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

Abstract

This paper proposes a method of quantum Monte-Carlo integration that retains the full quadratic quantum advantage, without requiring any arithmetic or the quantum Fourier transform 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].

A Survey of Quantum Computing for Finance

“…In addition, Plekhanov et al proposed a variational QAE [269] algorithm that combines the approximation of quantum states via variational methods and low-depth PQCs with the maximumlikelihood-estimation approach to QAE of Suzuki et al [314]. Potential methods for near-term implementations of payoff functions have been made by Herbert [166], who considered certain functions that can be approximated by truncated Fourier series. Also worth noting are non-unitary methods for efficient state preparation [144,275].…”

Quantum algorithm for calculating risk contributions in a credit portfolio

“…Generating many samples, whose number is typically of order 10 6 , for a large credit portfolio, which can contain millions of obligors for major banks, is of high computational cost. On the other hand, there are some quantum algorithms for Monte Carlo integration [7][8][9] based on quantum amplitude estimation [8,[10][11][12][13][14][15][16][17]. Although the classical Monte Carlo integration has sample complexity scaling as O(ǫ −2 ) on ǫ, the error tolerance for the integral, query complexity in the quantum counterparts scales as O(ǫ −1 ), which is often referred to as quantum quadratic speedup.…”

“…Then, this paper aims at quantum speedup of credit risk contribution calculation. Note that original quantum algorithms for Monte Carlo integration [7][8][9] output an estimate of a single expected value, and that sequentially applying such an algorithm to calculation of risk contributions of N gr obligor groups, which leads to O(N gr /ǫ) complexity, is not efficient. Instead, we resort to the recently proposed quantum algorithm for simultaneous calculation of expected values of multiple random variables [30] (see also [31]).…”

Bermudan option pricing by quantum amplitude estimation and Chebyshev interpolation

“…In this paper, we propose a new method for Bermudan option pricing, combining Chebyshev interpolation and quantum algorithm for Monte Carlo integration [30][31][32], which is based on quantum amplitude estimation (QAE) [31,[33][34][35][36][37][38][39][40][41][42]. As far as the author knows, this is the first proposal on the quantum method for Bermudan option pricing.…”