Approximating Hamiltonian dynamics with the Nyström method

Alessandro Rudi1, Leonard Wossnig2,3, Carlo Ciliberto2, Andrea Rocchetto2,4,5, Massimiliano Pontil6, and Simone Severini2

1INRIA - Sierra project team, Paris, France
2Department of Computer Science, University College London, London, United Kingdom
3Rahko Ltd., London, United Kingdom
4Department of Computer Science, University of Texas at Austin, Austin, United States
5Department of Computer Science, University of Oxford, Oxford, United Kingdom
6Computational Statistics and Machine Learning, IIT, Genoa, Italy

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Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nyström method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.

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[2] Sevag Gharibian and François Le Gall, "Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture", SIAM Journal on Computing 52 4, 1009 (2023).

[3] Ewin Tang, "Quantum Principal Component Analysis Only Achieves an Exponential Speedup Because of Its State Preparation Assumptions", Physical Review Letters 127 6, 060503 (2021).

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[5] Nai-Hui Chia, András Pal Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang, "Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning", Journal of the ACM 69 5, 1 (2022).

[6] Ewin Tang, "Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions", arXiv:1811.00414, (2018).

[7] Aram W. Harrow, "Small quantum computers and large classical data sets", arXiv:2004.00026, (2020).

[8] Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang, "Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning", arXiv:1910.06151, (2019).

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