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Bulletin of the American Physical Society

Quantum walk on a circle in phase space via superconducting circuit quantum electrodynamics

Barry C. Sanders, Peng Xue, Alexandre Blais, Kevin Lalumière

Published Mar 12, 2008• 1 citation
OpenAlex

Abstract

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.

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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).
The quincunx, or Galton board, has a long history as a tool for demonstrating and investigating random walk processes, but a quantum quincunx (QQ) for demonstrating a coined quantum walk (QW) is yet to be realized experimentally. We propose a variant of the QQ in cavity quantum electrodynamics, designed to eliminate the onerous requirement of directly flipping the coin. Instead, we propose driving the cavity in such a way that cavity field displacements are minimized and the coin is effectively flipped via this indirect process. An effect of this indirect flipping is that the walker's location is no longer confined to a single circle in the planar phase space, but we show that the phase distribution nonetheless shows quadratic enhancement of phase diffusion for the quantum versus classical walk despite this small complication. Thus our scheme leads to coined QW behaviour in cavity quantum electrodynamics without the need to flip the coin directly.
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 show how multi-walker quantum walks can be implemented in a quantum quincunx created via cavity quantum electrodynamics. The implementation of a quantum walk with a multi-walker opens up the interesting possibility to introduce entanglement and more advanced walks. With different coin tosses and initial states the multi-walker quantum walk shows different probability distributions which deviate strongly from the classical random walks with quadratic enhanced spreadings and localization effects. By introducing decoherence, the transition from quantum walks to the classical versions is observed. We introduce the average fidelity decay as a signature to investigate the decoherence-induced irreversibility of quantum walks.
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.