A Quantum Adaptive Importance Sampling Algorithm for Multidimensional Integration

This work presents Quantum Adaptive Importance Sampling (QAIS), a hybrid quantum-classical algorithmic workflow that performs Monte Carlo numerical integration of multivariate functions. This workflow extends standard Importance Sampling to a quantum computational system. Its primary purpose is to produce an unbiased estimate of integrals, with a limited number of measurements. By using the exponentially sized Hilbert space of a Parameterized Quantum Circuit (PQC), we manipulate the Probability Density Function (PDF) over a grid in its entirety. With adequate expressivity and entanglement, we effectively capture correlations among different integrand variables, thus bypassing the separable PDF assumption, as implemented in VEGAS. By optimizing the PQC, we adapt and ultimately load a PDF that approximates the integrand's behavior. Direct sampling from it, allocates samples to the important regions, allowing a highly accurate integral estimation. As an application, we study a benchmark two-dimensional integral, and compare the precision of the best grid of VEGAS against that of the quantum-generated proposal PDF, as a function of the number of samples.