The threshold for random 𝑘-SAT is 2^{𝑘}log2-𝑂(𝑘)

Achlioptas, Dimitris; Peres, Yuval

doi:10.1090/s0894-0347-04-00464-3

Cited by 138 publications

(50 citation statements)

References 20 publications

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“…With help of the cavity method, we are able to conjecture improved lower bounds for the satisfiability of QSAT on regular and Erdős-Rényi random graphs, canonical models for quantum constraint optimization problems. These statement hold just as well for classical satisfiabilitybut for some of these classical models better bounds are known [12,13].…”

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confidence: 86%

When a local Hamiltonian must be frustration-free

Sattath

Morampudi

Laumann

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

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A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustrationfree Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion-a sufficient condition-under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian's interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.quantum satisfiability | local Hamiltonian | hardcore lattice gas | critical exponents | universality A n overwhelming majority of systems of physical interest can be described via local Hamiltonians:[1]Here, the "k-local" operator Π i acts on a k-tuple of the microscopic degrees of freedom, best thought of as qudits for the computer scientists among our readers or spins for the physicists. The M operators define an interaction (hyper)graph G, displayed in Fig. 1. A surprisingly diverse and important class of such model Hamiltonians is defined by the additional property of being "frustrationfree": The ground state jψi of H is a simultaneous ground state of each and every Π i . This class comprises both commuting Hamiltonians-for which ½Π i , Π j = 0 ∀i, j-such as the toric code, general quantum error correcting codes, and Levin-Wen models (1-3), and noncommuting ones, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) and Rokhsar-Kivelson models (4-6). Their particular usefulness is also related to the fact that many of these examples can be viewed as "local parent Hamiltonians" for generalized matrix product states (7). In general, frustration-free conditions provide analytic control of ground state properties in otherwise largely inaccessible quantum problems.Determining whether a given Hamiltonian H is frustration-free is well known in quantum complexity theory as the quantum satisfiability...

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confidence: 86%

When a local Hamiltonian must be frustration-free

Sattath

Morampudi

Laumann

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

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“…On this scale the SAT/UNSAT phase transition occurs at α s = log 2 + O(2 −k ) [6,12]. We shall therefore assume α ∈ (0, log 2).…”

Section: B Intermediate Regimementioning

confidence: 99%

Clusters of solutions and replica symmetry breaking in randomk-satisfiability

Montanari

Ricci-Tersenghi

Semerjian³

2008

J. Stat. Mech.

126

209

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We study the set of solutions of random k-satisfiability formulae through the cavity method. It is known that, for an interval of the clause-to-variables ratio, this decomposes into an exponential number of pure states (clusters). We refine substantially this picture by: (i) determining the precise location of the clustering transition; (ii) uncovering a second 'condensation' phase transition in the structure of the solution set for k ≥ 4. These results both follow from computing the large deviation rate of the internal entropy of pure states. From a technical point of view our main contributions are a simplified version of the cavity formalism for special values of the Parisi replica symmetry breaking parameter m (in particular for m = 1 via a correspondence with the tree reconstruction problem) and new large-k expansions.

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“…The best upper bounds were derived using first moment methods [12,13]. Lower bounds can be found by analyzing some algorithms which find SAT assignments [14,15], but recently a new method, based on second moment methods, has found better and algorithm-independent lower bounds [16,17]. Using these bounds, it was shown that α c (K) scales as 2 K ln(2) when K → ∞.…”

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confidence: 99%

Clustering of Solutions in the Random Satisfiability Problem

2005

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Using elementary rigorous methods we prove the existence of a clustered phase in the random K-SAT problem, for K ≥ 8. In this phase the solutions are grouped into clusters which are far away from each other. The results are in agreement with previous predictions of the cavity method and give a rigorous confirmation to one of its main building blocks. It can be generalized to other systems of both physical and computational interest.

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The threshold for random 𝑘-SAT is 2^{𝑘}log2-𝑂(𝑘)

Abstract: Let F k ( n , m ) F_k(n,m) be a random k k -CNF formula formed by selecting uniformly and independently m m out of all possible k k -clauses on n n variables. It is well known that if r ≥ 2 k log ⁡ 2 r \geq 2^k \log 2 , then F k ( … Show more

Cited by 138 publications

References 20 publications

When a local Hamiltonian must be frustration-free

When a local Hamiltonian must be frustration-free

Clusters of solutions and replica symmetry breaking in randomk-satisfiability

Clustering of Solutions in the Random Satisfiability Problem

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