## Abstract

We propose an efficient protocol for optimizing the physical implementation of three-qubit quantum error correction with spatially separated quantum dot spins via virtual-photon-induced process. In the protocol, each quantum dot is trapped in an individual cavity and each two cavities are connected by an optical fiber. We propose the optimal quantum circuits and describe the physical implementation for correcting both the bit flip and phase flip errors by applying a series of one-bit unitary rotation gates and two-bit quantum iSWAP gates that are produced by the long-range interaction between two distributed quantum dot spins mediated by the vacuum fields of the fiber and cavity. The protocol opens promising perspectives for long distance quantum communication and distributed quantum computation networks.

© 2013 Optical Society of America

## 1. Introduction

Nowadays quantum information theory, which is based on fundamental quantum mechanical principle, has undergone a rapid development thanks to its attractive capabilities in contrast with classical information theory. Typical examples are quantum computation and quantum communication. The former could provide exponential or quadratic speedup for certain computational and mathematical problems that are not feasible on a classical computer, such as factoring problem [1], search problem [2], counting solution problem [3], phase estimation problem [4], hidden subgroup problem [5, 6], and so on. And the latter could achieve the faithful teleportation of unknown quantum states [7] and the secure communication of secret messages [8–12]. These quantum information processing tasks always require the implementation of quantum error correction and generation of entangled quantum states [13–16]. Quantum error correction, whose ability is to protect against the particular errors present in the respective quantum system making that the fragile quantum information is protected against corruption during computation, is one of the crucial requirements for universal quantum information processing. It has been shown that exploiting quantum error correction is advantageous even with imperfect encoding and recovery operations. In recent years, a large number of theoretical and experimental protocols have been proposed to implement quantum error correction [17–24]. Furthermore, there is an another effective way to solve the problem of the decoherence, such as entanglement purification and entanglement concentration. The goal of entanglement purification is used to obtain a subset of quantum systems in a maximally entangled state from an ensemble in a mixed entangled state. While for entanglement concentration, it is used to distill a subset system in a maximally entangled state from a set of systems in a partially entangled pure state. So far many schemes have been proposed to realize entanglement purification [25–28] and entanglement concentration [29–36] by using linear optical and optical microcavities systems.

Recently there has been growing interest in pursuing semiconductor quantum dots (QD) [37–41] as potential qubit candidates for solid-state-based quantum computation and quantum information processing due to their relative isolation from the environment [42] and the fact that the electronic spin degrees of freedom in semiconductors typically have decoherence times. Semiconductor QDs have potentially excellent scalability in the implementation of quantum computer and have wide applicability in the field of quantum information rooting in their unique advantages, such as large electric-dipole moments of intersubband transitions, high nonlinear optical susceptibility, and great flexibility in designing devices. For example, a quantum computation protocol based on electron spins in QDs has been previously proposed by Loss and DiVincenzo [43]. In this protocol, they described a universal set of one- and two-qubit gates using the spin states of coupled single-electron QDs based on local exchange interactions controlled by electrodes. Imamoḡlu *et al.*[44] proposed a new quantum information processing protocol based on QDs strongly coupled to a microcavity mode. They realized the parallel, long-range transverse spin-spin interactions between conduction-band electrons, mediated by the cavity mode, which is an effective way to carry out quantum computation. Hsieh and Hawrylak [45] proposed a microscopic theory of quantum circuits based on coded qubits encoded in chirality of electron spin complexes in lateral gated semiconductor triple QD molecules with one-electron spin in each dot. Their work established both single- and double-qubit operations necessarily used for performing quantum computation with chirality-based coded qubits. Hu and Rarity [46] proposed an efficient protocol for loss-resistant state teleportation and entanglement swapping using a single QD spin in an optical microcavity based on giant circular birefringence. Majumdar *et al.*[47] proposed a phonon-mediated off-resonant QD-cavity coupling model. Based on this model, they successfully explained recently observed resonant QD spectroscopic results and showed that phonon-mediated processes effectively extended the detuning range in which off-resonant QD-cavity coupling might occur beyond that given by pure dephasing processes.

