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arXiv:1505.04513 · Physical Review A

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

Tibor Rakovszky, János K. Asbóth

Published Nov 10, 2015• 63 citations
arXivHTMLPDF

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.

DOI
10.1103/physreva.92.052311

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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.
We investigate numerically and theoretically the effect of spatial disorder on two-dimensional split-step discrete-time quantum walks with two internal ``coin'' states. Spatial disorder can lead to Anderson localization, inhibiting the spread of quantum walks, putting them at a disadvantage against their diffusively spreading classical counterparts. We find that spatial disorder of the most general type, i.e., position-dependent Haar random coin operators, does not lead to Anderson localization but to a diffusive spread instead. This is a delocalization, which happens because disorder places the quantum walk to a critical point between different anomalous Floquet-Anderson insulating topological phases. We base this explanation on the relationship of this general quantum walk to a simpler case more studied in the literature and for which disorder-induced delocalization of a topological origin has been observed. We review topological delocalization for the simpler quantum walk, using time evolution of the wave functions and level spacing statistics. We apply scattering theory to two-dimensional quantum walks and thus calculate the topological invariants of disordered quantum walks, substantiating the topological interpretation of the delocalization and finding signatures of the delocalization in the finite-size scaling of transmission. We show criticality of the Haar random quantum walk by calculating the critical exponent $\ensuremath{\eta}$ in three different ways and find $\ensuremath{\eta}\ensuremath{\approx}0.52$ as in the integer quantum Hall effect. Our results showcase how theoretical ideas and numerical tools from solid-state physics can help us understand spatially random quantum walks.
We investigate time-independent disorder on several two-dimensional discrete-time quantum walks. We find numerically that, contrary to claims in the literature, random onsite phase disorder, spin-dependent or otherwise, cannot localize the Hadamard quantum walk; rather, it induces diffusive spreading of the walker. In contrast, split-step quantum walks are generically localized by phase disorder. We explain this difference by showing that the Hadamard walk is a special case of the split-step quantum walk, with parameters tuned to a critical point at a topological phase transition. We show that the topological phase transition can also be reached by introducing strong disorder in the rotation angles. We determine the critical exponent for the divergence of the localization length at the topological phase transition, and find $\ensuremath{\nu}=2.6$, in both cases. This places the two-dimensional split-step quantum walk in the universality class of the quantum Hall effect.
The interplay between disorder and topology triggers interesting effects for the propagation of quantum particles in low-dimensional network structures. For instance, the occurrence of Anderson localization, a commonly observed effect in a quantum network with static disorder, can be suppressed by introducing topology-a phenomenon dubbed as topological Anderson localization transition [1] . In one-dimensional systems, for example, disorder induces localization on a short length scale while topology counters that through long-range correlations, yielding divergent correlation length and extremely slow spreading at quantum criticality. However, its experimental demonstration still remains an outstanding challenge as that would require, in contrast to Anderson localization transition, addi¬tional control over internal degrees of freedom, which is hard to implement, for instance, in cold-atom systems [2] . Quantum walks (QWs), a quantum analog of random walks, comprising quantum coin toss and conditioned step operations are already established for studying quantum transport phenomena, such as, disorder and topology [5] . Based on our theoretical proposal [3] , here, we provide an experimental route using time-multiplexed quantum walk [4] for evidencing the targeted phenomenon.
We investigate one-dimensional (1D) discrete-time quantum walks (QWs) with spatially or temporally random defects as a consequence of interactions with random environments. We focus on the QWs with chiral symmetry in a topological phase, and reveal that chiral symmetry together with the bipartite nature of the QWs brings about intriguing behaviors such as coexistence of topologically protected edge states at zero energy and Anderson transitions in the 1D chiral class at nonzero energy in their dynamics. In contrast to results of previous studies, therefore, the spatially disordered QWs can avoid complete localization due to the Anderson transition. It is further confirmed that the edge states are robust to spatial disorder but not to temporal disorder.
We analyze the localization of quantum walks on a one-dimensional finite graph using vector-distance. We first vectorize the probability distribution of a quantum walker in each node. Then we compute out the probability distribution vectors of quantum walks in infinite and finite graphs in the presence of static disorder respectively, and get the distance between these two vectors. We find that when the steps taken are small and the boundary condition is tight, the localization between the infinite and finite cases is greatly different. However, the difference is negligible when the steps taken are large or the boundary condition is loose. It means quantum walks on a one-dimensional finite graph may also suffer from localization in the presence of static disorder. Our approach and results can be generalized to analyze the localization of quantum walks in higher-dimensional cases.
Discrete-time quantum walks, quantum generalizations of classical random walks, provide a framework for quantum information processing, quantum algorithms, and quantum simulation of condensed-matter systems. The key property of quantum walks, which lies at the heart of their quantum information applications, is the possibility for a parametric quantum speedup in propagation compared to classical random walks. In this work we study propagation of quantum walks on percolation-generated two-dimensional random lattices. In large-scale simulations of topological and trivial split-step walks, we identify distinct prediffusive and diffusive behaviors at different timescales. Importantly, we show that even arbitrarily weak concentrations of randomly removed lattice sites give rise to a complete breakdown of the superdiffusive quantum speedup, reducing the motion to ordinary diffusion. By increasing the randomness, quantum walks eventually stop spreading due to Anderson localization. Near the localization threshold, we find that the quantum walks become subdiffusive. The fragility of quantum speedup implies dramatic limitations for quantum information applications of quantum walks on random geometries and graphs.