Unlocking Multi-Dimensional Integration with Quantum Adaptive Importance Sampling

Abstract: We introduce a quantum algorithm that performs Quantum Adaptive Importance Sampling (QAIS) for Monte Carlo integration of multidimensional functions, targeting in particular the computational challenges of high-energy physics. In this domain, the fundamental ingredients for theoretical predictions such as multiloop Feynman diagrams and the phase-space require evaluating high-dimensional integrals that are computationally demanding due to divergences and complex mathematical structures. The established method of Adaptive Importance Sampling, as implemented in tools like VEGAS, uses a grid-based approach that is iteratively refined in a separable way, per dimension. This separable approach efficiently suppresses the exponentially growing grid-handling computational cost, but also introduces performance drawbacks whenever strong inter-variable correlations are present. To utilize sampling resources more efficiently, QAIS exploits the exponentially large Hilbert space of a Parameterised Quantum Circuit (PQC) to manipulate a non-separable Probability Density Function (PDF) defined on a multidimensional grid. In this setting, entanglement within the PQC captures the correlations and intricacies of the target integrand's structure. Then, the PQC strategically allocates samples across the multidimensional grid, focusing the samples in the small subspace where the important structures of the target integrand lie, and thus generates very precise integral estimations. As an application, we look at a very sharply peaked loop Feynman integral and at multi-modal benchmark integrals.