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 expression for the characteristic function of Xφ n . The expression clarifies the dependence of it on components of unitary matrix U and initial qubit state φ. As a consequence of the above results, we present a new ty...

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.

‎we investigate two constructions‎ - ‎the replacement and the zig-zag‎ ‎product of graphs‎ - ‎describing several fascinating connections‎ ‎with combinatorics‎, ‎via the notion of expander graph‎, ‎group‎ ‎theory‎, ‎via the notion of semidirect product and cayley graph‎, ‎and‎ ‎with markov chains‎, ‎via the lamplighter random walk‎. ‎many examples‎ ‎are provided‎.

A discrete-time quantum walk on a graph is the repeated application of a unitary evolution operator to a Hilbert space corresponding to the graph. Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. We derive an expression for hitting time using superoperators, and numerically evaluate it for the walk on the hypercube for various ...