Physical optimization of quantum error correction circuits with spatially separated quantum dot spins

Hong-Fu Wang; Ai-Dong Zhu; Shou Zhang

doi:10.1364/OE.21.012484

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 $Δ_{α}^{j} (α = c, L)$ .

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 $ω_{c}^{j}$ and illuminated by a laser field tuned to frequency $ω_{L}^{j}$ . 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 $Δ_{c}^{j}$ and $Δ_{L}^{j}$ . The QD qubit is defined by the conduction band states |m_x = −1/2〉 and |m_x = 1/2〉, which correspond the logical zero and one states, respectively, |m_x = −1/2〉 ≡ |0〉 and |m_x = 1/2〉 ≡ |1〉. In the short fiber limit Lν̄/(2πc) ≤ 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]

\hat{H} = {\hat{H}}_{C} + {\hat{H}}_{Q D} + {\hat{H}}_{int} + {\hat{H}}_{C F},

with

\begin{array}{l} {\hat{H}}_{C} = \sum_{j = A, B} ω_{c}^{j} {\hat{a}}_{c j}^{†} {\hat{a}}_{c j}, \\ {\hat{H}}_{Q D} = \sum_{j = A, B} (ε_{j}^{(0)} {\hat{σ}}_{j}^{00} + ε_{j}^{(1)} {\hat{σ}}_{j}^{11} + ε_{j}^{(v)} {\hat{σ}}_{j}^{v v}), \\ {\hat{H}}_{int} = \sum_{j = A, B} [g_{j} ({\hat{a}}_{c j} {\hat{σ}}_{j}^{1 v} + {\hat{a}}_{c j}^{†} {\hat{σ}}_{j}^{v 1}) + Ω_{j} (e^{- i ω_{L}^{j} t} {\hat{σ}}_{j}^{0 v} + e^{i ω_{L}^{j} t} {\hat{σ}}_{j}^{v 0})], \\ {\hat{H}}_{C F} = [η \hat{b} ({\hat{a}}_{c A}^{†} + {\hat{a}}_{c B}^{†}) + H . c .], \end{array}

where

{\hat{σ}}_{j}^{m n} = {| m 〉}_{j j} 〈 n |

with m, n ∈ {0, 1, v},

{\hat{a}}_{c j} ({\hat{a}}_{c j}^{†})

is the annihilation (creation) operator for the jth cavity mode, g_j and Ω_j denote the coupling constants of the jth dot interacting with the cavity mode and the classical laser field, respectively.

ε_{j}^{(k)}

(k ∈ {0, 1, v}) stands for the energy of level |k〉 of the jth dot, b̂ is the annihilation operator of the fiber mode, and η is the cavity-fiber coupling strength.

Fig. 1 (a) Schematic of QD-cavity-fiber-coupling system. (b) The relevant levels of QD.

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For the sake of convenience, we set g_j = g, Ω_j = Ω, $Δ_{c}^{j} = Δ_{c}$ , and $Δ_{L}^{j} = Δ_{L}$ . Under the large-detuning conditions $Δ_{c}^{j} ≫ g_{j}$ and $Δ_{L}^{j} ≫ Ω_{j}$ , the level |v〉_j can be eliminated adiabatically, the Hamiltonian describing the QD-cavity and cavity-fiber interactions in the interaction picture is thus given by

{\hat{H}}_{I} = {\hat{H}}_{I}^{0} + {\hat{H}}_{C F},

here Ĥ_CF is denoted by Eq. (2) and

{\hat{H}}_{I}^{0} = \sum_{j = A, B} [ξ {| 0 〉}_{j j} 〈 0 | + ζ {\hat{a}}_{c j} {\hat{a}}_{c j}^{†} {| 1 〉}_{j j} 〈 1 | + λ (e^{i δ t} {\hat{a}}_{c j} {\hat{σ}}_{j}^{10} + e^{- i δ t} {\hat{a}}_{c j}^{†} {\hat{σ}}_{j}^{01})],

where ξ = Ω²/Δ_L, ζ = g²/Δ_c, δ = Δ_c − Δ_L, and λ = gΩ(1/Δ_c + 1/Δ_L)/2. The first and second terms in Hamiltonian

