Quantum linear solvers for scientific computing: a comparison of VQLS, HHL and quantum annealing on time-fractional diffusion problems
Abstract
Time-fractional diffusion equations have emerged as powerful models for describing anomalous transport phenomena in physics, biology and engineering. To address the computational challenges arising from their non-local operators, we employ the WEB-spline finite element method, which provides a flexible and accurate discretization framework. The resulting linear system of equations are then explored in the context of quantum computing. Specifically, we investigate three prominent quantum linear solvers: the variational quantum linear solver (VQLS), the Harrow-Hassidim-Lloyd (HHL) algorithm and quantum annealing (QA). VQLS leverages hybrid variational techniques and shallow circuits, making it well-suited for noisy intermediate-scale quantum (NISQ) devices, while HHL offers a theoretically exponential speedup for sparse systems but requires deep fault-tolerant circuits. QA, in contrast, reformulates the problem into a quadratic unconstrained binary optimization (QUBO) instance, enabling approximate solutions through energy minimization on specialized hardware. We present a comparative analysis in terms of circuit depth, noise resilience, scalability and solution extraction, including the role of quantum state tomography in reconstructing classical information. Numerical experiments on a time-fractional diffusion problem highlight the complementary strengths and limitations of each method. This study bridges advanced numerical discretization with emerging quantum algorithms,…
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