We initiate the study of the generalization of random walks on nite graphs to the quantum world. Such quantum walks do not converge to any stationary distribution, as they are unitary and reversible. However, by suitably relaxing the denition, we can obtain a measure of how fast the quantum walk spreads or how conned the quantum walk stays in a small neighborhood. We give denitions of mixing time, lling time, dispersion time. We show that in all these measures, the quantum walk on the cycle is almost quadratically faster then its classical correspondent. On the other hand, we give a lower bound on the possible speed up by quantum walks for general graphs, showing that quantum walks can be at most polynomially faster than their classical counterparts. 1

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.

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

We present two new quantum algorithms that either find a triangle (a copy of K3) in an undirected graph G on n nodes, or reject if G is triangle free. The first algorithm uses combinatorial ideas with Grover Search and makes Õ(n10/7) queries. The second algorithm uses Õ(n13/10) queries, and it is based on a design concept of Ambainis [6] that incorporates the benefits of quantum walks into Grover search [18]. The first algorithm uses only O(log n) qubits in its quantum subroutines, whereas the second one uses O(n) qubits. The Triangle Problem was first treated in [12], where an algorithm with O(n + √ nm) query complexity was presented, where m is the number of edges of G.

We show that the hitting time of the discrete time quantum random walk on the n-bit hypercube from one corner to its opposite is polynomial in n. This gives the first exponential quantum-classical gap in the hitting time of discrete quantum random walks. We provide the framework for quantum hitting time and give two alternative definitions to set the ground for its study on general graphs. We then give an Random walks form one of the cornerstones of theoretical computer science as well as the basis of a broad variety of applications in mathematics, physics and the natural sciences. In computer science they are frequently used in the design and analysis of randomized algorithms. Markov chain simulations provide a paradigm for exploring an exponentially large set of combinatorial structures (such as assignments to a

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

Quantum walks are quantum counterparts of Markov chains. In this article, we give a brief overview of quantum walks, with emphasis on their algorithmic applications. 1

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 coin-flip 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 two-dimensional coin-space to achieve the time O ( √ N log N). 1

Abstract. This paper treats the quantum random walk on the line determined by a 2 × 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P,Q,R and S given by U. The dependence of the mth moment on U and initial qubit state ϕ is clarified. Furthermore a new type of limit theorems for the Hadamard walk is given. It shows that the behavior of quantum random walk is striking different from that of the classical ramdom walk. 1

Abstract. In this paper we study a one-dimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetry of probability distributions for the Hadamard walk.