Quantum Annealing for the Helmholtz Equation on D-Wave: QUBO Formulations and Encoding Strategies

Abstract

Quantum annealing has emerged as a promising paradigm for solving large-scale combinatorial optimization problems. Extending this framework to the numerical solution of differential equations requires the development of suitable optimization formulations that can be efficiently implemented on current quantum hardware. In this context, this thesis investigates the application of quantum annealing to the numerical solution of differential equations, using the one-dimensional Helmholtz equation as a benchmark problem. The approximate solution is represented through spectral basis function whose coefficients are determined by formulating the problem as QUBO (Quadratic Unconstrained Binary Optimization Problem). Different encoding strategies \sloppy are systematically investigated and evaluated from both algebraic and adiabatic perspectives, with their performance compared against the classical simulated annealing algorithm.

To assess the quality and reliability of the obtained solutions, several quantitative metrics are employed, including the minimum energy gap, dynamic range, and mean squared error. These measures provide insight into the adiabatic behavior of the system, numerical stability, and approximation accuracy. The results highlight the critical role of problem encoding and the resulting QUBO structure in achieving efficient quantum annealing performance. In particular, the algebraic properties of the discretized problem strongly determine the optimization landscape experienced by the quantum annealer.

Finally, the results highlight the potential of hybrid quantum–classical approaches that combine favorable algebraic properties with suitable adiabatic conditions. These findings contribute to the development of more effective QUBO formulations for differential equations and provide practical guidelines for designing quantum annealing algorithms on current and future hardware.