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Chinese Physics B

A quantum walk in phase space with resonator-assisted double quantum dots

Zhihao Bian, Hao Qin, Xiang Zhan, Jian Li, Peng Xue

Published Feb 1, 2016• 3 citations
Publisher

Abstract

We implement a quantum walk in phase space with a new mechanism based on the superconducting resonator-assisted double quantum dots. By analyzing the hybrid system, we obtain the necessary factors implementing a quantum walk in phase space: the walker, coin, coin flipping and conditional phase shift. The coin flipping is implemented by adding a driving field to the resonator. The interaction between the quantum dots and resonator is used to implement conditional phase shift. Furthermore, we show that with different driving fields the quantum walk in phase space exhibits a ballistic behavior over 25 steps and numerically analyze the factors influencing the spreading of the walker in phase space.

DOI
10.1088/1674-1056/25/2/020307

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We implement a quantum walk in phase space with a new mechanism based on the superconducting resonator-assisted double quantum dots. By analyzing the hybrid system, we obtain the necessary factors implementing a quantum walk in phase space: the walker, coin, coin flipping and conditional phase shift. The coin flipping is implemented by adding a driving field to the resonator. The interaction between the quantum dots and resonator is used to implement conditional phase shift. Furthermore, we show that with different driving fields the quantum walk in phase space exhibits a ballistic behavior over 25 steps and numerically analyze the factors influencing the spreading of the walker in phase space.
We implement a quantum walk in phase space with a new mechanism based on the superconducting resonator-assisted double quantum dots. By analyzing the hybrid system, we obtain the necessary factors of realization of a quantum walk in phase space: the walker, coin, coin flipping and conditional phase shift. In order to implement the coin flipping operator, we add a driving field to the resonator. The interaction between the quantum dots and resonator field is used to implement conditional phase shift. Furthermore, we show with different driving fields the quantum walk in phase space exhibits a ballistic behavior over 25 steps and numerically analyze the factors which influence the spreading of the walker in phase space.
We show how a quantum walk can be implemented in a quantum quincunx created via superconducting circuit quantum electrodynamics QED and how interpolation from a quantum to a random walk is implemented by controllable decoherence using a two-resonator system. Direct control over the coin qubit is difficult to achieve in either cavity or circuit QED, but we show that a Hadamard coin flip can be effected via direct driving of the cavity, with the result that the walker jumps between circles in phase space, but still exhibits quantum walk behavior over 15 steps.
We show how a quantum walk can be implemented in a quantum quincunx created via superconducting circuit quantum electrodynamics (QED) and how interpolation from a quantum to a random walk is implemented by controllable decoherence using a two-resonator system. Direct control over the coin qubit is difficult to achieve in either cavity or circuit QED, but we show that a Hadamard coin flip can be effected via direct driving of the cavity, with the result that the walker jumps between circles in phase space, but still exhibits quantum walk behavior over 15 steps.
Quantum walks have been investigated as they have remarkably different features in contrast to classical random walks. We present a quantum walk in a one-dimensional architecture, consisting of two coins and a walker whose evolution is in both position and phase spaces alternately controlled by the two coins respectively. By analyzing the dynamics evolution of the walker in both the position and phase spaces, we observe an influence on the quantum walk in one space from that in the other space, which behaves like decoherence. We propose an implementation of the two-coin quantum walk in both position and phase spaces via cavity quantum electrodynamics (QED).
We propose a scheme to implement the quantum walk for SU(1,1) in the phase space, which generalizes those associated with the Heisenberg-Weyl group. The movement of the walker described by the SU(1,1) coherent states can be visualized on the hyperboloid or the Poincar\'e disk. In both one-mode and two-mode realizations, we introduce the corresponding coin-flip and conditional-shift operators for the SU(1,1) group, whose relations with those for the Heisenberg-Weyl group are analyzed. The probability distribution, the standard deviation, and the von Neumann entropy are employed to describe the walking process. The nonorthogonality of the SU(1,1) coherent states precludes the quantum walk for SU(1,1) from the idealized one. However, the overlap between different SU(1,1) coherent states can be reduced by increasing the Bargmann index kkk, which indicates that the two-mode realization provides more possibilities to simulate the idealized quantum walk.
In this paper discrete quantum walks with different coins used for odd and even time steps are studied. These coins are called hybrid. The calculation results are compared with the most frequently used coin, the Hadamard transform. Furthermore, quantum walks on the line which involve two or more quantum walkers with hybrid coins are studied. Quantum walks with entangled walkers and hybrid coins are also studied. The results of these calculations show that the proposed types of quantum walks can be used for quantum search, because the walker can be directed towards preferred directions and can also be confined in certain segments of the line.