Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk

Abstract

Quantum walks are promising for information processing tasks because in regular graphs they spread quadratically more rapidly than random walks. Static disorder, however, can turn the tables: unlike random walks, quantum walks can suffer Anderson localization, with their wave function staying within a finite region even in the infinite time limit, with a probability exponentially close to 1. It is thus important to understand when a quantum walk will be Anderson localized and when we can expect it to spread to infinity even in the presence of disorder. In this work we analyze the response of a one-dimensional quantum walk-the split-step walk-to different forms of static disorder. We find that introducing static, symmetry-preserving disorder in the parameters of the walk leads to Anderson localization. In the completely disordered limit, however, a delocalization transition occurs, and the walk spreads subdiffusively to infinity. Using an efficient numerical algorithm, we calculate the bulk topological invariants of the disordered walk and find that the disorder-induced Anderson localization and delocalization transitions are governed by the topological phases of the quantum walk.

Unveiling hidden topological phases of a one-dimensional Hadamard quantum walk

“…For disordered quantum walks, e.g., where θ (x) is a random function of position, there are alternative formulations of the topological invariants, based on the scattering matrices [28,35]. These can be applied to disordered Hadamard walks using the mapping we presented in this paper.…”

Controlling quantum random walk with a step-dependent coin

“…It was shown, moreover, that QWs can be utilized as a universal computational primitives [10,11] in order to simulate other quantum processes. In addition, various experimental realization of QWs were demonstrated by means of ultracold atoms [12], photons [13,14] or ions [15,16] could simulate a nontrivial one-dimensional topological phase [17][18][19].…”

Edge states in a two-dimensional quantum walk with disorder

“…as "nonlinear disorder". However, the values of φ at different sites are correlated by (14) with the walker's state, making the nonlinear disorder essentially different to the rotation or magnetic disorder types. One important consequence of the introduction of the coin operator (13) with respect to the random rotation one of the previous section, is that the form R J (θ, φ) breaks the so-called particle-hole symmetry.…”

“…Experiments [11,12,13] show that in the presence of disorder, the quantum walk losses its coherent interference pattern and, depending on the nature of the noise, can localize or transit to a classical diffusion regime. Noise also affects the topological phases and the localization-delocalization properties of one dimensional [14] and two dimensional quantum walks [15]. In the absence of disorder, a one dimensional quantum walk can still localize at the interface between two distinct topological regions [16].…”