Expressivity Limits in Quantum Walk-based Optimization

TL;DR

This paper establishes upper bounds on the expressivity of quantum walk-based optimization algorithms by analyzing the dynamic Lie algebra, revealing polynomial scaling and conditions for overparameterization in solving complex optimization problems.

Contribution

It derives novel upper bounds on the dynamic Lie algebra dimension for QWOA, linking expressivity to problem complexity and overparameterization requirements.

Findings

DLA dimension scales at most quadratically with eigenvalues

Polynomial DLA dimension for NPO-PB problems

Overparameterization needed for optimal solutions in certain cases

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 of the ansatz has proven critical for evaluating their performance. A key method to study this aspect 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. Specifically, we show that the DLA dimension scales at most quadratically with the number of distinct eigenvalues of the problem Hamiltonian. As a consequence, our bound guarantees a polynomial DLA dimension with respect to the input size for optimization problems in the class $\mathsf{NPO}\text{-}\mathsf{PB}$. This result, coupled with recently established performance bounds for QWOA, allows us to identify complexity-theoretic conditions under which QWOA must be overparameterized to obtain optimal or approximate solutions for $\mathsf{NPO}\text{-}\mathsf{PB}$ problems.