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191
Quantum walk algorithms for element distinctness
 In: 45th Annual IEEE Symposium on Foundations of Computer Science, OCT 1719, 2004. IEEE Computer Society Press, Los Alamitos, CA
, 2004
"... We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrm ..."
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Cited by 174 (13 self)
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We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among N given items), we get an O(N 2/3) query quantum algorithm. This improves the previous O(N 3/4) quantum algorithm of Buhrman et al. [11] and matches the lower bound by [1]. We also give an O(N k/(k+1) ) query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among N items. 1
Coins make quantum walks faster
 Proceedings of SODA’05. Also quantph/0402107
"... We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently ..."
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Cited by 61 (8 self)
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We show how to search N items arranged on a √ N × √ N grid in time O ( √ N log N), using a discrete time quantum walk. This result for the first time exhibits a significant difference between discrete time and continuous time walks without coin degrees of freedom, since it has been shown recently that such a continuous time walk needs time Ω(N) to perform the same task. Our result furthermore improves on a previous bound for quantum local search by Aaronson and Ambainis. We generalize our result to 3 and more dimensions where the walk yields the optimal performance of O ( √ N) and give several extensions of quantum walk search algorithms for general graphs. The coinflip operation needs to be chosen judiciously: we show that another “natural ” choice of coin gives a walk that takes Ω(N) steps. We also show that in 2 dimensions it is sufficient to have a twodimensional coinspace to achieve the time O ( √ N log N). 1
Continuoustime quantum walks on the symmetric group
 Proc. RANDOMAPPROX (Sanjeev
, 2003
"... In this paper we study continuoustime quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quan ..."
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Cited by 28 (0 self)
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In this paper we study continuoustime quantum walks on Cayley graphs of the symmetric group, and prove various facts concerning such walks that demonstrate significant differences from their classical analogues. In particular, we show that for several natural choices for generating sets, these quantum walks do not have uniform limiting distributions, and are effectively blind to large areas of the graphs due to destructive interference. 1
Quantum Search Algorithms
, 2005
"... We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1 ..."
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Cited by 27 (1 self)
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We review some of quantum algorithms for search problems: Grover’s search algorithm, its generalization to amplitude amplification, the applications of amplitude amplification to various problems and the recent quantum algorithms based on quantum walks. 1
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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Cited by 21 (8 self)
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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.
Onedimensional discretetime quantum walks on random environments
, 2009
"... We consider discretetime nearestneighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk. ..."
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Cited by 14 (3 self)
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We consider discretetime nearestneighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk.
Quantum speedup of classical mixing processes. ArXiv: quantph/0609204. Kendon 52
 Physica A
, 2004
"... It is known that repeated measurements performed at uniformly random times enable the continuoustime quantum walk on a finite set S (using a stochastic transition matrix P as the timeindependent Hamiltonian) to sample almost uniformly from S provided that P does. Here we show that the same phenome ..."
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Cited by 14 (1 self)
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It is known that repeated measurements performed at uniformly random times enable the continuoustime quantum walk on a finite set S (using a stochastic transition matrix P as the timeindependent Hamiltonian) to sample almost uniformly from S provided that P does. Here we show that the same phenomenon holds for other (discretetime) walk variants and more general measurements types, then focus our attention on two questions: How are these repeatedlymeasured walks related to the decohering quantum walks proposed by Kendon/Tregenna and Alagic/Russell? And, when do they yield a speedup over their classical counterparts? We answer the first question with a proof that the two quantum walk models are essentially equivalent (in that they sample almost uniformly from S with nearly the same efficiency) by relating the spectral gaps of the Markov chains describing their action on S. We answer the second question (in part) by showing that these quantum walks sample almost uniformly from the torus Z d n in time O(n log ǫ−1). This represents a quadratic speedup over classical and for d = 1 confirms a conjecture of Kendon and Tregenna based on numerical experiments. 1