Connecting the Hamiltonian structure to the QAOA energy and Fourier landscape structure

TL;DR

This paper investigates how the structure of Hamiltonians affects the energy landscape of QAOA, using Fourier analysis and roughness metrics to understand optimization challenges in variational quantum algorithms.

Contribution

It introduces a detailed analysis of 1-layer QAOA landscapes for Hamiltonians with up to 5-local terms, including Fourier transforms and landscape roughness metrics.

Findings

Hamiltonian structure influences landscape roughness

Fourier analysis reveals relationships between Hamiltonian terms and landscape features

Predicting VQA landscapes from first principles remains highly challenging

Abstract

In this paper, we aim to expand the understanding of the relationship between the composition of the Hamiltonian in the Quantum Approximate Optimization Algorithm (QAOA) and the corresponding cost landscape characteristics. QAOA is a prominent example of a Variational Quantum Algorithm (VQA), which is most commonly used for combinatorial optimization. The success of QAOA heavily relies on parameter optimization, which is a great challenge, especially on scarce noisy quantum hardware. Thus understanding the cost function landscape can aid in designing better optimization heuristics and therefore potentially provide eventual value. We consider the case of 1-layer QAOA for Hamiltonians with up to 5-local terms and up to 20 qubits. In addition to visualizing the cost landscapes, we calculate their Fourier transform to study the relationship with the structure of the Hamiltonians from a complementary perspective. Furthermore, we introduce metrics to quantify the roughness of the landscape, which provide valuable insights into the nature of high-dimensional parametrized landscapes. While these techniques allow us to elucidate the role of Hamiltonian structure, order of the terms and their coefficients on the roughness of the optimization landscape, we also find that predicting the intricate landscapes of VQAs from first principles is very challenging and unlikely to be feasible in general.