Connecting the Hamiltonian structure to the QAOA performance and energy landscape

TL;DR

This paper explores how the structure of the Hamiltonian, especially sparsity, influences QAOA's effectiveness and energy landscape in solving optimization problems on NISQ quantum devices.

Contribution

It provides an analysis of the impact of Hamiltonian sparsity on QAOA performance, highlighting the algorithm's robustness across different matrix densities.

Findings

QAOA performance is largely unaffected by matrix sparsity.

Denser matrices increase the complexity of the energy landscape.

The study offers insights into designing Hamiltonians for better QAOA results.

Abstract

Quantum computing holds promise for outperforming classical computing in specialized applications such as optimization. With current Noisy Intermediate Scale Quantum (NISQ) devices, only variational quantum algorithms like the Quantum Alternating Operator Ansatz (QAOA) can be practically run. QAOA is effective for solving Quadratic Unconstrained Binary Optimization (QUBO) problems by approximating Quantum Annealing via Trotterization. Successful implementation on NISQ devices requires shallow circuits, influenced by the number of variables and the sparsity of the augmented interaction matrix. This paper investigates the necessary sparsity levels for augmented interaction matrices to ensure solvability with QAOA. By analyzing the Max-Cut problem with varying sparsity, we provide insights into how the Hamiltonian density affects the QAOA performance. Our findings highlight that, while denser matrices complicate the energy landscape, the performance of QAOA remains largely unaffected by sparsity variations. This study emphasizes the algorithm's robustness and potential for optimization tasks on near-term quantum devices, suggesting avenues for future research in enhancing QAOA for practical applications.