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arXiv:2401.03023 · Physical review. D/Physical review. D.

Loop Feynman integration on a quantum computer

Jorge J. Martínez de Lejarza, Leandro Cieri, Michele Grossi, Sofia Vallecorsa, Germán Rodrigo A

Published Oct 25, 2024• 10 citations
arXivHTMLPDF

Abstract

This work investigates in detail the performance and advantages of a new quantum Monte Carlo integrator, dubbed quantum Fourier iterative amplitude estimation (QFIAE), to numerically evaluate for the first time loop Feynman integrals in a near-term quantum computer and a quantum simulator. In order to achieve a quadratic speedup, QFIAE introduces a quantum neural network (QNN) that efficiently decomposes the multidimensional integrand into its Fourier series. For a one-loop tadpole Feynman diagram, we have successfully implemented the quantum algorithm on a real quantum computer and obtained a reasonable agreement with the analytical values. One-loop Feynman diagrams with more external legs have been analyzed in a quantum simulator. These results thoroughly illustrate how our quantum algorithm effectively estimates loop Feynman integrals and the method employed could also find applications in other fields such as finance, artificial intelligence, or other physical sciences. Published by the American Physical Society 2024

DOI
10.1103/physrevd.110.074031

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Monte Carlo integration is a widely used numerical method for approximating integrals, which is often computationally expensive. In recent years, quantum computing has shown promise for speeding up Monte Carlo integration, and several quantum algorithms have been proposed to achieve this goal. In this paper, we present an application of Quantum Machine Learning (QML) and Grover's amplification to build a new quantum algorithm for estimating Monte Carlo integrals. Our method, which we call Quantum Fourier Iterative Amplitude Estimation (QFIAE), decomposes the target function into its Fourier series using a Parametrized Quantum Circuit (PQC), specifically a Quantum Neural Network (QNN), and then inte- grates each trigonometric component using Iterative Quantum Amplitude Estimation (IQAE). This approach builds on Fourier Quantum Monte Carlo Integration (FQMCI) method, which also decomposes the target function into its Fourier series, but QFIAE avoids the need for numerical integration of Fourier coefficients. This approach reduces the computational load while maintaining the quadratic speedup achieved by IQAE. To evaluate the performance of QFIAE, we apply it to a test function corresponding to a particle physics scattering process and compare its accuracy with other quantum integration methods and the analytic result. Our results show that QFIAE achieves comparable accuracy while being suitable for execution on real hardware. We also demonstrate how the accuracy of QFIAE improves by increasing the number of terms in the Fourier series.
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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].