Quantum Monte Carlo Integration: The Full Advantage in Minimal Circuit Depth

Steven Herbert

Quantinuum (Cambridge Quantum), Terrington House, 13-15 Hills Rd, Cambridge, CB2 1NL, UK
Department of Computer Science and Technology, University of Cambridge, UK

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Abstract

This paper proposes a method of quantum Monte Carlo integration that retains the full quadratic quantum advantage, without requiring any arithmetic or quantum phase estimation to be performed on the quantum computer. No previous proposal for quantum Monte Carlo integration has achieved all of these at once. The heart of the proposed method is a Fourier series decomposition of the sum that approximates the expectation in Monte Carlo integration, with each component then estimated individually using quantum amplitude estimation. The main result is presented as theoretical statement of asymptotic advantage, and numerical results are also included to illustrate the practical benefits of the proposed method. The method presented in this paper is the subject of a patent application [Quantum Computing System and Method: Patent application GB2102902.0 and SE2130060-3].

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Cited by

[1] Dylan Herman, Cody Googin, Xiaoyuan Liu, Alexey Galda, Ilya Safro, Yue Sun, Marco Pistoia, and Yuri Alexeev, "A Survey of Quantum Computing for Finance", arXiv:2201.02773.

[2] Garrett T. Floyd, David P. Landau, and Michael R. Geller, "Quantum algorithm for Wang-Landau sampling", arXiv:2208.09543.

[3] Kirill Plekhanov, Matthias Rosenkranz, Mattia Fiorentini, and Michael Lubasch, "Variational quantum amplitude estimation", arXiv:2109.03687.

[4] M. C. Braun, T. Decker, N. Hegemann, and S. F. Kerstan, "Error Resilient Quantum Amplitude Estimation from Parallel Quantum Phase Estimation", arXiv:2204.01337.

[5] Koichi Miyamoto, "Quantum algorithm for calculating risk contributions in a credit portfolio", arXiv:2201.11394.

[6] Koichi Miyamoto, "Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation", arXiv:2108.09014.

[7] Koichi Miyamoto, "Quantum algorithms for numerical differentiation of expected values with respect to parameters", Quantum Information Processing 21 3, 109 (2022).

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