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

TL;DR

The paper investigates solving linear systems via a QUBO formulation implemented on quantum annealing, focusing on the box algorithm which iteratively translates or contracts a solution box. The key contribution is showing that the conventional contraction ratio $\beta = 0.5$ is sub-optimal and that $\beta \approx 0.2$ substantially reduces the total number of box iterations, yielding a speed-up of roughly $20\%$ to $60\%$ in practice. The authors derive analytic bounds for contraction and translation steps, extend the analysis from 1D to multi-dimensional problems, and validate the predictions with simulated annealing and quantum annealing experiments on positive-definite matrices and a 1D Poisson discretization. These results suggest meaningful efficiency gains for QUBO-based linear-system solvers on near-term quantum hardware and motivate further exploration of low-precision, high-iteration-box methods for larger systems.

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.

Figures (7)

• Figure 1: The box representation for $d = 2$.
• Figure 2: Box algorithm: (a) translation, (b) contraction.
• Figure 3: Typical box convergence in 2D.
• Figure 4: Upper bound $\hat{N}$ on the number of box iterations.
• Figure 5: Average number of box iterations $\overline N$.