A new construction for a QMA complete 3-local Hamiltonian

We present a new way of encoding a quantum computation into a 3-local Hamiltonian. Our construction is novel in that it does not include any terms that induce legal-illegal clock transitions. Therefore, the weights of the terms in the Hamiltonian do not scale with the size of the problem as in previous constructions. This improves the construction by Kempe and Regev, who were the first to prove that 3-local Hamiltonian is complete for the complexity class QMA, the quantum analogue of NP. Quantum k-SAT, a restricted version of the local Hamiltonian problem using only projector terms, was introduced by Bravyi as an analogue of the classical k-SAT problem. Bravyi proved that quantum 4-SAT is complete for the class QMA with one-sided error (QMA_1) and that quantum 2-SAT is in P. We give an encoding of a quantum circuit into a quantum 4-SAT Hamiltonian using only 3-local terms. As an intermediate step to this 3-local construction, we show that quantum 3-SAT for particles with dimensions 3x2x2 (a qutrit and two qubits) is QMA_1 complete. The complexity of quantum 3-SAT with qubits remains an open question.Comment: 11 pages, 4 figure

On Perfect Completeness for QMA

Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a "quantum oracle" relative to which they are different. As a byproduct, we find that there are facts about quantum complexity classes that are classically relativizing but not quantumly relativizing, among them such "trivial" containments as BQP in ZQEXP.Comment: 9 pages. To appear in Quantum Information & Computatio

Fast Universal Quantum Computation with Railroad-switch Local Hamiltonians

We present two universal models of quantum computation with a time-independent, frustration-free Hamiltonian. The first construction uses 3-local (qubit) projectors, and the second one requires only 2-local qubit-qutrit projectors. We build on Feynman's Hamiltonian computer idea and use a railroad-switch type clock register. The resources required to simulate a quantum circuit with L gates in this model are O(L) small-dimensional quantum systems (qubits or qutrits), a time-independent Hamiltonian composed of O(L) local, constant norm, projector terms, the possibility to prepare computational basis product states, a running time O(L log^2 L), and the possibility to measure a few qubits in the computational basis. Our models also give a simplified proof of the universality of 3-local Adiabatic Quantum Computation.Comment: Added references to work by de Falco et al., and realized that Feynman's '85 paper already contained the idea of a switch in i