Quantum Fourier Iterative Amplitude Estimation

Abstract

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 algorithm to build a new tool 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 integrates 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 that corresponds with 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. In conclusion, QFIAE is a promising end-to-end quantum algorithm for Monte Carlo integrals that combines the power of PQC with Fourier analysis and IQAE to offer a new approach for efficiently approximating integrals with high accuracy.

17 pages, 5 figures, 2 tables

References in corpus (14)

• Quantum algorithm for solving linear systems of equations
• Variational Quantum Algorithms
• The Variational Quantum Eigensolver: a review of methods and best practices
• Connecting ansatz expressibility to gradient magnitudes and barren plateaus
• A rigorous and robust quantum speed-up in supervised machine learning
• Application-Oriented Performance Benchmarks for Quantum Computing
• Qibo: a framework for quantum simulation with hardware acceleration
• Quantum Monte Carlo Integration: The Full Advantage in Minimal Circuit Depth
• Multidimensional Fourier series with quantum circuits
• Power and limitations of single-qubit native quantum neural networks
• Quantum Fourier Iterative Amplitude Estimation
• A Survey of Quantum Alternatives to Randomized Algorithms: Monte Carlo Integration and Beyond
• Linear Regression by Quantum Amplitude Estimation and its Extension to Convex Optimization
• Energy risk analysis with Dynamic Amplitude Estimation and Piecewise Approximate Quantum Compiling

Cited by in corpus (6)

• Quantum Fourier Iterative Amplitude Estimation
• Loop Feynman integration on a quantum computer
• Energy risk analysis with Dynamic Amplitude Estimation and Piecewise Approximate Quantum Compiling
• Quantum integration of decay rates at second order in perturbation theory
• A general approach to quantum integration of cross sections in high-energy physics
• Quantum Chebyshev Probabilistic Models for Fragmentation Functions