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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.

References

  1. Itai Arad, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. Linear Time Algorithm for Quantum 2SAT. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 15:1-15:14, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  2. M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM J. Comput., 23(1):97-119, 1994. Google Scholar
  3. S. Bravyi. Efficient algorithm for a quantum analogue of 2-SAT. Available at arXiv.org e-Print quant-ph/0602108v1, 2006. Google Scholar
  4. Niel de Beaudrap and Sevag Gharibian. A Linear Time Algorithm for Quantum 2-SAT. In Ran Raz, editor, 31st Conference on Computational Complexity (CCC 2016), volume 50 of Leibniz International Proceedings in Informatics (LIPIcs), pages 27:1-27:21, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. Google Scholar
  5. Rodney G. Downey and M. R. Fellows. Parameterized Complexity. Springer Publishing Company, Incorporated, 2012. Google Scholar
  6. E. Fischer, J.A. Makowsky, and E.V. Ravve. Counting truth assignments of formulas of bounded tree-width or clique-width. Discrete Applied Mathematics, 156(4):511-529, 2008. Third Haifa Workshop on Interdisciplinary Applications of Graph Theory, Combinatorics &Algorithm. Google Scholar
  7. Robert Ganian, Petr Hliněný, and Jan Obdržálek. Better algorithms for satisfiability problems for formulas of bounded rank-width. Fundam. Inf., 123(1):59-76, 2013. Google Scholar
  8. Serge Gaspers, Christos H. Papadimitriou, Sigve Hortemo Sæther, and Jan Arne Telle. On satisfiability problems with a linear structure. In IPEC, 2016. Google Scholar
  9. Sevag Gharibian, Yichen Huang, Zeph Landau, and Seung Woo Shin. Quantum Hamiltonian complexity. Foundations and Trends® in Theoretical Computer Science, 10(3):159-282, 2014. Google Scholar
  10. D. Gosset and D. Nagaj. Quantum 3-SAT is QMA1-complete. In Proceedings of the 54th IEEE Symposium on Foundations of Computer Science (FOCS 2013), pages 756-765, 2013. Google Scholar
  11. L. R. Ford Jr. and D. R. Fulkerson. Maximal flow through a network. Canadian Journal of Mathematics, 8:399-404, 1956. Google Scholar
  12. S. Jukna. Extremal Combinatorics With Applications in Computer Science. Springer, second edition, 2011. Google Scholar
  13. R. Karp. Reducibility among combinatorial problems. In Complexity of Computer Computations, pages 85-103. New York: Plenum, 1972. Google Scholar
  14. Gyula Y. Katona and Péter G.N. Szabó. Bounds on the number of edges in hypertrees. Discrete Mathematics, 339(7):1884-1891, 2016. 7th Cracow Conference on Graph Theory, Rytro 2014. Google Scholar
  15. J. Kempe, A. Kitaev, and O. Regev. The complexity of the local Hamiltonian problem. SIAM Journal on Computing, 35(5):1070-1097, 2006. Google Scholar
  16. A. Kitaev, A. Shen, and M. Vyalyi. Classical and Quantum Computation. American Mathematical Society, 2002. Google Scholar
  17. C. R. Laumann, A. M. Läuchli, R. Moessner, A. Scardicchio, and S. L. Sondhi. Product, generic, and random generic quantum satisfiability. Physical Review A, 81:062345, 2010. Google Scholar
  18. C. R. Laumann, R. Moessner, A. Scardicchio, and S. L. Sondhi. Phase transitions and random quantum satisfiability. Quantum Information &Computation, 10:1-15, 2010. Google Scholar
  19. Ranjan N. Naik, S.B. Rao, S.S. Shrikhande, and N.M. Singhi. Intersection graphs of k-uniform linear hypergraphs. European Journal of Combinatorics, 3(2):159-172, 1982. Google Scholar
  20. T. J. Osborne. Hamiltonian complexity. Reports on Progress in Physics, 75(2):022001, 2012. Google Scholar
  21. Daniel Paulusma, Friedrich Slivovsky, and Stefan Szeider. Model counting for cnf formulas of bounded modular treewidth. Algorithmica, 76(1):168-194, 2016. Google Scholar
  22. Sigve Hortemo Sæther, Jan Arne Telle, and Martin Vatshelle. Solving #sat and maxsat by dynamic programming. J. Artif. Int. Res., 54(1):59-82, sep 2015. Google Scholar
  23. Marko Samer and Stefan Szeider. Algorithms for propositional model counting. Journal of Discrete Algorithms, 8(1):50-64, 2010. Google Scholar
  24. A. Schönhage. Equation solving in terms of computational complexity. In Proceedings of the International Congress of Mathematicians, pages 131-153, 1986. Google Scholar
  25. Arnold Schönhage. Quasi-GCD computations. Journal of Complexity, 1(1):118-137, 1985. Google Scholar
  26. Stefan Szeider. On Fixed-Parameter Tractable Parameterizations of SAT, pages 188-202. Springer Berlin Heidelberg, Berlin, Heidelberg, 2004. Google Scholar
  27. D. A. Wolfram. Solving generalized fibonacci recurrences. Fibonacci Quart., 36(2):129-145, 1998. Google Scholar
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