Robust hyperparallel photonic quantum entangling gate with cavity QED

Bao-Cang Ren; Fu-Guo Deng

doi:10.1364/OE.25.010863

1. Introduction

Quantum computation has improved the method of solving computational tasks with parallel quantum computing, resorting to the superposition principle of quantum mechanics theory [1]. Quantum logic gates are very useful for solving computational tasks and manipulating quantum states in quantum information processing [2]. Quantum entangling gate is a key element in the scalable quantum computation, as it can be used to implement universal quantum computing together with the single-qubit gates [3]. The two-qubit controlled-not (CNOT) gate and controlled-phase-flip (CPF) gate are both universal quantum entangling gates for universal quantum computing [4]. Many works have been done for constructing these universal quantum logic gates with several quantum systems, including ion trap [5], nuclear magnetic resonance [6,7], quantum dots [8–12], superconducting charge qubits [13,14], diamond nitrogen vacancy (NV) centers [15], photon systems [16–22], photon-atom hybrid systems [23], and so on.

Photon is a natural quantum information carrier. It is easy to manipulate the single-photon state in quantum information processing. In the scalable quantum computation, the strong interaction between photons is an essential task to be accomplished for the quantum entangling gate operations. In 2001, Knill, Laflamme, and Milburn [16] proposed the first CNOT gate for photon system with a maximal success probability 3/4, where the interaction between photons is accomplished by the linear optical elements and auxiliary photons. Since this pioneering work, many improved schemes have been demonstrated in experiment for photonic CNOT and CPF gates [24–26] with linear optical elements. In order to obtain the near-deterministic interaction between photons, nonlinear optical elements are introduced to construct the quantum entangling gates. In 2004, Nemoto and Munro [18] proposed a near-deterministic photonic quantum CNOT gate with the cross-Kerr nonlinearity. Duan and Kimble [19] proposed a near-deterministic CPF gate for a two-photon system by using the nonlinearity of a single atom trapped in an optical cavity in the same year. Subsequently, some interesting proposals have been proposed for quantum entangling gates with nonlinear optical elements [20,27,28].

There are more than one degree of freedom (DOF) in photon systems, such as the polarization, spatial-mode, and time-bin DOFs. The multiple DOFs of photon systems are very useful in quantum communication protocols for increasing the channel capacity and the security [29], such as hyper-teleportation of quantum states in multiple DOFs and hyperentanglement swapping with polarization-spatial hyperentangled states analysis [30–33], high-capacity quantum repeaters with hyperentanglement concentration [34–37] and hyperentanglement purification [38, 39], and quantum hyperdense coding [40]. In quantum computation, the multiple DOFs of photon systems are also very useful for solving computational tasks. The spatial-mode DOF of photon systems has been used to assist the implementation of the polarization quantum logic gate operations. Moreover, both of the spatial-mode and polarization DOFs can be used to carry information in quantum computation [41–43]. The hyperparallel photonic quantum computation, which is used to perform the quantum logic gate operations on the multiple DOFs simultaneously, can reduce the resource consumption and depress the photonic dispassion noise in quantum circuit [44,45].

Cavity quantum electrodynamics (QED) is a useful technique for obtaining the photon-photon interaction, dipole emitter-dipole emitter interaction, and photon-dipole emitter interaction in quantum information processing [46–48], including the photon-photon interaction in both the polarization and spatial-mode DOFs. In 2016, Hu [49] proposed the deterministic and robust quantum gates based on the optics linearity (rather than optics nonlinearity) of a single quantum dot in optical microcavity. Nitrogen vacancy (NV) center in diamond is a promising candidate for the dipole emitter in cavity QED with its long electron-spin coherence time [50,51]. The entanglement between a single photon and the electron-spin in a diamond NV center has been demonstrated in experiment [52], and this effect can be enhanced by coupling the diamond NV center to an optical cavity (or nanomechanical resonator) [53, 54]. With the nonlinearity of the diamond NV center coupled to an optical cavity, many quantum information tasks can be accomplished, such as quantum entangling gates for the electron-spins [55, 56], hybrid systems [57, 58], and photon systems [28]. In these protocols, the inequality of the reflection coefficients of the coupled and uncoupled cavity-dipole emitter systems [59, 60] is a main factor that may decrease the fidelity of the quantum gate operation.

In this article, we investigate the possibility of constructing robust hyperparallel quantum entangling gate for a two-photon system in the balance condition of diamond NV center embedded in an optical cavity (one-sided cavity-NV-center system), where the noise caused by the inequality of two reflection coefficients of the coupled and uncoupled cavity-NV-center systems can be depressed efficiently. We present a robust photonic hyperparallel CPF (hyper-CPF) gate with the balance condition of one-sided cavity-NV-center system and the waveform corrector (WFC), in which double CPF gate operations are performed synchronously on the polarization and spatial-mode DOFs of a two-photon system with the near unit fidelity. Compared with the two cascade quantum entangling gates in one DOF, this hyperparallel quantum entangling gate can reduce the resources consumed largely and depress the photonic dissipation efficiently. We analyze the balance condition and the efficiency of this robust universal hyperparallel quantum entangling gate, finding that the robust quantum gate operation can be accomplished in both the strong coupling regime and the weak coupling regime, which makes this gate easier to be realized in experiment than the ones in the ideal condition.

