Quantum linear solvers for scientific computing: a comparison of VQLS, HHL and quantum annealing on time-fractional diffusion problems
Amir Hossein Salehi Shayegan
TL;DR
This paper compares three quantum methods for solving time-fractional diffusion equations, analyzing their strengths and limitations in scientific computing.
Contribution
The study bridges numerical discretization with quantum algorithms, offering a comparative analysis of VQLS, HHL, and quantum annealing for fractional PDEs.
Findings
VQLS is suitable for NISQ devices due to its shallow circuits and variational approach.
HHL provides exponential speedup but requires deep fault-tolerant circuits.
Quantum annealing reformulates the problem as QUBO, enabling approximate solutions on specialized hardware.
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,…
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Taxonomy
TopicsQuantum Computing Algorithms and Architecture · Tensor decomposition and applications · Fractional Differential Equations Solutions