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.Comment: 17 pages, 5 figures, 2 table