Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems

TL;DR

This paper introduces the first classical approximation algorithms that outperform the trivial random assignment for quantum 2-Local Hamiltonian problems, achieving better approximation ratios and connecting quantum problems to classical CSP techniques.

Contribution

It presents novel classical approximation algorithms for quantum 2-Local Hamiltonian problems that beat previous bounds, including for maximally entangled instances and special bipartite cases.

Findings

First polynomial-time 0.764-approximation for Max 2-Local Hamiltonian with rank 3 projectors

Numerical evidence suggests a 0.821-approximation for strictly quadratic instances

Improved classical approximation for bipartite strictly quadratic traceless 2-Local Hamiltonians

Abstract

The quantum k-Local Hamiltonian problem is a natural generalization of classical constraint satisfaction problems (k-CSP) and is complete for QMA, a quantum analog of NP. Although the complexity of k-Local Hamiltonian problems has been well studied, only a handful of approximation results are known. For Max 2-Local Hamiltonian where each term is a rank 3 projector, a natural quantum generalization of classical Max 2-SAT, the best known approximation algorithm was the trivial random assignment, yielding a 0.75-approximation. We present the first approximation algorithm beating this bound, a classical polynomial-time 0.764-approximation. For strictly quadratic instances, which are maximally entangled instances, we provide a 0.801 approximation algorithm, and numerically demonstrate that our algorithm is likely a 0.821-approximation. We conjecture these are the hardest instances to approximate. We also give improved approximations for quantum generalizations of other related classical 2-CSPs. Finally, we exploit quantum connections to a generalization of the Grothendieck problem to obtain a classical constant-factor approximation for the physically relevant special case of strictly quadratic traceless 2-Local Hamiltonians on bipartite interaction graphs, where a inverse logarithmic approximation was the best previously known (for general interaction graphs). Our work employs recently developed techniques for analyzing classical approximations of CSPs and is intended to be accessible to both quantum information scientists and classical computer scientists.