Expressivity Limits and Trainability Guarantees in Quantum Walk-based Optimization

Abstract: Quantum algorithms have emerged as a promising tool to solve combinatorial optimization problems. The quantum walk optimization algorithm (QWOA) is one such variational approach that has recently gained attention. In the broader context of variational quantum algorithms (VQAs), understanding the expressivity and trainability of the ansatz has proven critical for evaluating their performance. A key method to study both these aspects involves analyzing the dimension of the dynamic Lie algebra (DLA). In this work, we derive novel upper bounds on the DLA dimension for QWOA applied to arbitrary optimization problems. The consequence of our result is twofold: (a) it allows us to identify complexity-theoretic conditions under which QWOA must be overparameterized to obtain optimal or approximate solutions, and (b) it implies the absence of barren plateaus in the loss landscape of QWOA for $\mathsf{NP}$ optimization problems with polynomially bounded cost functions ($\mathsf{NPO}\text{-}\mathsf{PB}$).