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

Abstract

In this expository work, we discuss spatially inhomogeneous quantum walks in one dimension and describe a genre of mathematical methods that enables one to translate information about the time-independent eigenvalue equation for the unitary generator into dynamical estimates for the corresponding quantum walk. To illustrate the general methods, we show how to apply them to a 1D coined quantum walk whose coins are distributed according to an element of the Thue-Morse subshift.

Quantum walks with sequential aperiodic jumps

“…Specifically, we consider binary aperiodic sequences as the generator of the jumps performed by quantum particles on the chain. In spite of the fact that aperiodic sequences have been used as a source of disorder in the coin operator in both theoretical [28][29][30][31][32][33][34][35] and experimental optical setups [36][37][38], this kind of protocol has not been embedded into the step operator. In this work, we fill that gap by letting the steps of the quantum walker follow one out of three paradigmatic aperiodic sequences, namely Fibonacci, Thue-Morse or Rudin-Shapiro, as previously mentioned.…”