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An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem

Authors Sean Hallgren, Eunou Lee, Ojas Parekh

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Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Semidefinite programming
  • Theory of computation → Quantum complexity theory
Keywords
  • approximation algorithm
  • quantum computing
  • local Hamiltonian
  • mean-field theory
  • randomized rounding

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Abstract

We present a classical approximation algorithm for the MAX-2-Local Hamiltonian problem. This is a maximization version of the QMA-complete 2-Local Hamiltonian problem in quantum computing, with the additional assumption that each local term is positive semidefinite. The MAX-2-Local Hamiltonian problem generalizes NP-hard constraint satisfaction problems, and our results may be viewed as generalizations of approximation approaches for the MAX-2-CSP problem. We work in the product state space and extend the framework of Goemans and Williamson for approximating MAX-2-CSPs. The key difference is that in the product state setting, a solution consists of a set of normalized 3-dimensional vectors rather than boolean numbers, and we leverage approximation results for rank-constrained Grothendieck inequalities. For MAX-2-Local Hamiltonian we achieve an approximation ratio of 0.328. This is the first example of an approximation algorithm beating the random quantum assignment ratio of 0.25 by a constant factor.

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Sean Hallgren, Eunou Lee, and Ojas Parekh. An Approximation Algorithm for the MAX-2-Local Hamiltonian Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.59

Author Details

Sean Hallgren
  • Pennsylvania State University, State College, University Park, PA, USA
Eunou Lee
  • Pennsylvania State University, State College, University Park, PA, USA
Ojas Parekh
  • Sandia National Laboratories, Albuquerque, NM, USA

Funding

  • Hallgren, Sean: Partially supported by National Science Foundation awards CCF-1618287, CCF-1617710 and CNS-1617802, by a Vannevar Bush Faculty Fellowship from the US Department of Defense, and in part while the author was visiting the Simons Institute for the Theory of Computing.
  • Parekh, Ojas: Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525. Supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research for Quantum Computing and Quantum Algorithms Teams programs.

Acknowledgements

Thanks to Jianqiang Li and Sevag Gharibian for interesting discussions and useful comments.

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