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arXiv:2406.18768

Dynamical localization in 2D topological quantum random walks

D. O. Oriekhov, Guliuxin Jin, Eliška Greplová

Published Jun 26, 2024• 0 citations
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Abstract

We study the dynamical localization of discrete time evolution of topological split-step quantum random walk (QRW) on a single-site defect starting from a uniform distribution. Using analytical and numerical calculations, we determine the high localization probability regions in the parameter space of the quantum walker. These regions contain two or more pairs of trapped states, forming near a lattice defect. By investigating the spectral properties of the discrete-time evolution operators, we show that these trapped states have large overlap with the initial uniformly distributed state, thus offering a simple interpretation of the localization effect. As this localization scheme could be interpreted as a variation of spatial quantum search algorithm, we compare the localization probability and time with other types of two-dimensional quantum walks that do not have topological phases and realize localization time scaling similar to Grover's algorithm. Finally we show that mechanism of localization we identified is robust against the influence of disorder.

DOI
10.48550/arxiv.2406.18768

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We study the dynamical localization of discrete time evolution of topological split-step quantum random walk (QRW) on a single-site defect starting from a uniform distribution. Using analytical and numerical calculations, we determine the high localization probability regions in the parameter space of the quantum walker. These regions contain two or more pairs of trapped states, forming near a lattice defect. By investigating the spectral properties of the discrete-time evolution operators, we show that these trapped states have large overlap with the initial uniformly distributed state, thus offering a simple interpretation of the localization effect. As this localization scheme could be interpreted as a variation of spatial quantum search algorithm, we compare the localization probability and time with other types of two-dimensional quantum walks that do not have topological phases and realize localization time scaling similar to Grover's algorithm. Finally we show that mechanism of localization we identified is robust against the influence of disorder.
The Grover walk, which is related to Grover's search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.
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.
We systematically study the localization effect in discrete-time quantum walks on a honeycomb network and establish the mathematical framework. We focus on the Grover walk first and rigorously derive the limit form of the walker's state, showing it has a certain probability to be localized at the starting position. The relationship between localization and the initial coin state is concisely represented by a linear map. We also define and calculate the average probability of localization by generating random initial states. Further, coin operators varying with positions are considered and the sufficient condition for localization is discussed. We also similarly analyze another four-state Grover walk. Theoretical predictions are all in accord with numerical simulation results. Finally, our results are compared with previous works to demonstrate the unusual trapping effect of quantum walks on a honeycomb network, as well as the advantages of our method.
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 study the localization of probability distribution in a discrete quantum random walk on an infinite chain. With a phase defect introduced in any position of the quantum random walk (QRW), we have found that the localization of the probability distribution in the QRW emerges. Different localized behaviors of the probability distribution in the QRW are presented when the defect occupies different positions. Given that the coefficients of the localized stationary eigenstates relies on the coin operator, we reveal that when the defect occupies different positions, the amplitude of localized probability distribution in the QRW exhibits a non-trivial dependence on the coin operator.