Multi-stage quantum walks for finding Ising ground states

TL;DR

The paper tackles finding Ising ground states by using multi-stage quantum walks (MSQW) to approximate quantum annealing schedules. It develops a polynomial-time heuristic for selecting stage parameters $\gamma_k$ and times $t_i$, and demonstrates that MSQW achieves polynomial-time scaling with the number of stages on easy SK-like problems, while revealing exponential scaling for harder cases as stages increase. Through analytical and numerical analysis, including infinite-time and short-time heuristics, the work shows potential performance advantages over traditional quantum annealing and QAOA in certain regimes, and discusses hardware considerations and general applicability to optimization problems. Overall, MSQW with the proposed heuristics provides a versatile framework for designing annealing schedules across a range of Ising-encoded optimization tasks.

Abstract

One way to approximate a quantum annealing schedule is to use multiple quantum walks chained together, without intermediate measurements, to produce a multi-stage quantum walk (MSQW). Previous work has shown that MSQW is better than QAOA (quantum alternating operator ansatz) for solving optimization tasks using multiple stages [Gerblich et al, arXiv:2407.06663]. In this work, we develop an efficient heuristic for choosing the free parameters in MSQW, and use it to obtain improved scaling compared to single stage quantum walks. We show numerically that the heuristic works well for easy problems with a large minimum energy gap, giving a scaling polynomial in the number of stages, leading to an overall algorithm that scales polynomially in time. For harder problems, the scaling breaks down such that adding more stages decreases the success probability, leading to an overall scaling that is exponential in time, as expected. Our methods are general and can be applied to any optimization problem to obtain good annealing schedules.

Figures (7)

• Figure 1: The evolution of a state for the 10 qubit problem 'aaaufeflfwqdwhthcrcnynopihzciv', showing how well $t_s$ (dotted blue line) predicts the end of the fast growth period for the energy of $\hat{H}_G$ (solid blue line). The 2nd order approximation to $E_G$ is shown in orange.
• Figure 2: The slope of the regression line in \ref{['regress']} with number of stages, indicating that adding more stages will improve the scaling of the method. The blue line corresponds to typical problems, the orange line to hard problems, and the green line to the infinite-time average. Errors shown are the standard errors provided by scipy.stats.linregress.
• Figure 3: For hard problems, the intercept of the regression line in \ref{['regress']} decreases drastically as more stages are added, which can decrease the overall success probability if too many stages are added. The blue line corresponds to typical problems, the orange line to hard problems, and the green line to the infinite-time average. Errors shown are the standard errors provided by scipy.stats.linregress
• Figure 4: The median success probability across a range of system sizes for typical problem instances. Each line represents a multi-stage quantum walk with a different number of stages. The stages shown are 1 (blue), 2 (orange), 5 (green), 10 (red) and 20 (purple). Errors shown are standard errors on the median and have been calculated via bootstrap sampling with 1000 samples per point.
• Figure 5: The median success probability in the infinite time average. Each line represents a multi-stage quantum walk with a different number of stages. The stages shown are 1 (blue), 2 (orange), 5 (green), 10 (red) and 20 (purple). Errors shown are standard errors on the median and have been calculated via bootstrap sampling with 1000 samples per point.