Adiabatic quantum state generation and statistical zero knowledge
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
The Complexity of the Local Hamiltonian Problem
SIAM Journal on Computing
The PCP theorem by gap amplification
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Improved Gap Estimates for Simulating Quantum Circuits by Adiabatic Evolution
Quantum Information Processing
Circuit lower bounds for Merlin-Arthur classes
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation
SIAM Journal on Computing
The Power of Quantum Systems on a Line
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Finding cliques by quantum adiabatic evolution
Quantum Information & Computation
The complexity of stoquastic local Hamiltonian problems
Quantum Information & Computation
The complexity of quantum spin systems on a two-dimensional square lattice
Quantum Information & Computation
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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.