On the role of overparametrization in Quantum Approximate Optimization

TL;DR

This paper investigates how overparameterization affects the performance of the quantum approximate optimization algorithm (QAOA) in solving combinatorial problems, revealing that overparameterization is necessary for MAX-CUT but not for MAX-2-SAT.

Contribution

It provides the first detailed analysis of overparameterization in QAOA, demonstrating its necessity for MAX-CUT and sufficiency for MAX-2-SAT, with both numerical and analytical evidence.

Findings

Overparameterization is necessary for exact MAX-CUT solutions.

Underparameterized QAOA can solve most MAX-2-SAT instances.

Analytical results confirm the optimal depth for MAX-CUT on 2-regular graphs.

Abstract

Variational quantum algorithms have emerged as a cornerstone of contemporary quantum algorithms research. While they have demonstrated considerable promise in solving problems of practical interest, efficiently determining the minimal quantum resources necessary to obtain such a solution remains an open question. In this work, inspired by concepts from classical machine learning, we investigate the impact of overparameterization on the performance of variational algorithms. Our study focuses on the quantum approximate optimization algorithm (QAOA) -- a prominent variational quantum algorithm designed to solve combinatorial optimization problems. We investigate if circuit overparametrization is necessary and sufficient to solve such problems in QAOA, considering two representative problems -- MAX-CUT and MAX-2-SAT. For MAX-CUT we observe that overparametriation is both sufficient and (statistically) necessary for attaining exact solutions, as confirmed numerically for up to $20$ qubits. In fact, for MAX-CUT on 2-regular graphs we show the necessity to be exact, based on the analytically found optimal depth. In sharp contrast, for MAX-2-SAT, underparametrized circuits suffice to solve most instances. This result highlights the potential of QAOA in the underparametrized regime, supporting its utility for current noisy devices.