Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations

Abstract

Quantum walks subject to decoherence generically suffer the loss of their genuine quantum feature, a quadratically faster spreading compared to classical random walks. This intuitive statement has been verified analytically for certain models and is also supported by numerical studies of a variety of examples. In this paper we analyze the long-time behavior of a particular class of decoherent quantum walks, which, to the best of our knowledge, was only studied at the level of numerical simulations before. We consider a local coin operation which is randomly and independently chosen for each time step and each lattice site and prove that, under rather mild conditions, this leads to classical behavior: With the same scaling as needed for a classical diffusion the position distribution converges to a Gaussian, which is independent of the initial state. Our method is based on non-degenerate perturbation theory and yields an explicit expression for the covariance matrix of the asymptotic Gaussian in terms of the randomness parameters.

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Correspondence to Andre Ahlbrecht.

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Ahlbrecht, A., Cedzich, C., Matjeschk, R. et al. Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations. Quantum Inf Process 11, 1219–1249 (2012). https://doi.org/10.1007/s11128-012-0389-4

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Keywords

  • Quantum walk
  • Spatio-temporal coin fluctuation
  • Asymptotic behavior
  • Perturbation theory