Multi-variable integration with a variational quantum circuit

TL;DR

This paper introduces a quantum circuit-based method for evaluating multi-variable integrals by encoding variables, deriving circuits, and using quantum machine learning techniques to estimate integrals efficiently.

Contribution

It presents a novel quantum algorithm that encodes multiple variables into circuits and uses derivative-based predictions for integral estimation, advancing quantum numerical integration methods.

Findings

Efficient encoding of multi-variable functions into quantum circuits.

Use of the parameter shift rule for derivative computation within quantum circuits.

Potential for rapid partial integration and parametric integrand evaluation.

Abstract

In this work we present a novel strategy to evaluate multi-variable integrals with quantum circuits. The procedure first encodes the integration variables into a parametric circuit. The obtained circuit is then derived with respect to the integration variables using the parameter shift rule technique. The observable representing the derivative is then used as the predictor of the target integrand function following a quantum machine learning approach. The integral is then estimated using the fundamental theorem of integral calculus by evaluating the original circuit. Embedding data according to a reuploading strategy, multi-dimensional variables can be easily encoded into the circuit's gates and then individually taken as targets while deriving the circuit. These techniques can be exploited to partially integrate a function or to quickly compute parametric integrands within the training hyperspace.