2. Balance condition of one-sided cavity-NV-center system

The negatively charged NV center is a defect in diamond that consists of a substitutional nitrogen atom, an adjacent vacancy, and six electrons, where the six electrons come from the nitrogen atom and three carbon atoms surrounding the vacancy. The ground state of the NV center is a spin triple split with 2.88 GHz between the magnetic sublevels |0〉 (m_s = 0) and |± 1〉 (m_s = ±1) due to spin-spin interaction [52]. There are six excited states according to the Hamilton including spin-orbit and spin-spin interactions and C_3v symmetry [61]. The specifically excited state $| A_{2} 〉 = \frac{1}{\sqrt{2}} (| E_{-} 〉 | + 1 〉 + | E_{+} 〉 | - 1 〉)$ , which is robust with the stable symmetric properties, decays with an equal probability to the two ground states |− 1〉 and |+ 1〉 with emission of left (|L〉) and right (|R〉) circularly polarized photons, respectively, according to spin-conserving optical transitions as shown in Fig. 1(b) [52]. Here the photon polarization of optical transition is dependent on the electronic orbital angular momentum change. |E_±〉 are the orbit states with the angular momentum projections ±1 along the NV axis, and |E₀〉 is the orbit state of the ground state with the angular momentum projection 0 along the NV axis.

Fig. 1 (a) A one-sided cavity-NV-center system. The optical cavity consists of two concave mirrors. The bottom mirror is 100% reflective and the top mirror is partially reflective. (b) The optical transitions between the ground states (|E₀〉 |∓1〉) and the excited state |A₂〉. The photon state |L〉 corresponds to σ⁺ = 1 and the photon state |R〉 corresponds to σ⁻ = −1.

Download Full Size | PDF

The optical transition process of an NV center can be enhanced by coupling the diamond NV center to an optical cavity, where the optical cavity is required to be a frequency-degenerate but polarization-nondegenerate (σ⁺ and σ⁻ polarizations) two-mode one [62]. The input-output optical process of the cavity-NV-center system shown in Fig. 1(a) can be described by Heisenberg equations for the cavity field operator â and the dipole operator σ̂₋ in the interaction picture [63],

\begin{array}{l} \frac{d \hat{a}}{d t} & = - [i (ω_{c} - ω) + \frac{η}{2} + \frac{κ}{2}] \hat{a} - g {\hat{σ}}_{-} - \sqrt{η} {\hat{a}}_{in}, \\ \frac{d {\hat{σ}}_{-}}{d t} & = - [i (ω_{e} - ω) + \frac{γ}{2}] {\hat{σ}}_{-} - g {\hat{σ}}_{z} \hat{a}, \end{array}

where ω_e, ω_c, and ω are the frequencies of the energy transition between the ground states and the excited state, the cavity field mode, and the input light, respectively. γ/2 is the decay rate of the electron-spin state in NV center. η/2 and κ/2 are the decay rates of the cavity field mode to the output light and the cavity intrinsic loss mode, respectively. g is the coupling strength of the cavity-NV-center system. â_in and â_out denote the input and output field operators, and they satisfy the boundary relation

{\hat{a}}_{out} = {\hat{a}}_{in} + \sqrt{η} \hat{a}

. In the weak excitation limit where the electron-spin state in the NV center is dominantly in the ground state with 〈σ_z〉 = −1, the reflection coefficient of the one-sided cavity-NV-center system is expressed as [9,64]

r (ω) = \frac{{\hat{a}}_{out}}{{\hat{a}}_{in}} = \frac{[i (ω_{e} - ω) + \frac{γ}{2}] [i (ω_{c} - ω) - \frac{η}{2} + \frac{κ}{2}] + g^{2}}{[i (ω_{e} - ω) + \frac{γ}{2}] [i (ω_{c} - ω) + \frac{η}{2} + \frac{κ}{2}] + g^{2}} .

If the frequency of the cavity field mode is resonant to the one of the optical transition in the NV center (ω_c = ω_e = ω), the reflection coefficient of the one-sided cavity-NV-center system can be expressed as r = (F_P + r₀)/(F_P + 1). Here, F_P = 4g²/[γ(η + κ)] is the Purcell factor, and r₀ = (κ − η)/(κ + η) is the reflection coefficient for the case g = 0. In the case κ < η and F_P > |r₀|, there is a π phase shift between the two reflection coefficients r and r₀, which is very useful in quantum information processing. Usually, the two reflection coefficients |r| and |r₀| are unequal, which may decrease the fidelity of the quantum gate operation. Therefore, the two reflection coefficients |r| and |r₀| are required to be near equal to obtain the high-fidelity quantum gate operation. For example, the reflection coefficients are r ≃ 1 for g ≫ κ, η and r₀ ≃ −1 for g = 0 in the ideal condition with η ≫ κ.

In experiment, the ideal condition (η ≫ κ, g ≫ κ, η) is hard to achieve, and the two reflection coefficients are always unequal (|r| ≠ |r₀|), which has decreased the fidelity of the quantum gate operation. If the reflection coefficients are set to r₀ ≃ −r, the infidelity caused by the two unequal reflection coefficients can be efficiently depressed, which requires r₀ = −r = −F_P/(F_P + 2) and F_P = (1 − λ)/λ. Here, λ = κ/η. In this balance condition, the input-output optical property of one-sided cavity-NV-center system interacted with circularly polarized photon is expressed as

\begin{array}{l} | L, - 1 〉 \to r | L, - 1 〉, | L, + 1 〉 \to - r | L, + 1 〉, \\ | R, - 1 〉 \to - r | R, - 1 〉, | R, + 1 〉 \to r | R, + 1 〉 . \end{array}

The relationship between Purcell factor F_P and λ in the balance condition is shown in Fig. 2, which shows that the balance condition can be achieved in both the strong coupling regime and the weak coupling regime. Here the coupling strength g of the one-sided cavity-NV-center system can be engineered to some certain values that satisfy the relaion 4g²/[γ(η + κ)] = (1− κ/η)/(κ/η). In the balance condition, the noise caused by the unequal reflection coefficients can be depressed efficiently, and the fidelity of the quantum gate operation can be increased largely. In this article, we will use this balance condition of one-sided cavity-NV-center system (r₀ ≃ −r) to construct the robust universal hyperparallel quantum entangling gate operating on both the polarization and spatial-mode DOFs of photon systems.

