When a local Hamiltonian must be frustration-free

Significance Quantum computers promise computational power qualitatively superior to that achievable classically. This power will not be unlimited: Beyond much-touted applications, such as breaking encryption schemes, entire classes of problems are known to be intractable even for quantum computers. This work addresses a question of great practical relevance: In between these two extremes of certain (in)tractability, how can one efficiently diagnose the nature and properties of a given problem instance? To achieve this, we adopt a strategy of transferring insights from statistical physics and classical computing into the quantum realm. This provides a new perspective on the complexity of quantum problems and allows us to analyze a canonical class of quantum optimization problems in unprecedented explicit detail. 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. Frustration-free 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.