In this paper, a study on discrete-time coined quantum walks on the line is
presented. Clear mathematical foundations are still lacking for this quantum
walk model. As a step towards this objective, the following question is being
addressed: {\it Given a graph, what is the probability that a quantum walk
arrives at a given vertex after some number of steps?} This is a very natural
question, and for random walks it can be answered by several different
combinatorial arguments. For quantum walks this is a highly non-trivial task.
Furthermore, this was only achieved before for one specific coin operator
(Hadamard operator) for walks on the line. Even considering only walks on
lines, generalizing these computations to a general SU(2) coin operator is a
complex task. The main contribution is a closed-form formula for the amplitudes
of the state of the walk (which includes the question above) for a general
symmetric SU(2) operator for walks on the line. To this end, a coin operator
with parameters that alters the phase of the state of the walk is defined.
Then, closed-form solutions are computed by means of Fourier analysis and
asymptotic approximation methods. We also present some basic properties of the
walk which can be deducted using weak convergence theorems for quantum walks.
In particular, the support of the induced probability distribution of the walk
is calculated. Then, it is shown how changing the parameters in the coin
operator affects the resulting probability distribution.Comment: In v2 a small typo was fixed. The exponent in the definition of N_j
in Theorem 3 was changed from -1/2 to 1. 20 pages, 3 figures. Presented at
10th Asian Conference on Quantum Information Science (AQIS'10). Tokyo, Japan.
August 27-31, 201