Fig. 2 The cavity mode decay rate λ vs the Purcell factor F_P in the balance condition of a one-sided cavity-NV-center system.

Download Full Size | PDF

3. Robust photonic hyperparallel quantum entangling gate

In this section, we show how to construct the robust quantum entangling gate in the balance condition of one-sided cavity-NV-center system, and we introduce the construction of a robust photonic hyper-CPF gate as shown in Fig. 3.

Fig. 3 Schematic diagram for the robust hyper-CPF gate operating on the spatial-mode and polarization DOFs of a two-photon system in the balance condition. NV₁ and NV₂ represent two one-sided cavity-NV-center systems with reflection coefficient r. X₁ and X₂ represent two half-wave plates, which can perform the bit-flip operations $σ_{x}^{P} = | R 〉〈 L | + | L 〉〈 R |$ on the polarization DOF of a photon. CPBS_k (k = 1, 2, 3, 4) represents a polarizing beam splitter in the circular basis, which can transmit the photon in polarization state |R〉 and reflect the photon in polarization state |L〉. DL represents a time-delay device, which can make |R〉 arrive simultaneously with |L〉. WFC represents a waveform corrector, which can map |i₂〉 to r|i₂〉. i₁ and i₂ denote two spatial modes of photon i (i = a, b).

Download Full Size | PDF

The photonic hyper-CPF gate is used to perform CPF gate operations on both the spatial-mode and polarization DOFs of a two-photon system simultaneously. The two cavity-NV-center systems NV₁ and NV₂ are both in the balance condition with the reflection coefficient r (r = −r₀). The electron-spin states of NV₁ and NV₂ are prepared in |+〉_e₁ and |+〉_e₂, which can be accomplished with the Hadamard operation [|− 1〉 → |+〉, |+ 1〉 → |−〉]. Here, $| \pm 〉 = \frac{1}{\sqrt{2}} (| - 1 〉 \pm | + 1 〉)$ . The initial states of two photons a and b are prepared in |ψ_a〉 = (α₁|R〉 + β₁|L〉)_a(γ₁|a₁〉 + δ₁|a₂〉) and |ψ_b〉 = (α₂|R〉 + β₂|L〉)_b (γ₂|b₁〉 + δ₂|b₂〉). Here, i₁ and i₂ denote two spatial modes of photon i (i = a, b).

First, we put photon a into the quantum circuit shown in Fig. 3. After the wavepackets from two spatial modes a₁ and a₂ pass through NV₁ and waveform corrector (WFC), the state of the system consists of the electron spin e₁ (in NV₁) and the photon a is changed from |ψ_a〉 ⊗ |+〉_e₁ to |Ψ_ae₁〉. Here,

\begin{array}{l} | Ψ_{a e 1} 〉 = & \frac{1}{\sqrt{2}} {γ_{1} [{| - 1 〉}_{e_{1}} {(- α_{1} | R 〉 + β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(α_{1} | R 〉 - β_{1} | L 〉)}_{a}] | a_{1} 〉 \\ + δ_{1} [{| - 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a}] | a_{2} 〉} . \end{array}

WFC in the spatial mode i₂ is used to map |i₂〉 to r|i₂〉 [65], which can leave the fidelity of quantum gate operation intact. Then we put the wavepackets from two spatial modes a₁ and a₂ into X₁, CPBS₁, CPBS₂, DL, NV₂, WFC, CPBS₃, CPBS₄, and X₂ in sequence as shown in Fig. 3. The state of the system consists of the electron spins e₁, e₂ (in NV₁ and NV₂) and the photon a is changed from |Ψ_ae₁〉 ⊗ |+〉_e₂ to |Ψ_ae₁e₂〉. Here,

\begin{array}{l} | Ψ_{a e_{1} e_{2}} 〉 = & \frac{1}{2} {[- {| - 1 〉}_{e_{1}} (γ_{1} | a_{1} 〉 - δ_{1} | a_{2} 〉) + {| + 1 〉}_{e_{1}} (γ_{1} | a_{1} 〉 + δ_{1} | a_{2} 〉)] \otimes [{| - 1 〉}_{e_{2}} (α_{1} | R 〉 \\ {+ β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{2}} {(α_{1} | R 〉 - β_{1} | L 〉)}_{a}]} . \end{array}

Now, we can obtain the result of the robust hybrid CPF gate. That is, the electron spins e₁ and e₂ are used as the control qubits, and the spatial-mode and polarization DOFs of photon a are used as the target qubits.