In this paper, we propose an efficient protocol for optimal implementation of three-qubit quantum error correction with QD spins trapped in spatially separated cavities. We propose the optimal quantum circuits and describe the physical implementation for correcting both the bit flip and phase flip errors by using one-bit unitary rotation gates and two-bit quantum iSWAP gates that are produced by the long-range interaction between two distributed QD spins mediated by the vacuum fields of the fiber and cavity via virtual-photon-induced process. The present protocol has the following merits: (i) quantum error correction and entanglement among spatially separated quantum nodes is very useful for distributed quantum computation and quantum communication networks; (ii) the selective coupling between two QD spins is much easily manipulated by controlling the optical switch and it is unnecessary to address each QD selectively; (iii) the proposed protocol is robust with respect to systemic parameters, such as the fiber-cavity coupling strength *η* and frequency detunings
${\mathrm{\Delta}}_{\alpha}^{j}\left(\alpha =c,L\right)$.

## 2. Model and Hamiltonian

We consider a QD-cavity-fiber coupling system consisting of two distant cavities *A* and *B* connected by a single-transverse-mode optical fiber, as shown in Fig. 1(a). Each of two QDs is embedded inside a microdisk, put into a single-mode microcavity tuned to frequency
${\omega}_{c}^{j}$ and illuminated by a laser field tuned to frequency
${\omega}_{L}^{j}$. The QDs are doped such that each dot has a full valence band and a single conduction band electron, which is modeled by a three-level atom, as shown in Fig. 1(b), where each QD interacts with the cavity and the laser field with respective detunings
${\mathrm{\Delta}}_{c}^{j}$ and
${\mathrm{\Delta}}_{L}^{j}$. The QD qubit is defined by the conduction band states |*m _{x}* = −1/2〉 and |

*m*= 1/2〉, which correspond the logical zero and one states, respectively, |

_{x}*m*= −1/2〉 ≡ |0〉 and |

_{x}*m*= 1/2〉 ≡ |1〉. In the short fiber limit

_{x}*Lν̄*/(2

*πc*) ≤ 1, where

*L*is the length of the fiber and

*ν̄*is the decay rate of the cavity field into a continuum of the fiber modes, only one fiber mode essentially interacts with the cavity modes. In this case the total Hamiltonian of the system is written as (

*h̄*= 1) [44]

*m*,

*n*∈ {0, 1,

*v*}, ${\widehat{a}}_{cj}\left({\widehat{a}}_{cj}^{\u2020}\right)$ is the annihilation (creation) operator for the

*j*th cavity mode,

*g*and Ω

_{j}*denote the coupling constants of the*

_{j}*j*th dot interacting with the cavity mode and the classical laser field, respectively. ${\epsilon}_{j}^{\left(k\right)}$ (

*k*∈ {0, 1,

*v*}) stands for the energy of level |

*k*〉 of the

*j*th dot,

*b̂*is the annihilation operator of the fiber mode, and

*η*is the cavity-fiber coupling strength.

For the sake of convenience, we set *g _{j}* =

*g*, Ω

*= Ω, ${\mathrm{\Delta}}_{c}^{j}={\mathrm{\Delta}}_{c}$, and ${\mathrm{\Delta}}_{L}^{j}={\mathrm{\Delta}}_{L}$. Under the large-detuning conditions ${\mathrm{\Delta}}_{c}^{j}\gg {g}_{j}$ and ${\mathrm{\Delta}}_{L}^{j}\gg {\mathrm{\Omega}}_{j}$, the level |*

_{j}*v*〉

*can be eliminated adiabatically, the Hamiltonian describing the QD-cavity and cavity-fiber interactions in the interaction picture is thus given by*

_{j}*Ĥ*is denoted by Eq. (2) and

_{CF}*ξ*= Ω

^{2}/Δ

*,*

_{L}*ζ*=

*g*

^{2}/Δ

*,*

_{c}*δ*= Δ

*− Δ*

_{c}*, and*

_{L}*λ*=

*g*Ω(1/Δ

*+ 1/Δ*

_{c}*)/2. The first and second terms in Hamiltonian ${\widehat{H}}_{I}^{0}$ represent the Stark shifts for the states |1〉*

_{L}*and |0〉*

_{j}*, which are induced by the cavity fields and classical fields, respectively. The last term describes the Raman coupling of the two states |0〉*

_{j}*and |1〉*

_{j}*. As done in Refs. [48–51], we define*

_{j}*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}are three bosonic modes, which are linearly relative to the field modes of the cavities and fiber. Then the Hamiltonian of the whole system in Eq. (3) can be rewritten as

