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Title of Journal: Quantum Inf Process

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Abbravation: Quantum Information Processing

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Springer US

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DOI

10.1002/jmri.22600

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ISSN

1573-1332

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The CGMV method for quantum walks

Authors: M J Cantero F A Grünbaum L Moral L Velázquez
Publish Date: 2012/07/15
Volume: 11, Issue: 5, Pages: 1149-1192
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Abstract

We review the main aspects of a recent approach to quantum walks the CGMV method This method proceeds by reducing the unitary evolution to canonical form given by the socalled CMV matrices which act as a link to the theory of orthogonal polynomials on the unit circle This connection allows one to obtain results for quantum walks which are hard to tackle with other methods Behind the above connections lies the discovery of a new quantum dynamical interpretation for well known mathematical tools in complex analysis Among the standard examples which will illustrate the CGMV method are the famous Hadamard and Grover models but we will go further showing that CGMV can deal even with nontranslation invariant quantum walks CGMV is not only a useful technique to study quantum walks but also a method to construct quantum walks à la carte Following this idea a few more examples illustrate the versatility of the method In particular a quantum walk based on a construction of a measure on the unit circle due to F Riesz will point out possible nonstandard behaviours in quantum walks

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