When a local Hamiltonian must be frustration-free

Abstract

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...

Quantum Lovász local lemma: Shearer’s bound is tight

“…It turns out that Shearer's bound is generally not tight for variable version LLL (VLLL) [24]. Recently, Ambainis et al [3] introduced a quantum version LLL (QLLL), which was then shown to be powerful for the quantum satisfiability problem.In this paper, we prove that Shearer's bound is tight for QLLL, i.e., the relative dimension of the smallest satisfying subspace is completely characterized by the independent set polynomial, affirming a conjecture proposed by Sattath et al [34,39]. Our result also shows the tightness of Gilyén and Sattath's algorithm [18], and implies that the lattice gas partition function fully characterizes quantum satisfiability for almost all Hamiltonians with large enough qudits [39].Commuting LLL (CLLL), LLL for commuting local Hamiltonians which are widely studied in the literature, is also investigated here.…”

Correlation Length versus Gap in Frustration-Free Systems

“…Its quantum analogue, defined by Bravyi [10] (see also Refs. [11][12][13][14][15][16]), is a computational problem in which one is given a local Hamiltonian and asked to determine if it is frustration-free.…”

Clustering in Hilbert space of a quantum optimization problem

“…We refer to this as the "geometrization" property. The satisfying dimension R G may be lower bounded using the Quantum Shearer formula [13] and in certain limits it is conjectured that this formula gives a tight, and thus generic, result. For generic QSAT instances, zero energy product states exist if and only if the graph G admits "dimer coverings", which match qubits (circles) to interactions (squares) covering all of the interactions (see Fig.…”

“…The two generic possibilities are that there is either an (entangled) satisfiable phase, ENTSAT, as happens for k-QSAT for sufficiently large k [12,13]; or in its absence, a phase which is unsatisfiable, UNSAT, as happens for k = 2 [11]. For numerical studies, small k are more tractable than large ones, so that the question of the existence of an ENTSAT phase for k = 3 is of considerable practical relevance.…”

“…More precisely, the SAT phase in general contains two distinct phases -one where the ground states include product states (PRODSAT) [10,11] and another where the only ground states present are entangled (ENTSAT) [12]. For a more detailed review of this phase diagram, see [13][14][15].…”