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37
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.
Dynamical Localization of Quantum Walks in Random Environments,
 J. Stat. Phys.,
, 2010
"... Abstract The dynamics of a one dimensional quantum walker on the lattice with two internal degrees of freedom, the coin states, is considered. The discrete time unitary dynamics is determined by the repeated action of a coin operator in U (2) on the internal degrees of freedom followed by a one ste ..."
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Abstract The dynamics of a one dimensional quantum walker on the lattice with two internal degrees of freedom, the coin states, is considered. The discrete time unitary dynamics is determined by the repeated action of a coin operator in U (2) on the internal degrees of freedom followed by a one step shift to the right or left, conditioned on the state of the coin. For a fixed coin operator, the dynamics is known to be ballistic. We prove that when the coin operator depends on the position of the walker and is given by a certain i.i.d. random process, the phenomenon of Anderson localization takes place in its dynamical form. When the coin operator depends on the time variable only and is determined by an i.i.d. random process, the averaged motion is known to be diffusive and we compute the diffusion constants for all moments of the position.
Dynamical localization for dDimensional Random Quantum Walks
 Quantum Information Processing
, 2012
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Multiparty quantum computation
, 2001
"... We investigate definitions of and protocols for multiparty quantum computing in the scenario where the secret data are quantum systems. We work in the quantum informationtheoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of veri ..."
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Cited by 7 (1 self)
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We investigate definitions of and protocols for multiparty quantum computing in the scenario where the secret data are quantum systems. We work in the quantum informationtheoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t < n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to establish that any multiparty quantum computation can be securely performed as long as the number of dishonest players is less than n/6.
Merkle puzzles in a quantum world
 In Advances in Cryptology  CRYPTO ’11
, 2011
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Quantum algorithms for testing properties of distributions
, 2009
"... Suppose one has access to oracles generating samples from two unknown probability distributions p and q on some Nelement set. How many samples does one need to test whether the two distributions are close or far from each other in the L1norm? This and related questions have been extensively studie ..."
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Cited by 4 (0 self)
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Suppose one has access to oracles generating samples from two unknown probability distributions p and q on some Nelement set. How many samples does one need to test whether the two distributions are close or far from each other in the L1norm? This and related questions have been extensively studied during the last years in the field of property testing. In the present paper we study quantum algorithms for testing properties of distributions. It is shown that the L1distance ‖p − q‖1 can be estimated with a constant precision using only O(N 1/2) queries in the quantum settings, whereas classical computers need Ω(N 1−o(1) ) queries. We also describe quantum algorithms for testing Uniformity and Orthogonality with query complexity O(N 1/3). The classical query complexity of these problems is known to be Ω(N 1/2). A quantum algorithm for testing Uniformity has been recently independently discovered by Chakraborty et al [1].