Subsequently, we perform Hadamard operations on two electron spins e₁ and e₂. Then we put photon b into the quantum circuit shown in Fig. 3. After the wavepackets from two spatial modes b₁ and b₂ pass through the quantum circuit, we can also obtain the result of the robust hybrid CPF gate for the electron spins e₁, e₂ and the spatial-mode and polarization DOFs of photon b. The state of the system consists of the electron spins e₁, e₂ and the photons a, b is changed from |Ψ_ae₁e₂〉 ⊗ |ψ_b〉 to

\begin{array}{l} | Ψ_{a b e_{1} e_{2}} 〉 = & [{| - 1 〉}_{e_{1}} δ | a_{2} 〉 (γ_{2} | b_{1} 〉 - δ_{2} | b_{2} 〉) + {| + 1 〉}_{e_{1}} γ_{1} | a_{1} 〉 (γ_{2} | b_{1} 〉 + δ_{2} | b_{2} 〉)] \\ \otimes [{| - 1 〉}_{e_{2}} α_{1} {| R 〉}_{a} {(α_{2} | R 〉 + β_{2} | L 〉)}_{b} + {| + 1 〉}_{e_{2}} β_{1} {| L 〉}_{a} {(α_{2} | R 〉 - β_{2} | L 〉)}_{b}] . \end{array}

At last, we perform Hadamard operations on two electron spins e₁ and e₂ again and measure these two electron spins with orthogonal basis {|− 1〉, |+ 1〉}. If the states of two electron spins are |− 1〉_e₁ and |− 1〉_e₂, no additional operation is required to perform on photon a, and the state of photon system ab is transformed to

\begin{array}{l} | ψ_{a b} 〉 = & [γ_{1} | a_{1} 〉 (γ_{2} | b_{1} 〉 + δ_{2} | b_{2} 〉) + δ_{1} | a_{2} 〉 (γ_{2} | b_{1} 〉 - δ_{2} | b_{2} 〉)] \\ \otimes [α_{1} {| R 〉}_{a} {(α_{2} | R 〉 + β_{2} | L 〉)}_{b} + β_{1} {| L 〉}_{a} {(α_{2} | R 〉 - β_{2} | L 〉)}_{b}] . \end{array}

If the state of electron spin e₁ is |+ 1〉_e₁, a spatial-mode phase-flip operation

σ_{z}^{S} = | i_{1} 〉 〈 i_{1} | - | i_{2} 〉 〈 i_{2} |

is required to perform on photon a to obtain the state |ψ_ab〉. If the state of electron spin e₂ is |+ 1〉_e₂, a polarization phase-flip operation

σ_{z}^{P} = - | R 〉 〈 R | + | L 〉 〈 L |

is required to perform on photon a to obtain the state |ψ_ab〉. Now, we have obtained the result of the robust hyper-CPF gate in the balance condition, where the infidelity caused by the cavity-NV-center system is eliminated by the WFC. The robust hyperparallel quantum entangling gate for multi-photon system can be constructed in the same way by using the balance condition of one-sided cavity-NV-center system and WFC.

4. Discussion and summary

The multiple DOFs of photon system are very useful for increasing the capacity of carrying information in quantum information processing [66,67]. In this article, we have investigated the construction of robust universal hyperparallel quantum entangling gate operating on both the spatial-mode and polarization DOFs of photon systems simultaneously, resorting to the balance condition of one-sided cavity-NV-center system and WFC.

NV center in diamond is a promising candidate dipole emitter for cavity QED with its long spin coherence time even at the room temperature [52]. The electron spin in diamond NV center can be manipulated (∼subnanosecond) [50, 68] and readout (∼ 100μs) [69, 70] by using the microwave excitation. The entanglement between a single photon and the electron spin in diamond NV center is a basic technical in our proposal, which has been demonstrated in experiment [52]. The interaction of the photon and the electron spin can be enhanced by coupling the diamond NV center to optical cavity, because the spontaneous emission of dipole emitter into zero-phonon line is largely enhanced by the optical cavity [53, 54]. The investigations of the diamond NV center coupled to the optical cavity have been demonstrated either in strong coupling regime [71,72] or in weak coupling regime [73] in experiment, where the investigations of the strong coupling regime are mainly focused on the NV center ensemble coupled to the optical cavity and the strong coupling of a single NV center in a cavity is still a challenge. In the cavity-NV-center system, the spin-selective optical resonant transition time (∼ns) and the photon coherence time (∼ 10ns) are much shorter than the electron-spin coherence time (> 10ms) [74], and the fidelity of the quantum gate operation may be reduced slightly by the electronic spin preparation (< 1%) [70], spin decoherence (< 1%), off-resonant excitation errors (∼ 1%), spin-flip errors in the excited states (∼ 1%), and microwave pulse errors (∼ 3.5%) [74].

The inequality of the reflection coefficients of the coupled and uncoupled cavity-dipole emitter systems (|r| ≠ |r₀|) is a main factor that may decrease the fidelity of the quantum gate operation [59, 60]. Reiserer et al. [60] showed that the fidelity of the quantum gate operation is decreased by the inequality of the reflection coefficients of the coupled and uncoupled cavity-atom systems (|r| ∼ 0.66, |r₀| ∼ 0.7) in their experiment. Here, we investigate the balance condition (r ≃ −r₀) of one-sided cavity-NV-center system with F_P = (1 − λ)/λ, which can efficiently depress the noise caused by the inequality of two reflection coefficients and increase the fidelity of the quantum gate operation. For example, in the case |r| ≠ |r₀|, after photon a in the state |ψ_a〉 interacts with the one-sided cavity-NV-center system NV₁ in the state |+〉 as shown in Fig. 3, the state of the system ae₁ is transformed to

\begin{array}{l} | Ψ_{a e_{1}} 〉 = & \frac{1}{\sqrt{2}} {γ_{1} [{| - 1 〉}_{e_{1}} {(r_{0} α_{1} | R 〉 + r β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(r α_{1} | R 〉 + r_{0} β_{1} | L 〉)}_{a}] | a_{1} 〉 \\ + δ_{1} [{| - 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a}] | a_{2} 〉}, \end{array}

