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arXiv:1502.04700 · Physical Review A

Classical-quantum mixing in the random 2-satisfiability problem

Ionut-Dragos Potirniche, Chris R. Laumann, S. L. Sondhi

Published Oct 1, 2015• 0 citations
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Abstract

Classical satisfiability (SAT) and quantum satisfiability (QSAT) are complete problems for the complexity classes NP and QMA, respectively, and they are believed to be intractable for both classical and quantum computers. Statistical ensembles of instances of these problems have been studied previously in an attempt to elucidate their typical, as opposed to worst-case, behavior. In this paper, we introduce a statistical ensemble that interpolates between classical and quantum. For the simplest 2-SAT--2-QSAT ensemble, we find the exact boundary that separates SAT and UNSAT instances. We do so by establishing coincident lower and upper bounds, in the limit of large instances, on the extent of the UNSAT and SAT regions, respectively.

DOI
10.1103/physreva.92.040301

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Classical satisfiability (SAT) and quantum satisfiability (QSAT) are complete problems for the complexity classes NP and QMA, respectively, and they are believed to be intractable for both classical and quantum computers. Statistical ensembles of instances of these problems have been studied previously in an attempt to elucidate their typical, as opposed to worst-case, behavior. In this paper, we introduce a statistical ensemble that interpolates between classical and quantum. For the simplest 2-SAT--2-QSAT ensemble, we find the exact boundary that separates SAT and UNSAT instances. We do so by establishing coincident lower and upper bounds, in the limit of large instances, on the extent of the UNSAT and SAT regions, respectively.
Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. We study random ensembles of the QMA1_11​-complete quantum satisfiability (QSAT) problem introduced by Bravyi \cite{Bravyi:2006p4315}. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show that, as the density of clauses/projectors is varied, the ensembles exhibit quantum phase transitions between phases that are satisfiable and unsatisfiable. Remarkably, almost all instances of QSAT for \emph{any} hypergraph exhibit the same dimension of the satisfying manifold. This establishes the QSAT decision problem as equivalent to a, potentially new, graph theoretic problem and that the hardest typical instances are likely to be localized in a bounded range of clause density.
Alongside the effort underway to build quantum computers, it is important to better understand which classes of problems they will find easy and which others even they will find intractable. We study random ensembles of the QMA1_11​-complete quantum satisfiability (QSAT) problem introduced by Bravyi. QSAT appropriately generalizes the NP-complete classical satisfiability (SAT) problem. We show that, as the density of clauses/projectors is varied, the ensembles exhibit quantum phase transitions between phases that are satisfiable and unsatisfiable. Remarkably, almost all instances of QSAT for any hypergraph exhibit the same dimension of the satisfying manifold. This establishes the QSAT decision problem as equivalent to a, potentially new, graph theoretic problem and that the hardest typical instances are likely to be localized in a bounded range of clause density.
The local Hamiltonian problem is the canonical QMA\mathsf{QMA}QMA-complete problem, and O(2n)O(2^n)O(2n) time classical algorithms and O(2n/2)O(2^{n/2})O(2n/2) time quantum algorithms are known to solve the problem in the worst case. It is not clear how to improve these brute force strategies for a broad class of the problem because ground states are highly entangled in general, and we cannot directly apply known strategies for classical CSPs. In this work, we present exponentially faster classical and quantum algorithms under two mild assumptions: (1) the Hamiltonian is frustration-free on YES instances, and (2) it is approximately regular, meaning that every qubit is acted upon by approximately the same number of constraints. We complement these upper bounds by showing that, assuming (Q)SETH, quantum 5-SAT admits no non-trivial worst-case speedup. Our lower bound further demonstrates that the dependence of our algorithms on regularity is in some sense nearly optimal. Specifically, quantum 5-SAT remains (Q)SETH-hard even for Hamiltonians in which all but O(n)O(\sqrt{n})O(n​) qubits participate in only constantly many constraints, while the remaining O(n)O(\sqrt{n})O(n​) qubits each participate in O(n)O(\sqrt{n})O(n​) constraints. By contrast, if either the size of this high-degree subset or the degrees of its qubits is reduced by a factor of nδn^δnδ, for any δ>0δ>0δ>0, our algorithm solves the problem in time O(2(1−ε)n)O(2^{(1- \varepsilon)n})O(2(1−ε)n) for some ε>0\varepsilon>0ε>0. Together, our upper and lower bounds establish a fine-grained complexity dichotomy for quantum satisfiability.
We report a cluster of results on kkk-QSAT, the problem of quantum satisfiability for kkk-qubit projectors which generalizes classical satisfiability with kkk-bit clauses to the quantum setting. First we define the NP\mathit{NP}NP-complete problem of product satisfiability and give a geometrical criterion for deciding when a QSAT interaction graph is product satisfiable with positive probability. We show that the same criterion suffices to establish quantum satisfiability for all projectors. Second, we apply these results to the random graph ensemble with generic projectors and obtain improved lower bounds on the location of the SAT-unSAT transition. Third, we present numerical results on random, generic satisfiability which provide estimates for the location of the transition for k=3k=3k=3 and k=4k=4k=4 and mild evidence for the existence of a phase which is satisfiable by entangled states alone.
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].