*Ĥ*as the “free Hamiltonian” and perform the unitary transformation

_{CF}*e*

^{iĤCFt}, obtaining

*δ*≫

*λ*, $\left|\delta \pm \sqrt{2}\eta \right|\gg \lambda /2$, and $\sqrt{2}\eta \gg \lambda /2$,

*ζ*/4, the bosonic modes not only do not exchange quantum numbers with the QD system, but also do not exchange quantum numbers with each other. The Stark shifts and Heisenberg

*XY*coupling between the QDs are induced by the off-resonant Raman coupling. We thus have the effective Hamiltonian [49–51]

*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}are conserved during the interaction. Assume that the two cavity modes

*A*and

*B*and the fiber mode are all initially in the vacuum state. Then the three bosonic modes

*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}remain in the vacuum state during the evolution. In this case the effective Hamiltonian reduces to

*A*and

*B*can thus be expressed as

*|0〉*

_{A}*, |0〉*

_{B}*|1〉*

_{A}*, |1〉*

_{B}*|0〉*

_{A}*, and |1〉*

_{B}*|1〉*

_{A}*. With the choice of*

_{B}*ςt*=

*π*/2 and performing the single-qubit phase shifts: |0〉

*→*

_{j}*e*|0〉

^{i}^{ξ}^{t}*and |1〉*

_{j}*→*

_{j}*e*

^{i}^{(}

^{π}^{+}

^{ϑt}^{)}|1〉

*(*

_{j}*j*=

*A*,

*B*), an iSWAP gate is obtained such that

*α*=

*x*,

*z; i*= 1, 2) are the Pauli operators acting on the

*i*th qubit. In the following we show how to implement three-qubit quantum error correction for both bit flip and phase flip errors based on the iSWAP gate produced by the QD-cavity-fiber interaction model under Hamiltonian (10).

## 3. Optimal quantum circuit and physical implementation of three-qubit quantum error correction

Three-qubit quantum error correction can correct either one bit flip error or one phase flip error. The quantum circuit for the implementation of quantum error correction for correcting one bit flip error by applying iSWAP gates is shown in Fig. 2, including encoding, decoding, and error-correction steps. The top one qubit, which carries the information to be encoded, is initialized to the |Ψ〉_{1} = *α*|0〉_{1} +*β*|1〉_{1} state, and the bottom two qubits are initialized to the |0〉_{2}|0〉_{3} state. To protect the state |Ψ〉_{1} from external decoherence, the state |Ψ〉_{1} is encoded by applying one- and two-qubit gate operations to qubits 1, 2, and 3, as shown in Fig. 2, where

*is subject to external noise (i.e., (simulated) external decoherence partly disrupts the state), leading to a (partial) bit flip of one of the spins, |Ψ〉*

_{E}*→ |Ψ′〉*

_{E}*. After the decoding network, the state |Ψ〉*

_{E}*is decoded (recovered). Finally the qubits 2 and 3 are measured. If both qubits are in state |1〉, then the qubit 1 is flipped, otherwise it is left unchanged.*

_{E}For correcting phase errors instead of bit errors, the initial state *α*|0〉_{1}|0〉_{2}|0〉_{3} +*β*|1〉_{1}|0〉_{2}|0〉_{3} should be encoded as *α*|+〉_{2}|+〉_{3}|+〉_{1} + *β*|−〉_{2}|−〉_{3}|−〉_{1} in the encoding procedure, as shown in Fig. 3(a), where
$|\pm \u3009=\left(|0\u3009\pm |1\u3009\right)/\sqrt{2}$, and

We now show how to experimentally implement the quantum error correction circuits shown in Fig. 2 and Fig. 3. The schematic setup for correcting both the bit flip and phase flip errors is shown in Fig. 4. Each QD spin is trapped in a spatially separated cavity and each two cavities are connected by an optical fiber. The interaction between two QD spins is controlled by an optical switch. At the beginning, all the optical switches are closed and the cavity and fiber modes are in vacuum state. According to the orders of the gate operations shown in Fig. 2 and Fig. 3, the single-qubit gate operations required for implementing quantum error correction in the quantum circuits are realized by classical laser fields. Each time only the nearest-neighbor two QD spins interact with each other, while the others do not, by opening and closing the optical switches. The interaction time between two QD spins in the cavities is chosen as *τ* = *π*/2*ς* such that the two-qubit iSWAP gate operating on the nearest-neighbor two QD spins is achieved. Furthermore, the physical implementations of quantum measurements on spins in QDs have been proposed in previous Refs. [43, 53, 54]. Therefore, the three-qubit quantum error correction for correcting both the bit flip and phase flip errors can be implemented in a deterministic way by using the QD-cavity-fiber system.