where the parameters of the polarization state of photon a are changed. In the balance condition, the state of the system ae₁ is expressed as

\begin{array}{l} | Ψ_{a e_{1}} 〉 = & \frac{1}{\sqrt{2}} {r γ_{1} [{| - 1 〉}_{e_{1}} {(- α_{1} | R 〉 + β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a}] | a_{1} 〉 \\ + δ_{1} [{| - 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a}] | a_{2} 〉}, \end{array}

where the infidelity of the polarization state of photon a is transformed to the spatial-mode state of photon a. Then we can use the WFC (e.g., the unbalanced beam splitter with the reflection coefficient r [34] shown in Fig. 4) to eliminate the infidelity of the spatial-mode state of photon a, and the state of the system ae₁ is transformed to

\begin{array}{l} | Ψ_{a e_{1}} 〉 = & \frac{1}{\sqrt{2}} {γ_{1} [{| - 1 〉}_{e_{1}} {(- α_{1} | R 〉 + β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a}] | a_{1} 〉 \\ + δ_{1} [{| - 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a} + {| + 1 〉}_{e_{1}} {(α_{1} | R 〉 + β_{1} | L 〉)}_{a}] | a_{2} 〉} . \end{array}

Now, we can see that the infidelity caused by the one-sided cavity-NV-center system is eliminated. With this method, the robust universal hyperparallel quantum entangling gate for multi-photon system can also be accomplished. If a photon detector is put in the transmition port of the unbalanced beam splitter as shown in Fig. 4, we can obtain the heralded robust quantum gate operation, where the infidelity of the quantum gate operation is mapped to the photonic loss and detected by the photon detector.

Fig. 4 Schematic diagram for the unbalanced beam splitter. BS represents a 50:50 beam splitter. R_θ represents a wave plate, which is used to perform a phase shift on a spatial mode of a photon. D represents a photon detector.

Download Full Size | PDF

With the WFC, the fidelity of the quantum gate operation can achieve nearly unit in the balance condition, while the efficiency of the quantum gate operation may be decreased by this operation. In the balance condition, the reflection coefficient of the one-sided cavity-NV-center system is r = F_P/(F_P + 2), which is increased with the raise of the Purcell factor F_P as shown in Fig. 5. The efficiency of the robust quantum gate operation is dependent on the reflection coefficient r. In Fig. 6, the efficiency of the robust hyper-CPF gate is calculated vs Purcell factor F_P in the balance condition. From Fig. 6, we can see that the efficiency of the robust hyper-CPF gate is high with the raise of the Purcell factor F_P. Here, the efficiency is defined as the probability of a photon to be detected after it passes through the quantum circuit. In the weak coupling regime with F_P = 9 (e.g., g ∼ 2π × 0.5GHZ, η ∼ 2π × 7.77GHZ, γ ∼ 2π × 13MHZ, λ ∼ 0.1 [75]), the efficiency is E = 66.9% for the robust hyper-CPF gate. In the strong coupling regime with F_P = 400 (e.g., g ∼ 2π × 2GHZ, η ∼ 2π × 3GHZ, γ ∼ 2π × 13MHZ, λ ∼ 0.0025), the efficiency of the robust hyper-CPF gate is E = 99%. The efficiency of the robust quantum gate operation is high even in the weak coupling regime, so the robust quantum gate operation can be constructed in both the strong coupling regime and the weak coupling regime in the balance condition, which may be easier to realize in experiment than the ones in the ideal condition.

Fig. 5 The reflection cofficient r of the one-sided cavity-NV-center system vs the Purcell factor F_P in the balance condition.

Download Full Size | PDF

Fig. 6 Efficiency of the robust hyper-CPF gate vs the Purcell factor F_P in the balance condition.

Download Full Size | PDF

In summary, we have discussed the possibility of constructing robust hyperparallel quantum entangling gate for a two-photon system in the balance condition of a one-sided cavity-NV-center system. A robust hyper-CPF gate is constructed for a two-photon system in both the polarization and spatial-mode DOFs, which has a near unit fidelity in the balance condition. Compared with the two cascade quantum entangling gates in one DOF, this hyperparallel quantum entangling gate can reduce the resources consumed largely and depress the photonic dissipation efficiently. With this universal hyperparallel quantum entangling gate, the multi-photon hyperentangled state can be prepared and measured with less resource and less steps, which may speedup the quantum algorithm. Also, the universal hyperparallel quantum entangling gate can be used to solve computational tasks on both the spatial-mode and polarization DOFs of photon systems simultaneously, together with the single-photon manipulations.

In the balance condition, the reflection coefficients of the coupled and uncoupled one-sided cavity-NV-center systems are balanced (r ≃ −r₀), so the robust quantum gate operation can be constructed by using the WFC. The balance condition can be obtained in both the strong and the weak coupling regimes, so it is very useful for accomplishing high-fidelity quantum gate operations in quantum information processing. In experiment, the cavity-NV-center system is mainly demonstrated for the optical transitions with linear polarization. The optical cavity demonstrated for the optical transitions with circular polarization has been reported recently [62]. In the future, the cavity-NV-center system with circular polarization optical transitions could be demonstrated experimentally, which is a key requirement for the quantum information processing with the optical property of cavity-NV-center systems.

Funding

National Natural Science Foundation of China (NSFC) (11604226, 11474026, 11674033, 11547106); The Science and Technology Program Foundation of the Beijing Municipal Commission of Education of China (KM201710028005); Fundamental Research Funds for the Central Universities (2015KJJCA01).

References and links

1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University, 2000).

