Difficult Instances of the Counting Problem for 2-quantum-SAT are Very Atypical

Author Niel de Beaudrap



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Niel de Beaudrap

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Niel de Beaudrap. Difficult Instances of the Counting Problem for 2-quantum-SAT are Very Atypical. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 118-140, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)
https://doi.org/10.4230/LIPIcs.TQC.2014.118

Abstract

The problem 2-QUANTUM-SATISFIABILITY (QSAT[2]) is the generalisation of the 2-CNF-SAT problem to quantum bits, and is equivalent to determining whether or not a spin-1/2 Hamiltonian with two-body terms is frustration-free. imilarly to the classical problem #SAT[2], the counting problem #QSAT[2] of determining the size (i.e. the dimension) of the set of satisfying states is #P-complete. However, if we consider random instances of QSAT[2] in which constraints are sampled from the Haar measure, intractible instances have measure zero. An apparent reason for this is that almost all two-qubit constraints are entangled, which more readily give rise to long-range constraints. We investigate under which conditions product constraints also give rise to efficiently solvable families of #QSAT[2] instances. We consider #QSAT[2] involving only discrete distributions over tensor product operators, which interpolates between classical #SAT[2] and #QSAT[2] involving arbitrary product constraints. We find that such instances of #QSAT[2], defined on Erdös-Renyi graphs or bond-percolated lattices, are asymptotically almost surely efficiently solvable except to the extent that they are biased to resemble monotone instances of #SAT[2].
Keywords
  • Frustration-free
  • Hamiltonian
  • quantum
  • counting
  • satisfiability

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References

  1. B. Aspvall, M. F. Plass, and R. E. Tarjan. A linear-time algorithm for testing the truth of certain quantified boolean formulas. Information Processing Letters, 8(3):121-123, 1979. Google Scholar
  2. S. Bravyi. Efficient algorithm for a quantum analogue of 2-SAT. arXiv:quant-ph/0602108, 2006. Google Scholar
  3. S. Bravyi, C. Moore, and A. Russell. Bounds on the Quantum Satisfiability threshold. In Andrew Chi-Chih Yao, editor, ICS, pages 482-489. Tsinghua University Press, 2010. arXiv:0907.1297. Google Scholar
  4. V. Chvatal and B. Reed. Mick gets some (the odds are on his side). In Proc. 33rd Annual FOCS, pages 620-627, 1992. Google Scholar
  5. N. de Beaudrap, T. J. Osborne, and J. Eisert. Ground states of unfrustrated spin hamiltonians satisfy an area law. New J. Phys., 12:095007, 2010. arXiv:1009.3051. Google Scholar
  6. R. Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  7. P. Erdős and A. Rényi. On the evolution of random graphs. Publ. Math. Inst. Hungary. Acad. Sci., 5:17-61, 1960. Google Scholar
  8. E. Fischer, J. A. Makowsky, and E. V. Ravve. Counting truth assignments of formulas of bounded tree-width or clique-width. Discrete Applied Mathematics, 156:511-529, 2008. Google Scholar
  9. D. Gosset and D. Nagaj. Quantum 3-SAT is QMA1-complete. arXiv:1302.0290, 2013. Google Scholar
  10. G. Grimmett. Percolation. Springer, Berlin, 2nd edition, 1999. Google Scholar
  11. G. R. Grimmett, A. E. Holroyd, and G. Kozma. Percolation of finite clusters and infinite surfaces. Mathematical Proceedings of the Cambridge Philosophical Society, 156:263-279, 2014. arXiv:1303.1657. Google Scholar
  12. J. M. Hammersley. A generalization of McDiarmid’s theorem for mixed Bernoulli percolation. Math. Proc. Camb. Phil. Soc., 1980. Google Scholar
  13. Z. Ji, Z. Wei, and B. Zeng. Complete characterization of the ground space structure of two-body frustration-free hamiltonians for qubits. Phys. Rev. A, 84:042338, 2011. Google Scholar
  14. C. R. Laumann, R. Moessner, A. Scardicchio, and S. L. Sondhi. Phase transitions and random quantum satisfiability. Quant. Inf. and Comp., 10:1-15, 2010. arXiv:0903.1904. Google Scholar
  15. C. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Comp. Sys. Sci., 43:425-440, 1991. Google Scholar
  16. N. Robertson and P. D. Seymour. Graph minors. III. Planar tree-width. Journal of Combinatorial Theory, Series B, 36(1):49-64, 1984. Google Scholar
  17. L. Valiant. The complexity of enumeration and reliability problems. SIAM J. Computing, 8:410-421, 1979. Google Scholar
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