Optimal Box Contraction for Solving Linear Systems via Simulated and Quantum Annealing

TL;DR

This paper improves the efficiency of the box algorithm for solving linear systems on quantum-annealing machines by optimizing the box contraction ratio, leading to significant speed-ups in convergence.

Contribution

The paper demonstrates that a contraction ratio of 0.2 outperforms the traditional 0.5 ratio, providing a theoretical and empirical basis for optimizing the box algorithm.

Findings

Optimal contraction ratio of 0.2 speeds up convergence

Theoretical proof of sub-optimality of 0.5 ratio

Numerical experiments show 20-60% speed-up

Abstract

Solving linear systems of equations is an important problem in science and engineering. Many quantum algorithms, such as the Harrow-Hassidim-Lloyd (HHL) algorithm (for quantum-gate computers) and the box algorithm (for quantum-annealing machines), have been proposed for solving such systems. The focus of this paper is on improving the efficiency of the box algorithm. The basic principle behind this algorithm is to transform the linear system into a series of quadratic unconstrained binary optimization (QUBO) problems, which are then solved on annealing machines. The computational efficiency of the box algorithm is entirely determined by the number of iterations, which, in turn, depends on the box contraction ratio, typically set to 0.5. Here, we show through theory that a contraction ratio of 0.5 is sub-optimal and that we can achieve a speed-up with a contraction ratio of 0.2. This is confirmed through numerical experiments where a speed-up between $20 \%$ to $60 \%$ is observed when the optimal contraction ratio is used.