2. Y. Liu, G. L. Long, and Y. Sun, “Analytic one-bit and CNOT gate constructions of general n-qubit controlled gates,” Int. J. Quantum Inf. 6, 447–462 (2008). [CrossRef]

3. A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter, “Elementary gates for quantum computation,” Phys. Rev. A 52, 3457 (1995). [CrossRef] [PubMed]

4. Y. Y. Shi, “Both Toffoli and controlled-not need little help to do universal quantum computation,” Quantum Inf. Comput. 3, 084 (2003).

5. F. Schmidt-Kaler, H. Häffner, M. Riebe, S. Gulde, G. P. T. Lancaster, T. Deuschle, C. Becher, C. F. Roos, J. Eschner, and R. Blatt, “Realization of the Cirac-Zoller controlled-NOT quantum gate,” Nature 422, 408–411 (2003). [CrossRef] [PubMed]

6. N. A. Gershenfeld and I. L. Chuang, “Bulk spin-resonance quantum computation,” Science 275, 350–356 (1997). [CrossRef] [PubMed]

7. G. Feng, G. Xu, and G. Long, “Experimental realization of nonadiabatic Holonomic quantum computation,” Phys. Rev. Lett. 110, 190501 (2013). [CrossRef] [PubMed]

8. X. Li, Y. Wu, D. Steel, D. Gammon, T. H. Stievater, D. S. Katzer, D. Park, C. Piermarocchi, and L. J. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science 301, 809 (2003). [CrossRef] [PubMed]

9. C. Y. Hu, A. Young, J. L. O’Brien, W. J. Munro, and J. G. Rarity, “Giant optical Faraday rotation induced by a single-electron spin in a quantum dot: applications to entangling remote spins via a single photon,” Phys. Rev. B 78, 085307 (2008). [CrossRef]

10. C. Y. Hu, W. J. Munro, and J. G. Rarity, “Deterministic photon entangler using a charged quantum dot inside a microcavity,” Phys. Rev. B 78, 125318 (2008). [CrossRef]

11. C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80, 205326 (2009). [CrossRef]

12. C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,” Phys. Rev. B 83, 115303 (2011). [CrossRef]

13. T. Yamamoto, Y. A. Pashkin, O. Astafiev, Y. Nakamura, and J. S. Tsai, “Demonstration of conditional gate operation using superconducting charge qubits,” Nature 425, 941–944 (2003). [CrossRef] [PubMed]

14. Z. Y. Xue, Z. Q. Yin, Y. Chen, Z. D. Wang, and S. L. Zhu, “Topological quantum memory interfacing atomic and superconducting qubits,” Sci. China Phys. Mech. Astron. 59, 660301 (2016). [CrossRef]

15. T. van der Sar, Z. H. Wang, M. S. Blok, H. Bernien, T. H. Taminiau, D. M. Toyli, D. A. Lidar, D. D. Awschalom, R. Hanson, and V. V. Dobrovitski, “Decoherence-protected quantum gates for a hybrid solid-state spin register,” Nature 484, 82–86 (2012). [CrossRef] [PubMed]

16. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001). [CrossRef] [PubMed]

17. J. L. O’Brien, G. J. Pryde, A. G. White, T. C. Ralph, and D. Branning, “Demonstration of an all-optical quantum controlled-NOT gate,” Nature 426, 264–267 (2003). [CrossRef]

18. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-NOT gate,” Phys. Rev. Lett. 93, 250502 (2004). [CrossRef]

19. L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92, 127902 (2004). [CrossRef] [PubMed]

20. H. R. Wei and F. G. Deng, “Scalable photonic quantum computing assisted by quantum-dot spin in double-sided optical microcavity,” Opt. Express 21, 17671–17685 (2013). [CrossRef] [PubMed]

21. Q. Lin, “Optical parity gate and a wide range of entangled states generation,” Sci. China Phys. Mech. Astron. 58, 44201 (2015). [CrossRef]

22. Y. B. Sheng, J. Pan, R. Guo, L. Zhou, and L. Wang, “Efficient N-particle W state concentration with different parity check gates,” Sci. China Phys. Mech. Astron. 59, 660301 (2016).

23. L. Y. He, T. J. Wang, and C. Wang, “Construction of high-dimensional universal quantum logic gates using a Λ system coupled with a whispering-gallery-mode microresonator,” Opt. Express 24, 15429–15445 (2016). [CrossRef] [PubMed]

24. T. B. Pittman, M. J. Fitch, B. C. Jacobs, and J. D. Franson, “Experimental controlled-NOT logic gate for single photons in the coincidence basis,” Phys. Rev. A 68, 032316 (2003). [CrossRef]

25. S. Gasparoni, J. W. Pan, P. Walther, T. Rudolph, and A. Zeilinger, “Realization of a photonic controlled-NOT gate sufficient for quantum computation,” Phys. Rev. Lett. 93, 020504 (2004). [CrossRef] [PubMed]

26. Z. Zhao, A. N. Zhang, Y. A. Chen, H. Zhang, J. F. Du, T. Yang, and J. W. Pan, “Experimental demonstration of a nondestructive controlled-NOT quantum gate for two independent photon qubits,” Phys. Rev. Lett. 94, 030501 (2005). [CrossRef] [PubMed]

27. W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005). [CrossRef]

28. H. R. Wei and G. L. Long, “Universal photonic quantum gates assisted by ancilla diamond nitrogen-vacancy centers coupled to resonators,” Phys. Rev. A 91, 032324 (2015). [CrossRef]

