Multi-variable integration with a variational quantum circuit

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.