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

Authors Sevag Gharibian, Ojas Parekh

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


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.

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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)


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


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