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The local Hamiltonian problem plays the equivalent role of SAT in quantum complexity theory. Understanding the complexity of the intermediate case in which the constraints are quantum but all local terms in the Hamiltonian commute, is of importance for conceptual, physical and computational complexity reasons. Bravyi and Vyalyi showed in 2003 [8], using… (More)

We show that the quantum SAT problem is QMA 1-complete when restricted to interactions between a three-dimensional particle and a five-dimensional particle. The best previously known result is for particles of dimensions 4 and 9. The main novel ingredient of our proof is a certain Hamiltonian construction named the Triangle Hamiltonian. It allows to verify… (More)

We initiate the study of quantum Locally Testable Codes (qLTCs). We provide a definition together with a simplification, denoted sLTCs, for the special case of stabilizer codes, and provide some basic results using those definitions. The most crucial parameter of such codes is their sound-ness, R(δ), namely, the probability that a randomly chosen constraint… (More)

Recent results by Harrow et. al. [2], and by Ta-Shma [5], suggest that quantum computers may have an exponential advantage in solving a wealth of linear algebraic problems, over classical algorithms. Building on the quantum intuition of these results, we step back into the classical domain, and explore its usefulness in designing classical algorithms. We… (More)

Computing the eigenvectors and eigenvalues of a given Hermitian matrix is arguably one of the most well-studied computational problems. Yet despite its immense importance, and a vast array of heuristic techniques, there is no algorithm that can provably approximate the spectral decomposition of any Hermitian matrix in asymptotic bit-complexity o(n 3).… (More)

We introduce a new canonical form of lattices called the systematic normal form (SNF). We show that for every lattice there is an efficiently computable " nearby " SNF lattice, such that for any lattice one can solve lattice problems on its " nearby " SNF lattice, and translate the solutions back efficiently to the original lattice. The SNF provides direct… (More)

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