Complexity of Stoquastic Frustration-Free Hamiltonians
Sergey Bravyi, Barbara M. Terhal · SIAM Journal on Computing · 2009
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.