This paper treats absorption problems for the one-dimensional quantum 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 walk.be the probability that the particle hits 0 starting from k before it arrives at N. The absorption problem is well known as the Gambler's ruin problem.Recently, in quantum walks, various problems including absorption have been widely investigated by a number of groups in connection with quantum computing. Examples include Aharonov et al.Abbr. title: Absorption problems of quantum walks.

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.

No abstract

We count the number of nonzero states in the spatio-temporal or spatial patterns of cellular automata. By observing self-similar structures, we determine the number of nonzero states in the pattern. For Rule 90 and Rule 150 of one-dimensional elementary cellular automata, we provide an overview of previous studies on the number of nonzero states in spatio-temporal patterns until the finite time step 2n − 1. In this study, we calculate the numbers in spatial patterns for each time step t∈Z≥0 that allow us to compute future activities more efficiently than that by simulating the entire spatio-temporal pattern. We obtain results regarding the numbers for Rule 90, Rule 150, a generalization of Rule 90 to a two-dimensional case, and a modified Ulam’s model, which is a two-dimensional crystal growth model. In addition, we examine the relationship between spatio-temporal patterns generated by cellular automata and a singular function. Representing the number of nonzero states by specific numeric values, we show that they are described by a difference form of the singular function.

No abstract