{\hat{H}}_{I}^{0}

represent the Stark shifts for the states |1〉_j 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

\begin{array}{l} {\hat{c}}_{0} = \frac{1}{\sqrt{2}} ({\hat{a}}_{c A} - {\hat{a}}_{c B}), \\ {\hat{c}}_{1} = \frac{1}{2} ({\hat{a}}_{c A} + {\hat{a}}_{c B} + \sqrt{2} \hat{b}), \\ {\hat{c}}_{2} = \frac{1}{2} ({\hat{a}}_{c A} + {\hat{a}}_{c B} - \sqrt{2} \hat{b}), \end{array}

here ĉ₀, ĉ₁, and ĉ₂ 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

\begin{array}{l} {\hat{H}}_{C F} & = & \sqrt{2} η {\hat{c}}_{1}^{†} {\hat{c}}_{1} - \sqrt{2} η {\hat{c}}_{2}^{†} {\hat{c}}_{2}, \\ {\hat{H}}_{I}^{0} & = & \sum_{j = A, B} [ξ {| 0 〉}_{j j} 〈 0 | + \frac{ζ}{4} ({\hat{c}}_{1} {\hat{c}}_{1}^{†} + {\hat{c}}_{2} {\hat{c}}_{2}^{†} + {\hat{c}}_{2} {\hat{c}}_{2}^{†}) {| 1 〉}_{j j} 〈 1 |] \\ + \frac{1}{2} {λ e^{i δ t} [({\hat{c}}_{1} + {\hat{c}}_{2} + \sqrt{2} {\hat{c}}_{0}) {\hat{σ}}_{A}^{10} + ({\hat{c}}_{1} + {\hat{c}}_{2} - \sqrt{2} {\hat{c}}_{0}) {\hat{σ}}_{B}^{10}] \\ + \frac{ζ}{2} [({\hat{c}}_{1} {\hat{c}}_{2}^{†} + \sqrt{2} {\hat{c}}_{1} {\hat{c}}_{0}^{†} + \sqrt{2} {\hat{c}}_{2} {\hat{c}}_{0}^{†}) {| 1 〉}_{A A} 〈 1 | + ({\hat{c}}_{1} {\hat{c}}_{2}^{†} - \sqrt{2} {\hat{c}}_{1} {\hat{c}}_{0}^{†} - \sqrt{2} {\hat{c}}_{2} {\hat{c}}_{0}^{†}) {| 1 〉}_{B B} 〈 1 |] \\ + H . c .} . \end{array}

We now take Ĥ_CF as the “free Hamiltonian” and perform the unitary transformation e^iĤ_CFt, obtaining

\begin{array}{l} {\hat{H}}_{I}^{'} & = & \sum_{j = A, B} [ξ {| 0 〉}_{j j} 〈 0 | + \frac{ζ}{4} ({\hat{c}}_{1} {\hat{c}}_{1}^{†} + {\hat{c}}_{2} {\hat{c}}_{2}^{†} + {\hat{c}}_{2} {\hat{c}}_{2}^{†}) {| 1 〉}_{j j} 〈 1 |] \\ + {\frac{λ}{2} [e^{i (δ - \sqrt{2} η) t} {\hat{c}}_{1} + e^{i (δ + \sqrt{2} η) t} {\hat{c}}_{2} + \sqrt{2} e^{i δ t} {\hat{c}}_{0}] {\hat{σ}}_{A}^{10} \\ + \frac{λ}{2} [e^{i (δ - \sqrt{2} η) t} {\hat{c}}_{1} + e^{i (δ + \sqrt{2} η) t} {\hat{c}}_{2} - \sqrt{2} e^{i δ t} {\hat{c}}_{0}] {\hat{σ}}_{B}^{10} \\ + \frac{ζ}{4} [e^{- i 2 \sqrt{2} η t} {\hat{c}}_{1} {\hat{c}}_{2}^{†} + \sqrt{2} e^{- i \sqrt{2} η t} {\hat{c}}_{1} {\hat{c}}_{0}^{†} + \sqrt{2} e^{i \sqrt{2} η t} {\hat{c}}_{2} {\hat{c}}_{0}^{†}] {| 1 〉}_{A A} 〈 1 | \\ + \frac{ζ}{4} [e^{- i 2 \sqrt{2} η t} {\hat{c}}_{1} {\hat{c}}_{2}^{†} - \sqrt{2} e^{- i \sqrt{2} η t} {\hat{c}}_{1} {\hat{c}}_{0}^{†} - \sqrt{2} e^{i \sqrt{2} η t} {\hat{c}}_{2} {\hat{c}}_{0}^{†}] {| 1 〉}_{B B} 〈 1 | + H . c .} . \end{array}

