Localization for magnetic quantum walks

Abstract

Quantum walk was first proposed by Aharonov, Davidovich and Zagury [1]. It can be viewed as a quantum mechanical analogue of the classical random walk. Compared to the diffusive transport in the classical random walk, quantum walk leads to a ballistic spreading of the particle’s wave function. This fast spreading property has played a pivotal role in the development of quantum algorithms [2,36], including search algorithms, element distinctness and matrix product verification. Besides the applications in quantum information science, quantum walks are also very accessible and interesting to both experimental and theoretical studies for many complex quantum phenomena in physics. We refer interested readers to [34, 40] for a comprehensive review on quantum walks. In recent years, there have been an growing interest on quantum walk in mathematical community, see e.g. [9, 10, 13, 15, 21, 32, 33]. In particular, in [10], the authors discovered a beautiful connection between quantum walks and the CMV matrices, which is a class of unitary operators which arise in the theory of orthogonal polynomials on the unit circle (OPUC) [38, 39]. Recently, quantum walk model in electric fields also attracts a lot of attention in physics, see e.g. [12, 18, 43], and Anderson localization for that model was proved in [13] for a.e. electric field. In an upcoming work [45], we prove localization for electric quantum walks for all Diophantine fields. Now I will introduce the model that we study, which was given recently in [11] as a generalization of the model studied in [16]. This model arises from the two dimensional quantum walk on Z subject to a homogeneous magnetic field, see Section 3 of [11]. Let W : l(Z) ⊗ C → l(Z) ⊗ C =: H be a quantum walk defined by a quasi-periodic sequence of coins. We denote the standard basis of H