# Quantum linear solvers for scientific computing: a comparison of VQLS, HHL and quantum annealing on time-fractional diffusion problems

## 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.

## Key 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, 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, providing insights into the feasibility and future potential of quantum-enhanced solvers for fractional partial differential equations.