Quantum Relaxation for Linear Systems in Finite Element Analysis

TL;DR

This paper introduces Quantum Relaxation for Linear Systems (qRLS), an iterative quantum algorithm that improves finite element problem-solving efficiency by embedding stationary iterations into a block system, enabling exponential speedups on quantum hardware.

Contribution

The paper presents qRLS, a novel iterative quantum algorithm that maintains a well-conditioned system for finite element problems, overcoming previous limitations of quantum linear system algorithms.

Findings

Condition number scales linearly with iterations, independent of system size.

Solution time is independent of system size, requiring only O(log(N)) qubits.

Numerical results demonstrate practical feasibility on quantum simulators.

Abstract

Quantum linear system algorithms (QLSAs) for gate-based quantum computing can provide exponential speedups for solving linear systems but face challenges when applied to finite element problems due to the growth of the condition number with problem size. Furthermore, QLSAs cannot use an approximate solution or initial guess to output an improved solution. Here, we present Quantum Relaxation for Linear System (qRLS), as an iterative approach for gate-based quantum computers by embedding linear stationary iterations into a larger block linear system. The condition number of the block linear system scales linearly with the number of iterations independent of the size and condition number of the original system. The well-conditioned system enables a practical iterative solution of finite element problems using the state-of-the-art Quantum Signal Processing (QSP) variant of QLSAs, for which we provide numerical results using a quantum computer simulator. The iteration complexity demonstrates favorable scaling relative to classical architectures, as the solution time is independent of system size and requires O(log(N)) qubits. This represents an exponential efficiency gain, offering a new approach for iterative finite element problem-solving on quantum hardware.