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 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].

Quantum computing for data-centric engineering and science

“…The specific focus of the paper in question is on QMCI applications in finance, but the broad principle is likely to extend to other applications. We have estimated that our Fourier QMCI algorithm (Herbert, 2022) reduces resource requirements (number of quantum operations) by at least 30% to over 90% in some cases for the benchmarks set out, and if the circuits were re-constructed in a slightly different way we have estimated that the total number of physical superconducting qubits required would be in the 1,000s to 10,000 s. The exact value within this range depends on whether qubit qualities improve enough for low-overhead error-correction codes (Tomita and Svore, 2014) to be practical; and in particular, whether the overhead can be reduced by exploiting asymmetries in the noise (Ataides et al, 2021; Higgott et al, 2022)—both of which remain very active research topics.…”

“…Quadratic-advantage quantum algorithms are dominated by circuits of Toffoli gates, which are extremely expensive to implement using error-corrected quantum computation. This is certainly true for un-optimized algorithms, however, Fourier QMCI (Herbert, 2022) moves to classical post-processing exactly those Toffoli-heavy circuits, while upholding the full quantum advantage.…”

“…Two more, complementary, breakthroughs have further fueled the hope that QMCI can be a source of near-term quantum advantage: Noise-aware QAE (Herbert et al, 2021) takes an advantage of the fact that QAE circuits have a very specific structure to handle device noise as if it were estimation uncertainty. This suggests that significantly less error correction may be needed to achieve a useful advantage in QMCI, compared to other calculations of comparable size. Quantum Monte Carlo integration: the full advantage in minimal circuit depth (Herbert, 2022) shows how to decompose the Monte Carlo integral as a Fourier series such that the circuit , which may in general constitute an unreasonably large contribution to the total circuit depth, can be replaced by minimally deep circuits of rotation gates (this procedure is hereafter referred to as “Fourier QMCI”). …”

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( -2 ) on , the error tolerance for the integral, query complexity in the quantum counterparts scales as O( -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 / ) 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]).…”