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Quantum walks: a comprehensive review
, 2012
"... Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting ..."
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Cited by 24 (0 self)
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Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists and engineers. In this paper we review theoretical advances on the foundations of both discrete and continuoustime quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discretetime quantum walks. Furthermore, we have reviewed several algorithms based on both discrete and continuoustime quantum walks as well as a most important result: the computational universality of both continuous and discretetime quantum walks.
Recurrence for discrete time unitary evolutions
 A QUANTUM DYNAMICAL APPROACH TO MATRIX KHRUSHCHEV’S FORMULAS 41
"... Abstract. We consider quantum dynamical systems specified by a unitary operator U and an initial state vector φ. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens wit ..."
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Abstract. We consider quantum dynamical systems specified by a unitary operator U and an initial state vector φ. In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to φ. We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature. 1.
Dynamical localization for dDimensional Random Quantum Walks
 Quantum Information Processing
, 2012
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Fluid Pinchoff
"... This 4608 2 image of a combustion simulation result was rendered by a hybridparallel (MPI+pthreads) raycasting volume rendering implementation running on 216,000 cores of the JaguarPF supercomputer. Combustion simulation data courtesy of J. Bell and M. Day ..."
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This 4608 2 image of a combustion simulation result was rendered by a hybridparallel (MPI+pthreads) raycasting volume rendering implementation running on 216,000 cores of the JaguarPF supercomputer. Combustion simulation data courtesy of J. Bell and M. Day