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On Efficiently Solvable Cases of Quantum k-SAT

Authors Marco Aldi, Niel de Beaudrap, Sevag Gharibian, Seyran Saeedi

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Marco Aldi
  • Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA, USA
Niel de Beaudrap
  • Department of Computer Science, University of Oxford, UK
Sevag Gharibian
  • Department of Computer Science, University of Paderborn, Germany, and Virginia Commonwealth University, USA
Seyran Saeedi
  • Department of Computer Science, Virginia Commonwealth University, Richmond, VA, USA

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Marco Aldi, Niel de Beaudrap, Sevag Gharibian, and Seyran Saeedi. On Efficiently Solvable Cases of Quantum k-SAT. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 38:1-38:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open. Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry; we hope these prove useful elsewhere.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Quantum complexity theory
  • search complexity
  • local Hamiltonian
  • Quantum SAT
  • algebraic geometry


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