Complexity of Stoquastic Frustration-Free Hamiltonians

Abstract

We study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a “frustration-free” ground-state. A Hamiltonian belongs to this class iff it can be represented as $H=\sum_{a}H_{a}$ where (1) every term $H_{a}$ acts nontrivially on a constant number of qubits, (2) every term $H_{a}$ has real nonpositive off-diagonal matrix elements in the standard basis, and (3) the ground-state of $H$ is a ground-state of every term $H_{a}$. Second, we generalize the Cook-Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA (Merlin-Arthur games)—a probabilistic analogue of NP. Specifically, we construct a quantum version of the $k$-SAT problem which we call “stoquastic $k$-SAT” such that stoquastic $k$-SAT is contained in MA for any constant $k$, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first nontrivial example of a MA-complete promise problem.