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

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Abstract

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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.MFCS.2018.38

Author Details

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

Funding

This work was partially supported by the EPSRC National Quantum Technology Hub in Networked Quantum Information Processing, and NSF grants CCF-1526189 and CCF-1617710.
  • de Beaudrap, Niel: NdB acknowledges support from the EPSRC National Quantum Technology Hub in Networked Quantum Information Processing.
  • Gharibian, Sevag: SG acknowledges support from NSF grants CCF-1526189 and CCF-1617710.

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