Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices

Abstract

We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero-Grünbaum-Moral-Velázquez, this enables us to study spreading rates for quantum walks in one spatial dimension. We prove several general results which establish quantitative upper and lower bounds on the spreading of a quantum walk in terms of estimates on a pair of associated matrix cocycles. To demonstrate the power and utility of these methods, we apply them to several concrete cases of interest. In the case where the coins are distributed according to an element of the Fibonacci subshift, we are able to rather completely describe the dynamics in a particular asymptotic regime. As a pleasant consequence, this supplies the first concrete example of a quantum walk with anomalous transport, to the best of our knowledge. We also prove ballistic transport for a quantum walk whose coins are periodically distributed.

Singular continuous Cantor spectrum for magnetic quantum walks

“…From a mathematical point of view, quantum walks on a one-dimensional lattice can also be seen as a particular class of CMV matrices giving a link between quantum dynamical systems and orthogonal polynomials on the unit circle, a connection which has proved to be fruitful in both directions [12,15,16,19,24,39]. One can use spectral methods to deduce bounds on spreading [31,32]. On the other hand, estimates of a dynamical nature can be turned into estimates on spectral quantities, for example, quantitative regularity of the spectral measures [30].…”

Resolvent Methods for Quantum Walks with an Application to a Thue–Morse Quantum Walk

“…Given an initial state ψ ∈ H normalized by ψ = 1, we are interested in the time evolution of the vector ψ, that is, we want to study the evolution of ψ(ℓ) = U ℓ ψ as ℓ ∈ + grows. The most favorable situations are those in which the walk is translation-invariant, i.e., there exists q such that C n+q = C n for all n. In this case, one can explicitly solve the walk via a Floquet-Bloch transform, and one deduces strong ballistic motion and an explicit expression for the asymptotic group velocity; see [1,Theorem 4] and [8,Corollary 9.3]. In fact, ballistic motion with an explicit group velocity also holds for quantum walks that are rapidly and uniformly approximated by translation-invariant quantum walks [12,Remark 1.3.(4)].…”

“…Naturally, one wants to go beyond exactly solvable models. In this short note, we will give a brief introduction to the general methods of [8]. Specifically, we will describe a general method that enables one to study the time-dependent spreading characteristics of a quantum walk with spatially inhomogeneous coins by establishing suitable estimates on the time-independent eigenvalue equation.…”

“…The Fibonacci substitution, defined by S F (0) = 01 and S F (1) = 0 is another popular quasicrystal model. Quantum walks with Fibonacci coins were studied numerically in [20] and mathematically in [8][9][10]. Now, pick phases θ, φ ∈ Ê.…”

“…We remark that Theorem 1.2 is not new. Indeed, it is essentially [8,Theorem 8.2]. However, [8] produces this result as a corollary of a fairly general piece of machinery, [8, Theorem 2.1].…”