### Abstract

We study the asymptotic position distribution of general quantum walks on a lattice, including walks with a random coin, which is chosen from step to step by a general Markov chain. In the unitary (i.e., nonrandom) case, we allow any unitary operator which commutes with translations and couples only sites at a finite distance from each other. For example, a single step of the walk could be composed of any finite succession of different shift and coin operations in the usual sense, with any lattice dimension and coin dimension. We find ballistic scaling and establish a direct method for computing the asymptotic distribution of position divided by time, namely as the distribution of the discrete time analog of the group velocity. In the random case, we let a Markov chain (control process) pick in each step one of finitely many unitary walks, in the sense described above. In ballistic order, we find a nonrandom drift which depends only on the mean of the control process and not on the initial state. In diffusive scaling, the limiting distribution is asymptotically Gaussian, with a covariance matrix (diffusion matrix) depending on momentum. The diffusion matrix depends not only on the mean but also on the transition rates of the control process. In the nonrandom limit, i.e., when the coins chosen are all very close or the transition rates of the control process are small, leading to long intervals of ballistic evolution, the diffusion matrix diverges. Our method is based on spatial Fourier transforms, and the first and second order perturbation theory of the eigenvalue 1 of the transition operator for each value of the momentum.

Received 10 September 2010
Accepted 18 March 2011
Published online 19 April 2011

Acknowledgments:
We gratefully acknowledge the support of the DFG (Forschergruppe 635) and the EU projects CORNER, QUICS and CoQuit.

Article outline:

I. INTRODUCTION
II. THE SYSTEMS
A. General form of decoherent, translation invariant quantum walks
III. ASYMPTOTIC POSITION BY THE PERTURBATION METHOD
IV. UNITARY QUANTUM WALKS
A. Ballistic order
B. Higher orders
V. TIME DECOHERENT WALKS
A. Ballistic determinism for decoherent processes
B. Markovcontrolled walks in diffusive scaling
C. Higher orders without ballistic determinism
D. Memoryless decoherence
VI. EXAMPLES
A. The Hadamard walk with reflections
B. One dimensional quantum walks with equal position distribution
C. Nonunitary Kraus operators
D. Quantum walks with momentum shifts
E. Commuting Kraus operators

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