Some perspectives on the eigenvalue problem
SIAM Review
Journal of Approximation Theory
One-dimensional quantum walks with absorbing boundaries
Journal of Computer and System Sciences
CMV matrices: Five years after
Journal of Computational and Applied Mathematics
Localization of an inhomogeneous discrete-time quantum walk on the line
Quantum Information Processing
Localization of discrete-time quantum walks on a half line via the CGMV method
Quantum Information & Computation
Limit measures of inhomogeneous discrete-time quantum walks in one dimension
Quantum Information Processing
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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 so-called 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 non-translation 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 non-standard behaviours in quantum walks.