A Linear Time Algorithm for Quantum 2-SAT

Authors Niel de Beaudrap, Sevag Gharibian

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Niel de Beaudrap
Sevag Gharibian

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Niel de Beaudrap and Sevag Gharibian. A Linear Time Algorithm for Quantum 2-SAT. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


The Boolean constraint satisfaction problem 3-SAT is arguably the canonical NP-complete problem. In contrast, 2-SAT can not only be decided in polynomial time, but in fact in deterministic linear time. In 2006, Bravyi proposed a physically motivated generalization of k-SAT to the quantum setting, defining the problem "quantum k-SAT". He showed that quantum 2-SAT is also solvable in polynomial time on a classical computer, in particular in deterministic time O(n^4), assuming unit-cost arithmetic over a field extension of the rational numbers, where n is number of variables. In this paper, we present an algorithm for quantum 2-SAT which runs in linear time, i.e. deterministic time O(n+m) for n and m the number of variables and clauses, respectively. Our approach exploits the transfer matrix techniques of Laumann et al. [QIC, 2010] used in the study of phase transitions for random quantum 2-SAT, and bears similarities with both the linear time 2-SAT algorithms of Even, Itai, and Shamir (based on backtracking) [SICOMP, 1976] and Aspvall, Plass, and Tarjan (based on strongly connected components) [IPL, 1979].
  • quantum 2-SAT
  • transfer matrix
  • strongly connected components
  • limited backtracking
  • local Hamiltonian


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