Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations

• Published in: Quantum information processing Vol. 11; no. 5; pp. 1219 - 1249
• Main Authors: Ahlbrecht, Andre, Cedzich, Christopher, Matjeschk, Robert, Scholz, Volkher B., Werner, Albert H., Werner, Reinhard F.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.10.2012
• Subjects: Data Structures and Information Theory; Mathematical Physics; Physics; Physics and Astronomy; Quantum Computing; Quantum Information Technology; Quantum Physics; Spintronics; Perturbation theory; Quantum walk; Spatio-temporal coin fluctuation; Asymptotic behavior
• ISSN: 1570-0755; 1573-1332
• DOI: 10.1007/s11128-012-0389-4

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.