Generation and stabilization of a three-qubit entangled $W$ state in circuit QED via quantum feedback control

Shang-Yu Huang, Hsi-Sheng Goan, Xin-Qi Li, and G. J. Milburn

Phys. Rev. A 88, 062311 – Published 10 December 2013

Abstract

Circuit cavity quantum electrodynamics (QED) is proving to be a powerful platform to implement quantum feedback control schemes due to the ability to control superconducting qubits and microwaves in a circuit. Here, we present a simple and promising quantum feedback control scheme for deterministic generation and stabilization of a three-qubit $W$ state in the superconducting circuit QED system. The control scheme is based on continuous joint Zeno measurements of multiple qubits in a dispersive regime, which enables us not only to infer the state of the qubits for further information processing but also to create and stabilize the target $W$ state through adaptive quantum feedback control. We simulate the dynamics of the proposed quantum feedback control scheme using the quantum trajectory approach with an effective stochastic maser equation obtained by a polaron-type transformation method and demonstrate that in the presence of moderate environmental decoherence, the average state fidelity higher than $0.9$ can be achieved and maintained for a considerably long time (much longer than the single-qubit decoherence time). This control scheme is also shown to be robust against measurement inefficiency and individual qubit decay rate differences. Finally, the comparison of the polaron-type transformation method to the commonly used adiabatic elimination method to eliminate the cavity mode is presented.

Received 3 September 2013

DOI:https://doi.org/10.1103/PhysRevA.88.062311

Authors & Affiliations

Shang-Yu Huang^1,2, Hsi-Sheng Goan^1,2,*, Xin-Qi Li³, and G. J. Milburn⁴

¹Department of Physics and Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan
²Center for Quantum Science and Engineering, and National Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan
³Department of Physics, Beijing Normal University, Beijing 100875, China
⁴Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia QLD 4072, Australia

^*goan@phys.ntu.edu.tw

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Issue

Vol. 88, Iss. 6 — December 2013

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Images

Figure 1
Schematic illustration of three qubits in circuit QED quantum feedback control setup (MW: microwave drive).Reuse & Permissions
Figure 2
Measured homodyne currents ${〈 c_{0} + c_{0}^{†} 〉}_{c}$ (1000 realizations that yield roughly the same steady outcome values are grouped and averaged) for an initial cavity state in a vacuum state and an initial qubits’ state in the separable state $| ψ_{i} 〉$ of Eq. (32) with the qubits’ decay rates $γ_{j} = 0$ . The four measurement outcomes in solid lines are for the case of polaron-type transformation and they correspond to the qubits collapsing respectively onto states $| 111 〉$ , $| W^{+} 〉$ , $| W^{-} 〉$ , and $| 000 〉$ from the top to the bottom, while the measurement outcomes in dashed lines correspond to the case of adiabatic elimination. The parameters used are $ε = 2 κ$ , $χ = - 0.11 κ$ , $g = 10 κ$ , $η = 1$ , and $γ_{j} = 0$ .Reuse & Permissions
Figure 3
Time evolutions of the average fidelity $F$ of $| W^{-} 〉$ state (over 1000 realizations or trajectories) as a function of time for different initial states of $| 000 〉$ , $| W^{-} 〉$ , $| W^{+} 〉$ , $| 000 〉$ , and $| ψ_{i} 〉$ . The parameters used are $f = 2 κ$ , $ε = 2 κ$ , $χ = - 0.11 κ$ , $g = 10 κ$ , $η = 1$ , and $γ_{j} = γ = 4 \times 10^{- 3} κ$ .Reuse & Permissions
Figure 4
(a) Time evolutions of the average fidelity $F$ of the $| W^{-} 〉$ state (over 1000 trajectories) generated from the ground state $| 000 〉$ for different qubits’ decay rates of $γ_{j} = γ$ being $4 \times 10^{- 3} κ$ , $1 \times 10^{- 2} κ$ , $2 \times 10^{- 2} κ$ , and $4 \times 10^{- 2} κ$ (solid lines from top to bottom). The brown dashed line is the ensemble average fidelity without the application of feedback control for an initial qubits’ state being in $| W^{-} 〉$ . (b) A typical single trajectory of the estimated current ${〈 c_{0} + c_{0}^{†} 〉}_{c}$ for the qubits’ decay rates of $γ_{j} = γ = 4 \times 10^{- 3} κ$ . The qubits’ system is stabilized in the $| W^{-} 〉$ state with ${〈 c_{0} + c_{0}^{†} 〉}_{c} = - 1.68$ during the continuous feedback control process (in solid line), while it makes a sudden jump to the ground state $| 000 〉$ with ${〈 c_{0} + c_{0}^{†} 〉}_{c} = - 3.68$ without the application of feedback control (dashed line). Other parameters used are the same as those in Fig. 3.Reuse & Permissions
Figure 5
Dependence of the average fidelity $F$ of the $| W^{-} 〉$ state at $κ t = 350$ on (a) the dispersive coupling strength $χ$ , (b) the driving amplitude $ε$ , (c) the feedback strength $f$ , and (d) the measurement efficiency $η$ . The decay rate of the qubits is fixed at $γ_{j} = γ = 4 \times 10^{- 3} κ$ and the initial qubit state is the ground state $| 000 〉$ . The red dashed curve in (a) is the separation between the measurement outcomes of the ground state and the $| W^{-} 〉$ state (or the single-qubit excited state) with its vertical axis label shown on the right.Reuse & Permissions
Figure 6
Time evolutions of the average fidelity $F$ of the $| W^{-} 〉$ state with ( $f = 2 κ$ ) and without ( $f = 0$ ) quantum feedback control for three sets of different individual qubits’ decay rates. The initial state for the case with feedback control ( $f = 2 κ$ ) is the ground state $| 000 〉$ while it is the $| W^{-} 〉$ state for the case without feedback control ( $f = 0$ ). Other parameters used are $ε = 2 κ$ , $χ = - 0.11 κ$ , $g = 10 κ$ , $η = 1$ , and $γ = 10^{- 2} κ$ .Reuse & Permissions

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