29. F. G. Deng, B. C. Ren, and X. H. Li, “Quantum hyperentanglement and its applications in quantum information processing,” Sci. Bull. 62, 46–68 (2015). [CrossRef]

30. Y. B. Sheng, F. G. Deng, and G. L. Long, “Complete hyperentangled-Bell-state analysis for quantum communication,” Phys. Rev. A 82, 032318 (2010). [CrossRef]

31. B. C. Ren, H. R. Wei, M. Hua, T. Li, and F. G. Deng, “Complete hyperentangled-Bell-state analysis for photon systems assisted by quantum-dot spins in optical microcavities,” Opt. Express 20, 24664–24677 (2012). [CrossRef] [PubMed]

32. T. J. Wang, Y. Lu, and G. L. Long, “Generation and complete analysis of the hyperentangled Bell state for photons assisted by quantum-dot spins in optical microcavities,” Phys. Rev. A 86, 042337 (2012). [CrossRef]

33. Q. Liu and M. Zhang, “Generation and complete nondestructive analysis of hyperentanglement assisted by nitrogen-vacancy centers in resonators,” Phys. Rev. A 91, 062321 (2015). [CrossRef]

34. B. C. Ren, F. F. Du, and F. G. Deng, “Hyperentanglement concentration for two-photon four-qubit systems with linear optics,” Phys. Rev. A 88, 012302 (2013). [CrossRef]

35. T. J. Wang, C. Cao, and C. Wang, “Linear-optical implementation of hyperdistillation from photon loss,” Phys. Rev. A 89, 052303 (2014). [CrossRef]

36. X. H. Li and S. Ghose, “Efficient hyperconcentration of nonlocal multipartite entanglement via the cross-Kerr nonlinearity,” Opt. Express 23, 3550–3562 (2015). [CrossRef] [PubMed]

37. X. H. Li and S. Ghose, “Hyperentanglement concentration for time-bin and polarization hyperentangled photons,” Phys. Rev. A 91, 062302 (2015). [CrossRef]

38. B. C. Ren, F. F. Du, and F. G. Deng, “Two-step hyperentanglement purification with the quantum-state-joining method,” Phys. Rev. A 90, 052309 (2014). [CrossRef]

39. T. J. Wang, L. L. Liu, R. Zhang, C. Cao, and C. Wang, “One-step hyperentanglement purification and hyperdistillation with linear optics,” Opt. Express 23, 9284–9294 (2015). [CrossRef] [PubMed]

40. T. M. Graham and P. G. Kwiat, “Quantum hyperdense coding,” Proc. SPIE 9254, 92540B (2014).

41. M. Scholz, T. Aichele, S. Ramelow, and O. Benson, “Deutsch-Jozsa algorithm using triggered single photons from a single quantum dot,” Phys. Rev. Lett. 96, 180501 (2006). [CrossRef] [PubMed]

42. P. Zhang, R. F. Liu, Y. F. Huang, H. Gao, and F. L. Li, “Demonstration of Deutsch’s algorithm on a stable linear optical quantum computer,” Phys. Rev. A 82, 064302 (2010). [CrossRef]

43. C. Vitelli, N. Spagnolo, L. Aparo, F. Sciarrino, E. Santamato, and L. Marrucci, “Joining the quantum state of two photons into one,” Nat. Photonics 7, 521–526 (2013). [CrossRef]

44. B. C. Ren and F. G. Deng, “Hyper-parallel photonic quantum computation with coupled quantum dots,” Sci. Rep. 4, 4623 (2014). [CrossRef] [PubMed]

45. T. Li and G. L. Long, “Hyperparallel optical quantum computation assisted by atomic ensembles embedded in double-sided optical cavities,” Phys. Rev. A 94, 022343 (2016). [CrossRef]

46. T. Li and Z. Q. Yin, “Quantum superposition, entanglement, and state teleportation of a microorganism on an electromechanical oscillator,” Sci. Bull. 61, 163–171 (2016). [CrossRef]

47. R. Heilmann, M. Gräfe, S. Nolte, and A. Szameit, “A novel integrated quantum circuit for high-order W-state generation and its highly precise characterization,” Sci. Bull. 60, 96–100 (2015). [CrossRef]

48. M. Gao, F. C. Lei, C. G. Du, and G. L. Long, “Dynamics and entanglement of a membrane-in-the-middle optomechanical system in the extremely-large-amplitude regimes,” Sci. China Phys. Mech. Astron. 59, 610301 (2016). [CrossRef]

49. C. Y. Hu, “Spin-based single-photon transistor, dynamic random access memory, diodes, and routers in semiconductors,” Phys. Rev. B 94, 245307 (2016). [CrossRef]

50. B. B. Buckley, G. D. Fuchs, L. C. Bassett, and D. D. Awschalom, “Spin-light coherence for single-spin measurement and control in diamond,” Science 330, 1212–1215 (2010). [CrossRef] [PubMed]

51. G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hemmer, F. Jelezko, and J. Wrachtrup, “Ultralong spin coherence time in isotopically engineered diamond,” Nat. Mater. 8, 383–387 (2009). [CrossRef] [PubMed]

52. E. Togan, Y. Chu, A. S. Trifonov, L. Jiang, J. Maze, L. Childress, M. V. G. Dutt, A. S. Sørensen, P. R. Hemmer, A. S. Zibrov, and M. D. Lukin, “Quantum entanglement between an optical photon and a solid-state spin qubit,” Nature 466, 730–734 (2010). [CrossRef] [PubMed]

53. A. Faraon, C. Santori, Z. Huang, V. M. Acosta, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers to photonic crystal cavities in monocrystalline diamond,” Phys. Rev. Lett. 109, 033604 (2012). [CrossRef] [PubMed]

