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30
Exponential Algorithmic Speedup by a Quantum Walk
"... We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fouri ..."
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Cited by 102 (4 self)
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We construct a black box graph traversal problem that can be solved exponentially faster on a quantum computer than on a classical computer. The quantum algorithm is based on a continuous time quantum walk, and thus employs a di#erent technique from previous quantum algorithms based on quantum Fourier transforms. We show how to implement the quantum walk e#ciently in our black box setting. We then show how this quantum walk solves our problem by rapidly traversing a graph. Finally, we prove that no classical algorithm can solve the problem in subexponential time.
A new type of limit theorems for the onedimensional quantum random walk
, 2003
"... In this paper we consider the onedimensional quantum random walk Xϕ n at time n starting from initial qubit state ϕ determined by 2 × 2 unitary matrix U. We give a combinatorial. The expression clarifies the dependence of it on expression for the characteristic function of Xϕ n components of unit ..."
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Cited by 35 (15 self)
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In this paper we consider the onedimensional quantum random walk Xϕ n at time n starting from initial qubit state ϕ determined by 2 × 2 unitary matrix U. We give a combinatorial. The expression clarifies the dependence of it on expression for the characteristic function of Xϕ n components of unitary matrix U and initial qubit state ϕ. As a consequence of the above results, we present a new type of limit theorems for the quantum random walk. In contrast with the de MoivreLaplace limit theorem, our symmetric case implies that Xϕ n/n converges in distribution to a limit Zϕ as n → ∞ where Zϕ has a density 1/π(1−x 2) √ 1 − 2x2 for x ∈ ( − √ 2/2, √ 2/2). Moreover we discuss some known simulation results based on our limit theorems.
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 24 (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
Onedimensional quantum walks with absorbing boundaries. arXiv.org ePrint quantph/0207008
, 2002
"... In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the onedimensional case. We compute these probabilities both by employing generating functions and by use ..."
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Cited by 20 (2 self)
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In this paper we analyze the behavior of quantum random walks. In particular we present several new results for the absorption probabilities in systems with both one and two absorbing walls for the onedimensional case. We compute these probabilities both by employing generating functions and by use of an eigenfunction approach. The generating function method is used to determine some simple properties of the walks we consider, but appears to have limitations. The eigenfunction approach works by relating the problem of absorption to a unitary problem that has identical dynamics inside a certain domain, and can be used to compute several additional interesting properties, such as the time dependence of absorption. The eigenfunction method has the distinct advantage that it can be extended to arbitrary dimensionality. We outline the solution of the absorption probability problem of a (D − 1)dimensional wall in a Ddimensional space. 1
Analysis of absorbing times of quantum walks
 Physical Review A
, 2003
"... Quantum walks are expected to provide useful algorithmic tools for quantum computation. This paper introduces absorbing probability and time of quantum walks and gives both numerical simulation results and theoretical analyses on Hadamard walks on the line and symmetric walks on the hypercube from t ..."
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Cited by 8 (0 self)
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Quantum walks are expected to provide useful algorithmic tools for quantum computation. This paper introduces absorbing probability and time of quantum walks and gives both numerical simulation results and theoretical analyses on Hadamard walks on the line and symmetric walks on the hypercube from the viewpoint of absorbing probability and time. 1
Decoherence in a quantum walk on the line
 In Quantum Communication, Measurement & Computing (QCMC’02). Rinton
, 2002
"... We have studied how decoherence affects a quantum walk on the line. As expected, it is highly sensitive, consisting as it does of an extremely delocalized particle. We obtain an expression for the rate at which the standard deviation falls from the quantum value as decoherence increases and show tha ..."
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Cited by 7 (2 self)
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We have studied how decoherence affects a quantum walk on the line. As expected, it is highly sensitive, consisting as it does of an extremely delocalized particle. We obtain an expression for the rate at which the standard deviation falls from the quantum value as decoherence increases and show that it is proportional to the number of decoherence “events ” occuring during the walk. 1
Quantum walk based search algorithms
 In Proceedings of the 5th Conference on Theory and Applications of Models of Computation
, 2008
"... Abstract. In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We presen ..."
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Cited by 7 (1 self)
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Abstract. In this survey paper we give an intuitive treatment of the discrete time quantization of classical Markov chains. Grover search and the quantum walk based search algorithms of Ambainis, Szegedy and Magniez et al. will be stated as quantum analogues of classical search procedures. We present a rather detailed description of a somewhat simplified version of the MNRS algorithm. Finally, in the query complexity model, we show how quantum walks can be applied to the following
Almost uniform sampling via quantum walks
 New J. Phys
"... Many classical randomized algorithms (e.g., approximation algorithms for #Pcomplete problems) utilize the following random walk algorithm for almost uniform sampling from a state space S of cardinality N: run a symmetric ergodic Markov chain P on S for long enough to obtain a random state from with ..."
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Cited by 6 (2 self)
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Many classical randomized algorithms (e.g., approximation algorithms for #Pcomplete problems) utilize the following random walk algorithm for almost uniform sampling from a state space S of cardinality N: run a symmetric ergodic Markov chain P on S for long enough to obtain a random state from within ǫ total variation distance of the uniform distribution over S. The running time of this algorithm, the socalled mixing time of P, is O(δ −1 (log N+log ǫ −1)), where δ is the spectral gap of P. We present a natural quantum version of this algorithm based on repeated measurements of the quantum walk Ut = e −iPt. We show that it samples almost uniformly from S with logarithmic dependence on ǫ −1 just as the classical walk P does; previously, no such quantum walk algorithm was known. We then outline a framework for analyzing its running time and formulate two plausible conjectures which together would imply that it runs in time O(δ −1/2 log N log ǫ −1) when P is the standard transition matrix of a constantdegree graph. We prove each conjecture for a subclass of Cayley graphs. 1
A note on graphs resistant to quantum uniform mixing
, 2003
"... Continuoustime quantum walks on graphs is a generalization of continuoustime Markov chains on discrete structures. Moore and Russell proved that the continuoustime quantum walk on the¡cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs¢ ..."
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Cited by 5 (0 self)
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Continuoustime quantum walks on graphs is a generalization of continuoustime Markov chains on discrete structures. Moore and Russell proved that the continuoustime quantum walk on the¡cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs¢¤£, the continuoustime quantum walk is neither instantaneous (except for¡¦¥¨§�©���©��) nor average uniform mixing (except for¡�¥�§). We explore two natural grouptheoretic generalizations ofcirculant is the¡cube as a� and as a bunkbed�������,where� a finite group. Analyses of these classes suggest that the¡cube might be special in having instantaneous uniform mixing and that nonuniform average mixing is pervasive, i.e., no memoryless property for the average limiting distribution; an implication of these graphs having zero spectral gap. But on the bunkbeds, we note a memoryless property with respect to the two partitions. We also analyze average mixing on complete paths, where the spectral gaps are nonzero. 1
Quantum walks on directed graphs
 Quantum Inf. Comp
, 2007
"... We consider the definition of quantum walks on directed graphs. Call a directed graph reversible if, for each pair of vertices (vi, vj), if vi is connected to vj then there is a path from vj to vi. We show that reversibility is a necessary and sufficient condition for a directed graph to allow the n ..."
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Cited by 4 (0 self)
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We consider the definition of quantum walks on directed graphs. Call a directed graph reversible if, for each pair of vertices (vi, vj), if vi is connected to vj then there is a path from vj to vi. We show that reversibility is a necessary and sufficient condition for a directed graph to allow the notion of a discretetime quantum walk, and discuss some implications of this condition. We present a method for defining a “partially quantum ” walk on directed graphs that are not reversible. 1