with generative models. Nature Machine Intelligence, 1(3):155–161, 2019.
  19. Unitary-projective entanglement dynamics. Phys. Rev. B, 99:224307, Jun 2019.
  20. Sign problem in tensor network contraction. arXiv preprint arXiv:2404.19023, 2024.
  21. Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning. J. ACM, 69(5):Art. 33, 72, 2022.
  22. Two-dimensional frustrated J1−J2subscript𝐽1subscript𝐽2{J}_{1}\text{$-$}{J}_{2}italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT model studied with neural network quantum states. Phys. Rev. B, 100:125124, Sep 2019.
  23. Random quantum circuits anticoncentrate in log depth. PRX Quantum, 3:010333, Mar 2022.
  24. The DeepMind JAX Ecosystem, 2020.
  25. Neural gutzwiller-projected variational wave functions. Phys. Rev. B, 100:125131, Sep 2019.
  26. The ITensor Software Library for Tensor Network Calculations. SciPost Phys. Codebases, page 4, 2022.
  27. Efficient representation of quantum many-body states with deep neural networks. Nature communications, 8(1):662, 2017.
  28. Random insights into the complexity of two-dimensional tensor network calculations. Phys. Rev. B, 109:235102, Jun 2024.
  29. Johnnie Gray. quimb: a python library for quantum information and many-body calculations. Journal of Open Source Software, 3(29):819, 2018.
  30. Easing the monte carlo sign problem. Science advances, 6(33):eabb8341, 2020.
  31. Matthew B Hastings. An area law for one-dimensional quantum systems. Journal of Statistical Mechanics: Theory and Experiment, 2007(08):P08024, 2007.
  32. Matthew B Hastings. How quantum are non-negative wavefunctions? Journal of Mathematical Physics, 57(1):015210, 2016.
  33. An entropy inequality. Quantum Information & Computation, 9(7):622–627, 2009.
  34. Recurrent neural network wave functions. Phys. Rev. Res., 2:023358, Jun 2020.
  35. Investigating topological order using recurrent neural networks. Phys. Rev. B, 108:075152, Aug 2023.
  36. Entanglement of a pair of quantum bits. Phys. Rev. Lett., 78:5022–5025, Jun 1997.
  37. Predicting many properties of a quantum system from very few measurements. Nature Physics, 16(10):1050–1057, Oct 2020.
  38. Certifying almost all quantum states with few single-qubit measurements. arXiv preprint arXiv:2404.07281, 2024.
  39. Neural network enhanced measurement efficiency for molecular groundstates. Machine Learning: Science and Technology, 4(1):015016, 2023.
  40. Pseudorandom quantum states. In Advances in Cryptology–CRYPTO 2018: 38th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19–23, 2018, Proceedings, Part III 38, pages 126–152. Springer, 2018.
  41. Jiaqing Jiang. Local hamiltonian problem with succinct ground state is ma-complete. arXiv preprint arXiv:2309.10155, 2023.
  42. Positive bias makes tensor-network contraction tractable. arXiv preprint arXiv:2410.05414, 2024.
  43. Adam: A method for stochastic optimization. CoRR, abs/1412.6980, 2014.
  44. Entanglement entropy transitions with random tensor networks. arXiv preprint arXiv:2108.02225, 2021.
  45. Quantum zeno effect and the many-body entanglement transition. Phys. Rev. B, 98:205136, Nov 2018.
  46. Probing sign structure using measurement-induced entanglement. arXiv preprint arXiv:2205.05692, 2022.
  47. Measurement-induced entanglement transition in a two-dimensional shallow circuit. Phys. Rev. B, 106:144311, Oct 2022.
  48. Efficient representation of topologically ordered states with restricted boltzmann machines. Phys. Rev. B, 99:155136, Apr 2019.
  49. Measurement-induced entanglement and complexity in random constant-depth 2D circuits. To appear, 2024.
  50. Restricted boltzmann machines in quantum physics. Nature Physics, 15(9):887–892, 2019.
  51. Language models for quantum simulation. Nature Computational Science, pages 1–8, 2024.
  52. Yannick Meurice. Experimental entanglement entropy without twin copy. arXiv preprint arXiv:2404.09935, 2024.
  53. Ramis Movassagh. The hardness of random quantum circuits. Nature Physics, 19(11):1719–1724, 2023.
  54. Efficient classical simulation of random shallow 2d quantum circuits. Phys. Rev. X, 12:021021, Apr 2022.
  55. This is the same as changing X𝑋Xitalic_X and Z𝑍Zitalic_Z Pauli operators in the cluster state Hamiltonian to X→cos⁡(2⁢θ)⋅X−sin⁡(2⁢θ)⋅Z,Z→cos⁡(2⁢θ)⋅Z+sin⁡(2⁢θ)⋅Xformulae-sequence→𝑋⋅2𝜃𝑋⋅2𝜃𝑍→𝑍⋅2𝜃𝑍⋅2𝜃𝑋X\rightarrow\cos(2\theta)\cdot X-\sin(2\theta)\cdot Z,\hskip 10.00002ptZ% \rightarrow\cos(2\theta)\cdot Z+\sin(2\theta)\cdot Xitalic_X → roman_cos ( 2 italic_θ ) ⋅ italic_X - roman_sin ( 2 italic_θ ) ⋅ italic_Z , italic_Z → roman_cos ( 2 italic_θ ) ⋅ italic_Z + roman_sin ( 2 italic_θ ) ⋅ italic_X.
  56. Expressive power of complex-valued restricted boltzmann machines for solving nonstoquastic hamiltonians. Phys. Rev. B, 106:134437, Oct 2022.
  57. RWKV: Reinventing RNNs for the transformer era. In The 2023 Conference on Empirical Methods in Natural Language Processing, 2023.
  58. Ab initio solution of the many-electron schrödinger equation with deep neural networks. Phys. Rev. Res., 2:033429, Sep 2020.
  59. Localizable entanglement. Phys. Rev. A, 71:042306, Apr 2005.
  60. Markov entropy decomposition: a variational dual for quantum belief propagation. Physical Review Letters, 106(8):080403, 2011.
  61. Neural tensor contractions and the expressive power of deep neural quantum states. Phys. Rev. B, 106:205136, Nov 2022.
  62. Measurement-induced phase transitions in the dynamics of entanglement. Phys. Rev. X, 9:031009, Jul 2019.
  63. Sandro Sorella. Green function monte carlo with stochastic reconfiguration. Phys. Rev. Lett., 80:4558–4561, May 1998.
  64. Variational monte carlo with large patched transformers. Communications Physics, 7(1):90, 2024.
  65. Neural network wave functions and the sign problem. Phys. Rev. Res., 2:033075, Jul 2020.
  66. Ewin Tang. A quantum-inspired classical algorithm for recommendation systems. In STOC’19—Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 217–228. ACM, New York, 2019.
  67. Matus Telgarsky. Representation benefits of deep feedforward networks. arXiv preprint arXiv:1509.08101, 2015.
  68. Precise measurement of quantum observables with neural-network estimators. Phys. Rev. Res., 2:022060, Jun 2020.
  69. Neural-network quantum state tomography. Nature Physics, 14(5):447–450, 2018.
  70. Entanglement versus correlations in spin systems. Phys. Rev. Lett., 92:027901, Jan 2004.
  71. NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems. SciPost Phys. Codebases, page 7, 2022.
  72. William K. Wootters. Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett., 80:2245–2248, Mar 1998.
  73. From tensor-network quantum states to tensorial recurrent neural networks. Phys. Rev. Res., 5:L032001, Jul 2023.
  74. Dmitry Yarotsky. Error bounds for approximations with deep ReLU networks. Neural Networks, 94:103–114, 2017.
  75. Empirical sample complexity of neural network mixed state reconstruction. arXiv preprint arXiv:2307.01840, 2023.
  76. Spurious long-range entanglement and replica correlation length. Phys. Rev. B, 94:075151, Aug 2016.
Citations (2)
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Summary

