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Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut

Authors Sevag Gharibian, Ojas Parekh

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Author Details

Sevag Gharibian
  • University of Paderborn, Germany
  • Virginia Commonwealth University, Richmond, VA, USA
Ojas Parekh
  • Sandia National Laboratories, Albuquerque, New Mexico, USA

Funding

  • Gharibian, Sevag: NSF grants CCF-1526189 and CCF-1617710
  • Parekh, Ojas: Laboratory Directed Research and Development program at Sandia National Laboratories, 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. Also supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams program.

Acknowledgements

We thank David Gosset and Mark Wilde for helpful discussions, and an anonymous referee for catching a technical error in an earlier version of this draft.

Cite AsGet BibTex

Sevag Gharibian and Ojas Parekh. Almost Optimal Classical Approximation Algorithms for a Quantum Generalization of Max-Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.31

Abstract

Approximation algorithms for constraint satisfaction problems (CSPs) are a central direction of study in theoretical computer science. In this work, we study classical product state approximation algorithms for a physically motivated quantum generalization of Max-Cut, known as the quantum Heisenberg model. This model is notoriously difficult to solve exactly, even on bipartite graphs, in stark contrast to the classical setting of Max-Cut. Here we show, for any interaction graph, how to classically and efficiently obtain approximation ratios 0.649 (anti-feromagnetic XY model) and 0.498 (anti-ferromagnetic Heisenberg XYZ model). These are almost optimal; we show that the best possible ratios achievable by a product state for these models is 2/3 and 1/2, respectively.

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
  • Max-Cut
  • local Hamiltonian
  • QMA-hard
  • Heisenberg model
  • product state

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