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arXiv:1206.3178 · Physical Review A

Quantum walks on trees with disorder: Decay, diffusion, and localization

Steven R. Jackson, T. J. Khoo, Frederick W. Strauch

Published Aug 27, 2012• 28 citations
arXivHTMLPDF

Abstract

Quantum walks have been shown to have impressive transport properties compared to classical random walks. However, imperfections in the quantum-walk algorithm can destroy any quantum-mechanical speedup due to Anderson localization. We numerically study the effect of static disorder on a quantum walk on the glued trees graph. For small disorder, we find that the dominant effect is a type of quantum decay, and not quantum localization. For intermediate disorder, there is a crossover to diffusive transport, while a localization transition is observed at large disorder, in agreement with Anderson localization on the Cayley tree.

DOI
10.1103/physreva.86.022335

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Quantum walks have been shown to have impressive transport properties compared to classical random walks. However, imperfections in the quantum-walk algorithm can destroy any quantum-mechanical speedup due to Anderson localization. We numerically study the effect of static disorder on a quantum walk on the glued trees graph. For small disorder, we find that the dominant effect is a type of quantum decay, and not quantum localization. For intermediate disorder, there is a crossover to diffusive transport, while a localization transition is observed at large disorder, in agreement with Anderson localization on the Cayley tree.
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.
Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength WWW, where a quantum to classical crossover at time ${t}_{c}\ensuremath{\propto}{W}^{\ensuremath{-}2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution, we relate our results to previously studied models of decoherence for quantum walks, and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.
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.
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.
This thesis is devoted to the study of quantum disordered systems, with applications ranging from Anderson localization to quantum computation. We focus on models defined on tree graphs, or on their finite size regularizations: random graphs. This allows for exact mean-field approaches. The first part deals with the Anderson localization problem. We obtain analytical results on the fast decay of the density of states near its edge (the \\textit{Lifshitz tail regime}), as well as a rigorous estimate on the localization threshold in the large connectivity limit. We also study the analogous model for bosons. In a second part, we study the quantum adiabatic algorithm and obtain results on its efficiency in solving a realistic problem, the coloring one, as well as analytical predictions for its performance in solving more generic problems. Finally, we also discuss how this quantum algorithm compares with a classical approximation for it.