Non-convex optimization algorithm based on alternating quantum walk with potentials

Abstract

This paper proposes a new model, alternating quantum walk with potentials (AQWP), designed to solve high-dimensional non-convex optimization problems. The method integrates problem-dependent potential-induced phase modulation into an alternating discrete-time quantum walk, enabling directional interference bias toward descent directions while preserving coherent quantum dynamics. A formal analysis of the algorithmic mechanism demonstrates that potential-induced phases generate constructive interference along descent paths and destructive interference elsewhere, with finite potential barriers traversable via quantum tunneling. Under mild regularity assumptions, this yields probabilistic concentration near low-energy regions instead of trapping at local minima. Computational complexity analysis of AQWP, accounting for classical preprocessing and quantum evolution, shows the overall cost scales polynomially with problem dimension and iteration count. To address parameter sensitivity, an online local estimation strategy for the phase normalization parameter is introduced, revealing a broad robustness interval that obviates global landscape scanning. Extensive numerical experiments on benchmark non-convex functions and binary classification neural networks confirm AQWP’s stability under random initialization and favorable scaling with input dimension and network capacity. Compared with classical baselines, AQWP consistently achieves faster convergence and better solution quality, establishing it as a scalable, robust quantum-inspired optimization paradigm for non-convex learning tasks.