Asymptotic evolution of quantum walks with random coin

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., non-random) 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 non-random 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 non-random 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.Comment: 34 pages, 20 figure

Index Theory of One Dimensional Quantum Walks and Cellular Automata

“…The limit is in the strong resolvent sense. Hence the probability distribution for the selfadjoint operator G in a state ρ is equal to the asymptotic position distribution starting from ρ in "ballistic" scaling [1]. In particular, when the internal state is unpolarised, i.e., when the initial state is of the form ρ = σ ⊗ 1I/d, we find [12] X(t) = X(0) + t d ind (U ).…”

“…Formally, that is a consequence of demanding lim x→±∞ x ± = ±∞. It is easy to see that such a 1 We are indebted to a referee who drew our attention to the inflationary use of the word "local" in our manuscript, where, among other things, quantum walks were called "local" unitaries. We changed this to "causal" for the crucial finite propagation property of walks and cellular automata.…”

“…In this case the family of projections P n associated with the approximating algebras is also increasing, and converges strongly to P . When B 1 ⊂ A and B 2 ⊂ A are commuting matrix subalgebras, the corresponding matrix units, say e (1) ij and e (2) ab , can be taken together as a set of matrix units for B 1 ⊗ B 2 , and we get…”

“…(1) We are considering a doubled system in which the one-cell Hilbert space at x is H x ⊕H x . Let S x denote the unitary operator on the doubled system which swaps these two summands, and acts as the identity on the one-cell spaces of all other sites.…”

Convergence of quantum random walks with decoherence

“…Their scaling factor is 1 t . Recently in [2], the scaling limit of a quantum random walk with a Markov controlled coin process converges either with scaling factor 1 t or 1 √ t , depending on eigenvalue conditions of the walk operator.…”