A Quantum Approximate Optimization Algorithm for Local Hamiltonian Problems

TL;DR

This paper introduces HamQAOA, a quantum algorithm for local Hamiltonian problems, providing theoretical guarantees and demonstrating empirical advantages over prior methods, especially for quantum MaxCut and Heisenberg models.

Contribution

The paper proposes HamQAOA, a new quantum approximation algorithm with performance guarantees and heuristic strategies, suitable for near-term quantum hardware.

Findings

Rigorous bounds for QMC on high-girth graphs

Empirical outperformance of prior algorithms on QMC instances

Efficient ground state preparation for 1D Heisenberg chains

Abstract

Local Hamiltonian Problems (LHPs) are important problems that are computationally QMA-complete and physically relevant for many-body quantum systems. Quantum MaxCut (QMC), which equates to finding ground states of the quantum Heisenberg model, is the canonical LHP for which various algorithms have been proposed, including semidefinite programs and variational quantum algorithms. We propose and analyze a quantum approximation algorithm which we call the Hamiltonian Quantum Approximate Optimization Algorithm (HamQAOA), which builds on the well-known scheme for combinatorial optimization and is suitable for implementations on near-term hardware. We establish rigorous performance guarantees of the HamQAOA for QMC on high-girth regular graphs, and our result provides bounds on the ground energy density for quantum Heisenberg spin glasses in the infinite size limit that improve with depth. Furthermore, we develop heuristic strategies with which to efficiently obtain good HamQAOA parameters. Through numerical simulations, we show that the HamQAOA empirically outperforms prior algorithms on a wide variety of QMC instances. In particular, our results indicate that the linear-depth HamQAOA can deterministically prepare exact ground states of 1-dimensional antiferromagnetic Heisenberg spin chains described by the Bethe ansatz, in contrast to the exponential depths required in previous protocols for preparing Bethe states.