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.

翻译：求解线性方程组是科学与工程领域的重要问题。目前已提出多种量子算法用于求解此类系统，包括Harrow-Hassidim-Lloyd（HHL）算法（适用于量子门计算机）和盒算法（适用于量子退火机）。本文重点关注盒算法的效率提升。该算法的基本原理是将线性系统转化为一系列二次无约束二元优化（QUBO）问题，并在退火机上求解。盒算法的计算效率完全取决于迭代次数，而迭代次数又取决于盒收缩比率（通常设为0.5）。本文通过理论证明，0.5的收缩比率并非最优，采用0.2的收缩比率可实现加速。数值实验进一步证实，采用最优收缩比率时，可观测到$20\%$至$60\%$的加速效果。