Optimised Hybrid Classical-Quantum Algorithm for Accelerated Solution of Sparse Linear Systems

TL;DR

This paper presents a hybrid classical-quantum algorithm that combines GPU-accelerated preconditioning with quantum HHL to efficiently solve large sparse linear systems, enhanced by reinforcement learning for dynamic optimization.

Contribution

It introduces a novel hybrid approach integrating classical GPU preconditioning, quantum algorithms, and machine learning to improve the efficiency and scalability of solving sparse linear systems.

Findings

Outperforms traditional classical methods in speed and scalability.

Reduces quantum algorithm limitations through classical preprocessing.

Uses reinforcement learning to optimize system parameters dynamically.

Abstract

Efficiently solving large-scale sparse linear systems poses a significant challenge in computational science, especially in fields such as physics, engineering, machine learning, and finance. Traditional classical algorithms face scalability issues as the size of these systems increases, leading to performance degradation. On the other hand, quantum algorithms, like the Harrow-Hassidim-Lloyd (HHL) algorithm, offer exponential speedups for solving linear systems, yet they are constrained by the current state of quantum hardware and sensitivity to matrix condition numbers. This paper introduces a hybrid classical-quantum algorithm that combines CUDA-accelerated preconditioning techniques with the HHL algorithm to solve sparse linear systems more efficiently. The classical GPU parallelism is utilised to preprocess and precondition the matrix, reducing its condition number, while quantum computing is employed to solve the preconditioned system using the HHL algorithm. Additionally, the algorithm integrates machine learning models, particularly reinforcement learning, to dynamically optimise system parameters, such as block sizes and preconditioning stratgies, based on real-time performance data. Our experimental results show that the proposed approach not only surpasses traditional methods in speed and scalability but also mitigates some of the inherent limitations of quantum algorithms. This work pushes the boundaries of efficient computing and provides a foundation for future advancements in hybrid computational frameworks.