  • The paper demonstrates that feedforward neural networks can represent quantum states with short-range conditional correlations using polynomial computational resources.
  • The paper reveals how measurement-induced conditional correlations determine the tractability of approximating entangled quantum states.
  • The paper details the transition in rotated cluster states and addresses challenges posed by complex phase structures in neural quantum state learning.

An Examination of Classical Neural Network Representations of Quantum States

The paper "When Can Classical Neural Networks Represent Quantum States?" by Tai-Hsuan Yang, Mehdi Soleimanifar, Thiago Bergamaschi, and John Preskill seeks to explore the realms of computational quantum physics and machine learning by offering insights into the classical approximations of quantum states via neural networks. The focus is largely placed on identifying and understanding the constraints of classical representations in retaining the complex structures of quantum states. The discussion broadens the scope of traditional few-body correlations by introducing the concept of measurement-induced conditional correlations, which fundamentally determine the prowess of neural quantum state representations.

Theoretical Foundations and Results

One key theoretical proposal in this paper is the significance of conditional correlations, which are basis-dependent and resultant from measurement-induced entanglement. These correlations may vary from short-range to long-range based on the measurement's influence on the quantum state's entanglement and sign structure. The authors delve into the aspect that, while many-body states usually require exponential parameters for precision representation, certain neural network architectures can offer a polynomially efficient representation under optimal conditional correlation structures.

Highlights of the analysis bring forth several central results:

  • Feedforward Neural Networks for Short-Range Correlations: Through rigorous analysis, it is demonstrated that quantum states exhibiting short-range conditional correlations are effectively representable by feedforward neural networks with logarithmic depth and polynomial width. This assures computation within a tractable time frame, respecting the constraints of polynomial resource scaling for increasing qubits.
  • Entanglement and Conditional Correlations: The characteristics of states prepared by random shallow quantum circuits and tensor networks were elucidated, indicating short-range conditional correlations. These quantum states, constrained by their physical systems, allow an efficient classical rendition by leveraging these correlations, following significantly from the entanglement area law implications.
  • Transition in Rotated Cluster States: A significant theoretical insight is the treatment of rotated cluster states, where a variable rotation angle manipulates the conditional correlation length, consequently transitioning the state from short- to long-range conditional dependencies and affecting computational tractability.

Computational Implications and Future Directions

The paper suggests that while neural networks such as recurrent neural networks and restricted Boltzmann machines show impressive empirical success, the intricacies induced by phase complexities and sign structure present critical challenges. Their results intimate that the presence of complex phases, which exacerbate long-range conditional correlations, calls for refined architectural advancements or novel training strategies to mitigate representation inefficacies in various algorithms.

Looking ahead, the amalgamation of neural networks and quantum systems posits numerous questions pivotal to the theoretical and practical advancement of AI. Key among these is determining the full implications of non-trivial phase structures on the computational representations of quantum states, especially in native complex entanglement landscapes. Exploring potential parallels between long-known quantum challenges, like the sign problem, in QMC algorithms and expressivity bounds inherent in neural networks, could lead to innovative solutions and deeper understanding of quantum systems.

This paper lays a substantive groundwork for that exploration, opening pathways for enhancing VMC's performance, extending the techniques to higher dimensions, and testing advanced architectures such as Transformers and RWKV for quantum state learning. Such investigation aligns with the quest to achieve efficient quantum simulations, challenging the current bounds of classical computational models in accommodating the exponentially intricate nature of quantum phenomena.

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