Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider quantum walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the “Hadamard walk”. In particular, we show that after t time steps, the probability distribution on the line induced by the Hadamard walk ...

Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider quantum walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the “Hadamard walk”. We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost unifo...

Quantum walks on graphs [? ] are an extension of random walks to the quantum domain. Towards mixing, they can seemingly provide a quadratic speedup compared to reversible classical random walks. On the other hand, lifted randomwalks, as proposed by Diaconis et al. [? ], are a non-quantum extension proven to deliver the same speedup [? ]). However, the construction of lifted walks makes explicit...

This paper treats absorption problems for the one-dimensional quantum random walk determined by a 2× 2 unitary matrix U on a state space {0, 1, . . . ,N} where N is finite or infinite by using a new path integral approach based on an orthonormal basis P,Q,R and S of the vector space of complex 2× 2 matrices. Our method studied here is a natural extension of the approach in the classical random ...

We consider quantum random walks on congested lattices and contrast them to classical random walks. Congestion is modelled on lattices that contain static defects which reverse the walker's direction. We implement a dephasing process after each step which allows us to smoothly interpolate between classical and quantum random walks as well as study the effect of dephasing on the quantum walk. Ou...

In this paper we review our recent results on limit theorems and absorption problems for the one-dimensional quantum random walk determined by 2× 2 unitary matrix.

For a long time one has associated to the Quantum Heisenberg Ferromagnet on a lattice, a random walk on the permutation group of the lattice vertices. We here present a polymer expansion for the solution of the heat equation coupled to the random walk. We work on a finite lattice, there is no question of convergence. We leave to future work bounding terms in the expansion necessary to extend th...

Consider the random walk among N places with N(N - 1)/2 transports. We attach an exponential random variable Xij to each transport between places Pi and Pj and take these random variables mutually independent. If transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. For given two functions, f and g, as an oracle which have domains of size N and M (N ≤ M), respectively, and the same range, the goal of the problem is to find x and y such that f (x) = g(y). This problem has been considered in both quantum and classical settings in t...

Quantum walks can be considered as a generalized version of the classical random walk. There are two classes of quantum walks, that is, the discrete-time (or coined) and the continuous-time quantum walks. This manuscript treats the discrete case in Part I and continuous case in Part II, respectively. Most of the contents are based on our results. Furthermore, papers on quantum walks are listed ...