Under the conditions δ ≫ λ,

| δ \pm \sqrt{2} η | ≫ λ / 2

, and

\sqrt{2} η ≫ λ / 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]

\begin{array}{l} {\hat{H}}_{eff} & = & \sum_{j = A, B} {ξ {| 0 〉}_{j j} 〈 0 | + \frac{ζ}{4} ({\hat{c}}_{1} {\hat{c}}_{1}^{†} + {\hat{c}}_{2} {\hat{c}}_{2}^{†} + {\hat{c}}_{2} {\hat{c}}_{2}^{†}) {| 1 〉}_{j j} 〈 1 | \\ + \frac{λ^{2}}{4} [(\frac{2}{δ} {\hat{c}}_{0}^{†} {\hat{c}}_{0} + \frac{1}{δ - \sqrt{2} η} {\hat{c}}_{1}^{†} {\hat{c}}_{1} + \frac{1}{δ + \sqrt{2} η} {\hat{c}}_{2}^{†} {\hat{c}}_{2}) {| 0 〉}_{j j} 〈 0 | \\ + (\frac{2}{δ} {\hat{c}}_{0} {\hat{c}}_{0}^{†} + \frac{1}{δ - \sqrt{2} η} {\hat{c}}_{1} {\hat{c}}_{1}^{†} + \frac{1}{δ - \sqrt{2} η} {\hat{c}}_{2} {\hat{c}}_{2}^{†}) {| 1 〉}_{j j} 〈 1 |]} \\ + \frac{ζ^{2}}{32 \sqrt{2} η} [({\hat{c}}_{1} {\hat{c}}_{1}^{†} - {\hat{c}}_{2} {\hat{c}}_{2}^{†}) {({| 1 〉}_{A A} 〈 1 | + {| 1 〉}_{B B} 〈 1 |)}^{2} \\ + 4 ({\hat{c}}_{1} {\hat{c}}_{1}^{†} - {\hat{c}}_{2} {\hat{c}}_{2}^{†}) {({| 1 〉}_{A A} 〈 1 | - {| 1 〉}_{B B} 〈 1 |)}^{2}] - ς ({\hat{σ}}_{A}^{10} {\hat{σ}}_{B}^{01} + {\hat{σ}}_{A}^{01} {\hat{σ}}_{B}^{10}), \end{array}

where

ς = \frac{λ^{2}}{4} (\frac{2}{δ} - \frac{1}{δ - \sqrt{2} η} - \frac{1}{δ + \sqrt{2} η}) .

The quantum numbers of the bosonic modes ĉ₀, ĉ₁, and ĉ₂ 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 ĉ₀, ĉ₁, and ĉ₂ remain in the vacuum state during the evolution. In this case the effective Hamiltonian reduces to

{\hat{H}}_{eff}^{'} = \sum_{j = A, B} (ξ {| 0 〉}_{j j} 〈 0 | + ϑ {| 1 〉}_{j j} 〈 1 |) - ς ({\hat{σ}}_{A}^{10} {\hat{σ}}_{B}^{01} + {\hat{σ}}_{A}^{01} {\hat{σ}}_{B}^{10}),

where

ϑ = \frac{1}{4} [3 ζ + λ^{2} (\frac{2}{δ} + \frac{1}{δ - \sqrt{2} η} + \frac{1}{δ + \sqrt{2} η})] .

