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Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems

Authors Ojas Parekh, Kevin Thompson

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Author Details

Ojas Parekh
  • Sandia National Laboratories, Albuquerque, NM, USA
Kevin Thompson
  • Sandia National Laboratories, Albuquerque, NM, USA

Funding

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. This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Accelerated Research in Quantum Computing and Quantum Algorithms Teams programs.

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Ojas Parekh and Kevin Thompson. Beating Random Assignment for Approximating Quantum 2-Local Hamiltonian Problems. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 74:1-74:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ESA.2021.74

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Semidefinite programming
  • Theory of computation → Quantum complexity theory
Keywords
  • Quantum Approximation Algorithms
  • Local Hamiltonian

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