Performance of Variational Algorithms for Local Hamiltonian Problems on Random Regular Graphs

TL;DR

This paper introduces two variational algorithms inspired by QAOA for optimizing local Hamiltonians on random regular graphs, analyzing their performance and limitations, and comparing them to classical and worst-case algorithms.

Contribution

The paper develops formulae to analyze variational algorithms' energy performance on large random regular graphs and compares their effectiveness to classical and worst-case methods.

Findings

Algorithms outperform classical methods on EPR and bipartite QMC Hamiltonians.

Symmetry in the algorithms hinders optimization of the general QMC Hamiltonian.

Five layers suffice to approximate the ground state within 1.62% error for QMC on a 1D ring.

Abstract

We design two variational algorithms to optimize specific 2-local Hamiltonians defined on graphs. Our algorithms are inspired by the Quantum Approximate Optimization Algorithm. We develop formulae to analyze the energy achieved by these algorithms with high probability over random regular graphs in the infinite-size limit, using techniques from [arXiv:2110.14206]. The complexity of evaluating these formulae scales exponentially with the number of layers of the algorithms, so our numerical evaluation is limited to a small constant number of layers. We compare these algorithms to simple classical approaches and a state-of-the-art worst-case algorithm. We find that the symmetry inherent to these specific variational algorithms presents a major \emph{obstacle} to successfully optimizing the Quantum MaxCut (QMC) Hamiltonian on general graphs. Nonetheless, the algorithms outperform known methods to optimize the EPR Hamiltonian of [arXiv:2209.02589] on random regular graphs, and the QMC Hamiltonian when the graphs are also bipartite. As a special case, we show that with just five layers of our algorithm, we can already prepare states within 1.62% error of the ground state energy for QMC on an infinite 1D ring, corresponding to the antiferromagnetic Heisenberg spin chain.