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Quantum Walks: A Markovian Perspective

  • Diego de Falco
  • Dario Tamascelli
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4910)

Abstract

For a continuous-time quantum walk on a line the variance of the position observable grows quadratically in time, whereas, for its classical counterpart on the same graph, it exhibits a linear, diffusive, behaviour. A quantum walk, thus, propagates at a rate which is linear in time, as compared to the square root rate for a classical random walk. Indeed, it has been suggested that there are graphs that can be traversed by a quantum walker exponentially faster than by the classical random analogue. In this note we adopt the approach of exploring the conditions to impose on a Markov process in order to emulate its quantum counterpart: the central issue that emerges is the problem of taking into account, in the numerical generation of each sample path, the causative effect of the ensemble of trajectories to which it belongs. How to deal numerically with this problem is shown in a paradigmatic example.

Keywords

continuous-time quantum walks birth-and-death processes sample paths 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Diego de Falco
    • 1
    • 2
  • Dario Tamascelli
    • 1
    • 2
  1. 1.Dipartimento di Scienze dell’InformazioneUniversità degli Studi di MilanoMilanoItaly
  2. 2.CIMAINA, Centro Interdipartimentale Materiali e Interfacce NanostrutturatiUniversità degli Studi di Milano 

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