Solving wave equation problems on D-Wave quantum annealers

TL;DR

This paper explores solving the 1D Helmholtz equation using D-Wave quantum annealers within a pseudospectral framework, comparing encoding strategies and benchmarking against classical algorithms to improve stability and accuracy.

Contribution

It introduces novel encoding strategies for quantum annealers that improve solution stability and accuracy for wave equations, emphasizing the importance of algebraic conditioning.

Findings

Encoding strategies with full-rank algebraic systems perform better.

Small dynamic ranges enhance quantum annealer performance.

Hybrid quantum-classical schemes could meet algebraic and adiabatic conditions.

Abstract

We solve the one-dimensional Helmholtz equation in several scenarios using the quantum annealer provided by the D-Wave systems within a pseudospectral scheme, where its solution is encoded into certain set of suitable basis functions. We assess the performance of different strategies of encoding based on algebraic arguments and the adiabatic condition, and benchmark these against the classical heuristic simulating annealing algorithm. In particular, we compute the minimum energy gap, the so-called dynamic range and the mean squared error to assess the numerical stability, consistency and accuracy of the solutions returned by each strategy. Our work stresses out the importance of developing custom embedded techniques ensuring well-conditioned algebraic systems. In particular, we find out that encoding strategies retrieving algebraic systems exhibiting full-rank and small dynamic ranges enhance the performance of the quantum annealer even under polychromatic driving and for intricate initial conditions. We further discuss the prospect of developing hybrid quantum-classical schemes enable to meet suitable algebraic and adiabatic conditions simultaneously.