Approximation algorithms for QMA-complete problems

TL;DR

This paper introduces a natural approximation version of the QMA-complete local Hamiltonian problem, demonstrating classical approximation algorithms and methods for dense instances, advancing understanding of quantum constraint satisfaction problems.

Contribution

It defines a new approximation framework for QMA-complete problems and develops classical algorithms with provable approximation ratios for dense instances.

Findings

A non-trivial approximation ratio achievable in NP using product states.

A polynomial-time classical algorithm for dense instances with similar approximation guarantees.

Adaptation of the exhaustive sampling method to the quantum setting.

Abstract

Approximation algorithms for classical constraint satisfaction problems are one of the main research areas in theoretical computer science. Here we define a natural approximation version of the QMA-complete local Hamiltonian problem and initiate its study. We present two main results. The first shows that a non-trivial approximation ratio can be obtained in the class NP using product states. The second result (which builds on the first one), gives a polynomial time (classical) algorithm providing a similar approximation ratio for dense instances of the problem. The latter result is based on an adaptation of the "exhaustive sampling method" by Arora et al. [J. Comp. Sys. Sci. 58, p.193 (1999)] to the quantum setting, and might be of independent interest.