Quantum variational optimization: The role of entanglement and problem hardness

TL;DR

This paper systematically investigates how entanglement, circuit structure, and problem complexity influence the success of quantum variational algorithms like VQE on QUBO problems, revealing insights for improving quantum optimization.

Contribution

It provides a detailed analysis of entanglement and problem structure effects on VQE performance, and introduces strategies like topology-adapted entangling gates and CVaR cost functions.

Findings

Entanglement distribution aligned with problem topology enhances performance.

CVaR cost functions improve overlap with optimal solutions.

Problem hardness correlates with Hamming distance between ground and first excited states.

Abstract

Quantum variational optimization has been posed as an alternative to solve optimization problems faster and at a larger scale than what classical methods allow. In this paper we study systematically the role of entanglement, the structure of the variational quantum circuit, and the structure of the optimization problem, in the success and efficiency of these algorithms. For this purpose, our study focuses on the variational quantum eigensolver (VQE) algorithm, as applied to quadratic unconstrained binary optimization (QUBO) problems on random graphs with tunable density. Our numerical results indicate an advantage in adapting the distribution of entangling gates to the problem's topology, specially for problems defined on low-dimensional graphs. Furthermore, we find evidence that applying conditional value at risk type cost functions improves the optimization, increasing the probability of overlap with the optimal solutions. However, these techniques also improve the performance of Ans\"atze based on product states (no entanglement), suggesting that a new classical optimization method based on these could outperform existing NISQ architectures in certain regimes. Finally, our study also reveals a correlation between the hardness of a problem and the Hamming distance between the ground- and first-excited state, an idea that can be used to engineer benchmarks and understand the performance bottlenecks of optimization methods.