Heuristics for multi-stage quantum walks to find Ising ground states

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 a 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 problems with a large minimum energy gap, giving a polynomial scaling in the number of stages, and leading to an overall algorithm that scales polynomially in time. For problems with a small minimum gap, 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.