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Quantum linear solvers for scientific computing: a comparison of VQLS, HHL and quantum annealing on time-fractional
1Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran. ahsalehi.kau@gmail.com.
Scientific Reports
|February 23, 2026
Summary
This study explores quantum computing for time-fractional diffusion equations, comparing variational quantum linear solvers (VQLS), Harrow-Hassidim-Lloyd (HHL), and quantum annealing (QA). Findings reveal complementary strengths for these quantum algorithms in solving complex transport phenomena.
Area of Science:
- Computational physics and applied mathematics.
- Quantum information science and algorithms.
- Numerical analysis and scientific computing.
Background:
- Time-fractional diffusion equations model anomalous transport but pose computational challenges due to non-local operators.
- The WEB-spline finite element method offers a flexible and accurate discretization for these equations.
- Emerging quantum computing algorithms present potential solutions for the resulting large-scale linear systems.
Purpose of the Study:
- To investigate the application of quantum linear solvers to time-fractional diffusion problems.
- To comparatively analyze the performance of Variational Quantum Linear Solver (VQLS), Harrow-Hassidim-Lloyd (HHL), and Quantum Annealing (QA).
- To assess the feasibility and potential of quantum-enhanced methods for fractional partial differential equations.
Main Methods:
- Discretization of time-fractional diffusion equations using the WEB-spline finite element method.
- Implementation and analysis of three quantum linear solvers: VQLS, HHL, and QA.
- Comparative evaluation based on circuit depth, noise resilience, scalability, and solution extraction, including quantum state tomography.
Main Results:
- VQLS shows promise for Noisy Intermediate-Scale Quantum (NISQ) devices due to shallow circuits.
- HHL offers theoretical speedups for sparse systems but requires fault-tolerant quantum computers.
- QA provides approximate solutions via QUBO formulation, suitable for specialized hardware.
- Numerical experiments demonstrate the distinct advantages and limitations of each quantum approach.
Conclusions:
- Quantum algorithms offer diverse strategies for solving computationally intensive fractional diffusion problems.
- The choice of quantum solver depends on specific problem characteristics and available quantum hardware.
- This research bridges numerical methods with quantum computing, paving the way for future quantum-enhanced scientific discovery.
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