The quantum information in the present protocol is stored in the state |0〉 and |1〉. For QD-cavity-fiber interaction system shown in Fig. 1, two-qubit operation produced by Hamiltonian (10) acting on qubits A and B can thus be expressed as

{\hat{U}}_{eff} (t) = e^{- i H_{eff}^{'} t} = (\begin{matrix} e^{- i 2 ξ t} & 0 & 0 & 0 \\ 0 & e^{- i (ξ + ϑ) t} cos ς t & i e^{- i (ξ + ϑ) t} sin ς t & 0 \\ 0 & i e^{- i (ξ + ϑ) t} sin ς t & e^{- i (ξ + ϑ) t} cos ς t & 0 \\ 0 & 0 & 0 & e^{- i 2 ϑ t} \end{matrix}),

where the basis is ordered as |0〉_A|0〉_B, |0〉_A|1〉_B, |1〉_A|0〉_B, and |1〉_A|1〉_B. With the choice of ςt = π/2 and performing the single-qubit phase shifts: |0〉_j → eⁱ^ξ^t |0〉_j and |1〉_j → eⁱ⁽^π⁺^ϑt⁾|1〉_j (j = A, B), an iSWAP gate is obtained such that

\begin{array}{l} {| 0 〉}_{A} {| 0 〉}_{B} = {| 0 〉}_{A} {| 0 〉}_{B}, & {| 1 〉}_{A} {| 1 〉}_{B} = {| 1 〉}_{A} {| 1 〉}_{B}, \\ {| 0 〉}_{A} {| 1 〉}_{B} = - i {| 1 〉}_{A} {| 0 〉}_{B}, & {| 1 〉}_{A} {| 0 〉}_{B} = - i {| 0 〉}_{A} {| 1 〉}_{B} . \end{array}

The conventional two-qubit controlled-NOT (CNOT) gate can be constructed by applying iSWAP gate twice (together with six single-qubit rotation gates), namely

{\hat{U}}_{CNOT} = e^{i (π / 4)} e^{- i (π / 2) {\hat{σ}}_{z}^{1}} e^{i (π / 4) {\hat{σ}}_{z}^{2}} {\hat{U}}_{iSWAP} e^{- i (π / 4) {\hat{σ}}_{x}^{1}} {\hat{U}}_{iSWAP} e^{i (π / 4) {\hat{σ}}_{z}^{2}} e^{- i (π / 4) {\hat{σ}}_{x}^{2}} e^{- i (π / 4) {\hat{σ}}_{z}^{1}},

where

σ_{α}^{i}

(α = x, z; i = 1, 2) are the Pauli operators acting on the ith 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

\begin{array}{l} u_{11} = (\begin{matrix} e^{- i π / 4} & 0 \\ 0 & e^{i π / 4} \end{matrix}), \\ u_{12} = (\begin{matrix} cos (π / 4) & - sin (π / 4) \\ sin (π / 4) & cos (π / 4) \end{matrix}), \\ u_{13} = (\begin{matrix} e^{i 3 π / 4} cos (π / 4) & e^{- i 3 π / 4} sin (π / 4) \\ - e^{i 3 π / 4} sin (π / 4) & e^{- i 3 π / 4} cos (π / 4) \end{matrix}) . \end{array}

After that, a highly entangled three-qubit state is obtained,

\begin{array}{l} {| Ψ 〉}_{E} & = & \hat{E} ({| Ψ 〉}_{1} {| 0 〉}_{2} {| 0 〉}_{3}) \\ = & α {| 0 〉}_{2} {| 0 〉}_{3} {| 0 〉}_{1} + β {| 1 〉}_{2} {| 1 〉}_{3} {| 1 〉}_{1} . \end{array}

Then the encoded state |Ψ〉_E 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.

Fig. 2 Three-bit quantum circuit representation for correcting one bit flip error. Here the gate • − • corresponds to a two-qubit iSWAP gate and u_ij are single-qubit rotation gates.

