Dynamics of unitary operators

Abstract

Abstract. We consider the iteration of a unitary operator on a separable Hilbert space and study the spreading rates of the associated discrete-time dynamical system relative to a given orthonormal basis. We prove lower bounds for the transport exponents, which measure the time-averaged spreading on a power-law scale, in terms of dimensional properties of the spectral measure associated with the unitary operator and the initial state. These results are the unitary analog of results established in recent years for the dynamics of the Schrödinger equation, which is a continuum-time dynamical system associated with a self-adjoint operator. We discuss how these general results may be studied by means of subordinacy theory in cases where the unitary operator is given by a CMV matrix. An example of particular interest in which this scenario arises is given by a time-homogeneous quantum walk on the integers. For the particular case of the time-homogeneous Fibonacci quantum walk, we illustrate how these components work together and produce explicit lower bounds for the transport exponents associated with this model.Mathematics Subject Classification (2010). 42C05; 82B41.

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].…”

Purely singular continuous spectrum for limit-periodic CMV operators with applications to quantum walks

“…Extended CMV matrices comprise a natural playground for spectral theoretic techniques, as they are canonical unitary analogs of Schrödinger operators and Jacobi matrices. Moreover, CMV operators are interesting in their own right, since they naturally arise not just from 1D quantum walks [5,7,11,12], but also in the classical ferromagnetic Ising model [9,13], and OPUC (orthogonal polynomials on the unit circle) [35,36].…”

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

“…The astute observer will notice that both terms that comprise a(n, ℓ) correspond to the modulus squared of a Fourier coefficient of a spectral measure of U . Since regularity estimates on measures enable one to prove decay estimates on their Fourier coefficients, one can prove quantitative estimates on wavepacket propagation by investigating regularity and singularity of spectral measures of U , an approach that is proposed and executed in [9]. However, there are two nontrivial drawbacks to this method.…”

“…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.…”