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.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.

arXiv:2406.18768Jun 26, 20240 citations

Quantum walks on a random environment

Y. Yin, D. E. Katsanos, S. N. Evangelou

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 $W$ , 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.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 $W$, 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.

arXiv:0708.1137Physical Review AFeb 5, 2008119 citations

Quantum Random Walks in One Dimension

Norio Konno

arXiv:quant-ph/0206053Quantum Information ProcessingOct 1, 2002301 citations

One-dimensional discrete-time quantum walks on random environments

Norio Konno

arXiv:0904.0392Quantum Information ProcessingMay 26, 200963 citations

Localization of two dimensional quantum walks defined by generalized Grover coins

Amrita Mandal, Rohit Sarma Sarkar, Bibhas Adhikari

Abstract Localization phenomena of quantum walks makes the propagation dynamics of a walker strikingly different from that corresponding to classical random walks. In this paper, we study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices with coin operators as one-parameter orthogonal matrices that are also permutative, a combinatorial structure of the Grover matrix. We show that the proposed walks localize at its initial position for canonical initial coin states when the coin belongs to classes which contain the Grover matrix that we consider in this paper, however, the localization phenomena depends on the coin parameter when the class of parametric coins does not contain the Grover matrix.Abstract Localization phenomena of quantum walks makes the propagation dynamics of a walker strikingly different from that corresponding to classical random walks. In this paper, we study the localization phenomena of four-state discrete-time quantum walks on two-dimensional lattices with coin operators as one-parameter orthogonal matrices that are also permutative, a combinatorial structure of the Grover matrix. We show that the proposed walks localize at its initial position for canonical initial coin states when the coin belongs to classes which contain the Grover matrix that we consider in this paper, however, the localization phenomena depends on the coin parameter when the class of parametric coins does not contain the Grover matrix.

arXiv:2103.00515Journal of Physics A Mathematical and TheoreticalJan 13, 20233 citations

W

, 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.