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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 $| \pm 〉 = (| 0 〉 \pm | 1 〉) / \sqrt{2}$ , and

\begin{array}{l} u_{21} = u_{22} = u_{23} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix}), \\ u_{11}^{'} = v_{11} \cdot u_{13}, u_{22}^{'} = u_{12} \cdot v_{12} \cdot u_{13}, \\ u_{12}^{'} = v_{13} \cdot u_{13}, u_{24}^{'} = u_{12} \cdot v_{13}^{†}, \\ u_{13}^{'} = v_{13} \cdot u_{11}, u_{26}^{'} = u_{12} \cdot v_{12} \cdot u_{13}, \\ u_{14}^{'} = u_{12} \cdot v_{14}, u_{32}^{'} = u_{12} \cdot v_{12}^{†} \cdot u_{13}, \\ u_{21}^{'} = u_{23}^{'} = u_{25}^{'} = u_{31}^{'} = u_{11}, \end{array}

with

\begin{array}{l} v_{11} = (\begin{matrix} 1 & 0 \\ 0 & i \end{matrix}), v_{12} = \frac{1}{\sqrt{4 - 2 \sqrt{2}}} (\begin{matrix} 1 & 1 - \sqrt{2} \\ \sqrt{2} - 1 & 1 \end{matrix}), \\ v_{13} = (\begin{matrix} 1 & 0 \\ 0 & e^{- i π / 4} \end{matrix}), v_{14} = \frac{1}{\sqrt{4 - 2 \sqrt{2}}} (\begin{matrix} 1 & \sqrt{2} - 1 \\ i (\sqrt{2} - 1) & - i \end{matrix}) . \end{array}

After the decoding procedure, the original state of the first qubit can then be restored by applying a three-qubit Toffoli gate, whose quantum circuit is shown in Fig. 3(b) [52], while the other two qubits contain information about the error which occurred due to the effect of external noise. In this way the full error-correction process, including the final Toffoli gate, is implemented making that the expected state preservation of an arbitrary coherent input.

Fig. 3 (a) Three-bit quantum circuit representation for correcting one bit phase error. (b) Quantum circuit implementation of three-qubit Toffoli gate by applying two-qubit iSWAP gates and single-qubit rotation gates.

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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/2ς 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.

Fig. 4 Schematic setup to implement the quantum circuits shown in Fig. 2 and Fig. 3 for correcting both the bit flip error and phase flip error. Here s₁ and s₂ are optical switches, which control two adjacent cavities whether have interaction or not.

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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 = 10g, Δ_c = 11g, $η = \sqrt{2} g$ , and κ = γ = Γ = 0.001g, 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_c − Δ_L = g ≈ 10.5λ, which satisfy the conditions Δ_c ≫ g, Δ_L ≫ Ω, δ ≫ λ, $| δ \pm \sqrt{2} η | ≫ λ / 2$ , and $\sqrt{2} η ≫ λ / 2$ , ζ/4. Our calculations show that (i) the probabilities that QD undergo transitions from |0〉 and |1〉 states to |v〉 state are $P_{0 v} = Ω^{2} / Δ_{L}^{2} = 10^{- 2}$ and $P_{1 v} = g^{2} / Δ_{c}^{2} \approx 0.826 \times 10^{- 2}$ ; (ii) the probability that the three modes ĉ₀, ĉ₁, and ĉ₂ are excited due to nonresonant coupling with the classical modes is

P_{1} = \frac{λ^{2}}{4} [\frac{1}{{(δ - \sqrt{2} η)}^{2}} + \frac{1}{{(δ + \sqrt{2} η)}^{2}} + \frac{2}{δ^{2}}] \approx 0.709 \times 10^{- 2} .

Therefore, the effective Hamiltonian 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′ = P₁κ ≈ 0.709 × 10⁻⁵g, Γ′₀_v = P₀_vΓ = 10⁻⁵g, and Γ′₁_v = P₁_vΓ ≈ 0.826 × 10⁻⁵g, respectively; (iv) the time required for realizing the iSWAP gate is t = π/2ς ≈ 2.59 × 10²/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.

Fig. 5 The entanglement fidelities for one-qubit bit flip and phase flip errors correction, as the functions of pulse error ε.

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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.

References and links

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Physical optimization of quantum error correction circuits with spatially separated quantum dot spins

Abstract

1. Introduction

2. Model and Hamiltonian

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

4. Analysis and discussion

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (19)

Optics Express