Quantum Walks on the Line with Phase Parameters

Marcos VILLAGRA  Masaki NAKANISHI  Shigeru YAMASHITA  Yasuhiko NAKASHIMA  

Publication
IEICE TRANSACTIONS on Information and Systems   Vol.E95-D   No.3   pp.722-730
Publication Date: 2012/03/01
Online ISSN: 1745-1361
DOI: 10.1587/transinf.E95.D.722
Print ISSN: 0916-8532
Type of Manuscript: Special Section PAPER (Special Section on Foundations of Computer Science – Mathematical Foundations and Applications of Computer Science and Algorithms –)
Category: 
Keyword: 
quantum computation,  random walks,  quantum walks,  asymptotic approximation,  

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Summary: 
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 toward this objective, the following question is being addressed: 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.