Quantum Information Processing

, Volume 14, Issue 3, pp 839–866 | Cite as

Quantum walk with a general coin: exact solution and asymptotic properties

  • Miquel Montero


In this paper, we present closed-form expressions for the wave function that governs the evolution of the discrete-time quantum walk on the line when the coin operator is arbitrary. The formulas were derived assuming that the walker can either remain put in the place or proceed in a fixed direction but never move backward, although they can be easily modified to describe the case in which the particle can travel in both directions. We use these expressions to explore properties of magnitudes associated to the process, as the probability mass function or the probability current, even though we also consider the asymptotic behavior of the exact solution. Within this approximation, we will estimate upper and lower bounds, examine the origins of an emerging approximate symmetry, and deduce the general form of the stationary probability density of the relative location of the walker.


Quantum walks Closed-form solutions Limiting theorems 



The author acknowledges partial support from the former Spanish Ministerio de Ciencia e Innovación under Contract No. FIS2009-09689, from the Spanish Ministerio de Economía y Competitividad under Contract No. FIS2013-47532-C3-2-P, and from Generalitat de Catalunya, Contract No. 2014SGR608.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Departament de Física FonamentalUniversitat de Barcelona (UB)BarcelonaSpain

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