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arXiv:1604.05905 · Physical Review A

Localized state in a two-dimensional quantum walk on a disordered lattice

Peng Xue, Rong Zhang, Zhihao Bian, Xiang Zhan, Hao Qin, Barry C. Sanders

Published Oct 14, 2015• 29 citations
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Abstract

We realize a pair of simultaneous ten-step one-dimensional quantum walks with two walkers sharing coins, which we prove is analogous to the ten-step two-dimensional quantum walk with a single walker holding a four-dimensional coin. Our experiment demonstrates a ten-step quantum walk over an $11\ifmmode\times\else\texttimes\fi{}11$ two-dimensional lattice with a line defect, thereby realizing a localized walker state.

DOI
10.1103/physreva.92.042316

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We realize a pair of simultaneous ten-step one-dimensional quantum walks with two walkers sharing coins, which we prove is analogous to the ten-step two-dimensional quantum walk with a single walker holding a four-dimensional coin. Our experiment demonstrates a ten-step quantum walk over an 11\ifmmode×\else\texttimes\fi1111\ifmmode\times\else\texttimes\fi{}1111\ifmmode×\else\texttimes\fi11 two-dimensional lattice with a line defect, thereby realizing a localized walker state.
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.
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 consider two independent quantum walks on separate lines augmented by partial or full swapping of coins after each step. For classical random walks, swapping or not swapping coins makes little difference to the random-walk characteristics, but we show that quantum walks with partial swapping of coins have complicated yet elegant interwalker correlations. Specifically we study the joint position distribution of the reduced two-walker state after tracing out the coins and analyze total, classical, and quantum correlations in terms of mutual information, quantum mutual information, and measurement-induced disturbance. Our analysis shows intriguing quantum features without classical analogs.
Quantum walk (QW) on a disordered lattice leads to a multitude of interesting phenomena, such as Anderson localization. While QW has been realized in various optical and atomic systems, its implementation with superconducting qubits still remains pending. The major challenge in simulating QW with superconducting qubits emerges from the fact that on-chip superconducting qubits cannot hop between two adjacent lattice sites. Here we overcome this barrier and develop a gate-based scheme to realize the discrete time QW by placing a pair of qubits on each site of a one-dimensional (1D) lattice and treating an excitation as a walker. It is also shown that various lattice disorders can be introduced and fully controlled by tuning the qubit parameters in our quantum walk circuit. We observe a distinct signature of transition from the ballistic regime to a localized QW with an increasing strength of disorder. Finally, an eight-qubit experiment is proposed where the signatures of such localized and delocalized regimes can be detected with existing superconducting technology. Our proposal opens up the possibility of exploring various quantum transport processes with promising superconducting qubits.