Solving wave equation problems on D-Wave quantum annealers

We solve the one-dimensional Helmholtz equation in several scenarios using the quantum annealer provided by D-Wave Systems within a pseudospectral framework, where the solution is encoded in an appropriate set of basis functions. We evaluate the performance of different encoding strategies based on algebraic conditioning and adiabatic considerations, and benchmark their performance against the classical simulated annealing algorithm. In particular, we analyze 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 highlights the importance of developing custom embedding techniques ensuring well-conditioned algebraic systems. In particular, we show that encoding strategies producing full-rank matrices with reduced dynamic ranges enhance the performance of the quantum annealer even under polychromatic driving and nontrivial initial conditions. We further discuss the potential of developing hybrid quantum-classical schemes designed to simultaneously satisfy algebraic conditioning and adiabatic requirements.