When a local Hamiltonian must be frustration-free

TL;DR

This paper introduces a classical-quantum criterion to determine when local Hamiltonians are frustration-free, linking quantum complexity with classical statistical mechanics and providing new bounds for quantum satisfiability transitions.

Contribution

It extends Shearer's theorem to quantum systems, offering a practical criterion for frustration-freeness based on classical analysis, and applies it to various lattice models and satisfiability problems.

Findings

A sufficient condition for frustration-freeness of local Hamiltonians.

New bounds on SAT/UNSAT transition in random quantum satisfiability.

Identification of a universality notion in quantum satisfiability problems.

Abstract

A broad range of quantum optimisation problems can be phrased as the question 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 in order 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 hard-core 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 which permit us to obtain new bounds on the SAT/UNSAT transition in random quantum satisfiability. These also lead us 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 underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.