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arXiv:0910.2058 · Physical Review A

Product, generic, and random generic quantum satisfiability

Chris R. Laumann, Andreas Martin Läuchli, Roderich Moessner, Antonello Scardicchio, S. L. Sondhi

Published Jun 28, 2010• 35 citations
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Abstract

We report a cluster of results on $k$-QSAT, the problem of quantum satisfiability for $k$-qubit projectors which generalizes classical satisfiability with $k$-bit clauses to the quantum setting. First we define the $\mathit{NP}$-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 $k=3$ and $k=4$ and mild evidence for the existence of a phase which is satisfiable by entangled states alone.

DOI
10.1103/physreva.81.062345

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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.
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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.
The kkk-QSAT problem is a quantum analog of the famous kkk-SAT constraint satisfaction problem. We must determine the zero energy ground states of a Hamiltonian of NNN qubits consisting of a sum of MMM random kkk-local rank-one projectors. It is known that product states of zero energy exist with high probability if and only if the underlying factor graph has a clause-covering dimer configuration. This means that the threshold of the PRODSAT phase is a purely geometric quantity equal to the dimer covering threshold. We revisit and fully prove this result through a combination of complex analysis and algebraic methods based on Buchberger's algorithm for complex polynomial equations with random coefficients. We also discuss numerical experiments investigating the presence of entanglement in the PRODSAT phase in the sense that product states do not span the whole zero energy ground state space.
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.
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].
Random quantum satisfiability is a natural quantum analogue of random constraint satisfaction and a basic model for frustration-free local Hamiltonians. Despite extensive work on its satisfiable and unsatisfiable regimes, the quantitative location of the random quantum kkk-SAT threshold has remained poorly understood, with the best general upper bounds leaving a large gap to the known lower bounds. In this paper we prove a new upper bound on the satisfiability threshold of random quantum kkk-SAT. Our result improves the previously known asymptotic upper bound by a factor of order kkk, giving a bound of order 2k/k2^k/k2k/k. The improvement is also significant at small values of kkk; in particular, for random quantum 333-SAT we obtain a substantially smaller explicit upper bound than the one previously available. The proof combines the geometric formulation of generic quantum satisfiability with a dimension-decay analysis of the full satisfying subspace. The key input is a multiplicative Shearer-type inequality for tensor-product subspaces, which quantifies how global dimension forces nontrivial local dimension on typical sets of qubits.