## 4. Analysis and discussion

We now briefly analyze and discuss some practical issues in relation to the experimental feasibility of the present protocol. For the presently available technology in QD experiments, we set *g* = 0.5 meV, Ω = *g*, Δ* _{L}* = 10

*g*, Δ

*= 11*

_{c}*g*, $\eta =\sqrt{2}g$, and

*κ*=

*γ*= Γ = 0.001

*g*, with

*κ*,

*γ*, and Γ being the decay rates for the cavity modes, the fiber mode, and the excited state of QD, respectively. In this way we have

*δ*= Δ

*− Δ*

_{c}*=*

_{L}*g*≈ 10.5

*λ*, which satisfy the conditions Δ

*≫*

_{c}*g*, Δ

*≫ Ω,*

_{L}*δ*≫

*λ*, $\left|\delta \pm \sqrt{2}\eta \right|\gg \lambda /2$, and $\sqrt{2}\eta \gg \lambda /2$,

*ζ*/4. Our calculations show that (i) the probabilities that QD undergo transitions from |0〉 and |1〉 states to |

*v*〉 state are ${P}_{0v}={\mathrm{\Omega}}^{2}/{\mathrm{\Delta}}_{L}^{2}={10}^{-2}$ and ${P}_{1v}={g}^{2}/{\mathrm{\Delta}}_{c}^{2}\approx 0.826\times {10}^{-2}$; (ii) the probability that the three modes

*ĉ*

_{0},

*ĉ*

_{1}, and

*ĉ*

_{2}are excited due to nonresonant coupling with the classical modes is

*Ĥ*′

_{eff}in Eq. (10) is valid; (iii) the effective decoherence rates due to the decay of the bosonic modes and the spontaneous emissions of QD are

*κ*′ =

*P*

_{1}

*κ*≈ 0.709 × 10

^{−5}

*g*, Γ′

_{0}

*=*

_{v}*P*

_{0}

*Γ = 10*

_{v}^{−5}

*g*, and Γ′

_{1}

*=*

_{v}*P*

_{1}

*Γ ≈ 0.826 × 10*

_{v}^{−5}

*g*, respectively; (iv) the time required for realizing the iSWAP gate is

*t*=

*π*/2

*ς*≈ 2.59 × 10

^{2}/

*g*, the average infidelity induced by the decoherence is thus about

*F̄*

_{inf}= 1.62 × 10

^{−3}.

For the quantum error correction process, ideally the output would be identical to the input. To examine the performance of the proposed protocol, we theoretically calculate the entanglement fidelity [55], which is a useful measurement of how well the quantum information in the input is preserved, as shown in Fig. 5. We can see from Fig. 5 that even when the pulse error is chosen as *ε* = 0.1, the entanglement fidelities *F*_{bf} > 99% corresponding to correct bit flip error and *F*_{pf} > 97% to phase flip error. On the other hand, when the thermal photons in the environment can be negligible, the protocol is insensitive to the cavity decay, fiber loss, and the spontaneous emission of QD since the long-range interaction between two distributed QDs is mediated by the vacuum fields of the fiber and cavity and the total system evolves in the decoherence-free subspace in which neither of the subsystems is excited. These features make the protocol more feasible for experimental realization and very promising for the implementations of scalable quantum communication networks and distributed quantum computation. Therefore, the proposed protocol is efficient and feasible.

## 5. Conclusions

In conclusion, we have proposed the optimal quantum circuits and described the physical implementation of quantum error correction by applying iSWAP gate produced by the long-range interaction between distributed QD spins mediated by the vacuum fields of the fiber and cavity. During the process of implementing quantum error correction, neither subsystem is excited and the protocol is insensitive to the cavity decay, the spontaneous emission of QD, and the fiber loss. The protocol is simple and feasible and the experimental realization of the protocol would be a useful step towards the implementation of more complex quantum computation in solid-state qubits.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 11264042, 61068001, and 11165015; China Postdoctoral Science Foundation under Grant No. 2012M520612; the Program for Chun Miao Excellent Talents of Jilin Provincial Department of Education under Grant No. 201316; and the Talent Program of Yanbian University of China under Grant No. 950010001.

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