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.

Abstract. In this paper we consider the one-dimensional 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 Moivre-Laplace 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. 1

In this paper we study continuous-time 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

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 one-dimensional 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 D-dimensional space. 1

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

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

Continuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the¡-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs¢¤£, the continuous-time quantum walk is neither instantaneous (except for¡¦¥¨§�©���©��) nor average uniform mixing (except for¡�¥�§). We explore two natural group-theoretic generalizations of-circulant 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

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

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 discrete-time 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

The present paper deals with both instantaneous uniform mixing property and temporal standard deviation for continuous-time quantum random walks on circles in order to study their fluctuations comparing with discrete-time quantum random walks, and continuous- and discrete-time classical random walks. Keywords: Continuous-time quantum random walks; temporal fluctuations; circle. 1.