54. S. Schietinger, M. Barth, T. Aichele, and O. Benson, “Plasmon-enhanced single photon emission from a nanoassembled metal-diamond hybrid structure at room temperature,” Nano Lett. 9, 1694–1698 (2009). [CrossRef] [PubMed]

55. Q. Chen, W. Yang, M. Feng, and J. Du, “Entangling separate nitrogen-vacancy centers in a scalable fashion via coupling to microtoroidal resonators,” Phys. Rev. A 83, 054305 (2011). [CrossRef]

56. A. Zheng, J. Li, R. Yu, X. Y. Lü, and Y. Wu, “Generation of Greenberger-Horne-Zeilinger state of distant diamond nitrogen-vacancy centers via nanocavity input-output process,” Opt. Express 20, 16902–16912 (2012). [CrossRef]

57. T. J. Wang and C. Wang, “Universal hybrid three-qubit quantum gates assisted by a nitrogen-vacancy center coupled with a whispering-gallery-mode microresonator,” Phys. Rev. A 90, 052310 (2014). [CrossRef]

58. H. R. Wei and G. L. Long, “Hybrid quantum gates between flying photon and diamond nitrogen-vacancy centers assisted by optical microcavities,” Sci. Rep. 5, 12918 (2015). [CrossRef] [PubMed]

59. T. G. Tiecke, J. D. Thompson, N. P. de Leon, L. R. Liu, V. Vuletić, and M. D. Lukin, “Nanophotonic quantum phase switch with a single atom,” Nature 508, 241–244 (2014). [CrossRef] [PubMed]

60. A. Reiserer, N. Kalb, G. Rempe, and S. Ritter, “A quantum gate between a flying optical photon and a single trapped atom,” Nature 508, 237–240 (2014). [CrossRef] [PubMed]

61. A. Lenef and S. C. Rand, “Electronic structure of the N-V center in diamond: Theory,” Phys. Rev. B 53, 13441–13455 (1996). [CrossRef]

62. Y. Eto, A. Noguchi, P. Zhang, M. Ueda, and M. Kozuma, “Projective measurement of a single nuclear spin qubit by using two-mode cavity QED,” Phys. Rev. Lett. 106, 160501 (2011). [CrossRef] [PubMed]

63. D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, 1994). [CrossRef]

64. J. H. An, M. Feng, and C. H. Oh, “Quantum-information processing with a single photon by an input-output process with respect to low-Q cavities,” Phys. Rev. A 79, 032303 (2009). [CrossRef]

65. Y. Li, L. Aolita, D. E. Chang, and L. C. Kwek, “Robust-fidelity atom-photon entangling gates in the weak-coupling regime,” Phys. Rev. Lett. 109, 160504 (2012). [CrossRef] [PubMed]

66. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Beating the channel capacity limit for linear photonic superdense coding,” Nat. Phys. 4, 282–286 (2008). [CrossRef]

67. X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015). [CrossRef] [PubMed]

68. G. D. Fuchs, V. V. Dobrovitski, D. M. Toyli, F. J. Heremans, and D. D. Awschalom, “Gigahertz dynamics of a strongly driven single quantum spin,” Science 326, 1520–1522 (2009). [CrossRef] [PubMed]

69. L. Jiang, J. S. Hodges, J. R. Maze, P. Maurer, J. M. Taylor, D. G. Cory, P. R. Hemmer, R. L. Walsworth, A. Yacoby, A. S. Zibrov, and M. D. Lukin, “Repetitive readout of a single electronic spin via quantum logic with nuclear spin ancillae,” Science 326, 267–272 (2009). [CrossRef] [PubMed]

70. L. Robledo, L. Childress, H. Bernien, B. Hensen, P. F. A. Alkemade, and R. Hanson, “High-fidelity projective read-out of a solid-state spin quantum register,” Nature 477, 574–578 (2011). [CrossRef] [PubMed]

71. Y. S. Park, A. K. Cook, and H. Wang, “Cavity QED with diamond nanocrystals and silica microspheres,” Nano Lett. 6, 2075–2079 (2006). [CrossRef] [PubMed]

72. J. Teissier, A. Barfuss, P. Appel, E. Neu, and P. Maletinsky, “Strain coupling of a nitrogen-vacancy center spin to diamond mechanical oscillator,” Phys. Rev. Lett. 113, 020503 (2014). [CrossRef]

73. P. E. Barclay, K. M. C. Fu, C. Santori, and R. G. Beausoleil, “Chip-based microcavities coupled to nitrogen-vacancy centers in single crystal diamond,” Appl. Phys. Lett. 95, 191115 (2009). [CrossRef]

74. H. Bernien, B. Hensen, W. Pfaff, G. Koolstra, M. S. Blok, L. Robledo, T. H. Taminiau, M. Markham, D. J. Twitchen, L. Childress, and R. Hanson, “Heralded entanglement between solid-state qubits separated by three metres,” Nature 497, 86–90 (2013). [CrossRef] [PubMed]

75. D. O’Shea, C. Junge, J. Volz, and A. Rauschenbeutel, “Fiber-optical switch controlled by a single atom,” Phys. Rev. Lett. 111, 193601 (2013). [CrossRef]

Robust hyperparallel photonic quantum entangling gate with cavity QED

Abstract

1. Introduction

2. Balance condition of one-sided cavity-NV-center system

3. Robust photonic hyperparallel quantum entangling gate

4. Discussion and summary

Funding

References and links

Cited By

Figures (6)

Equations (10)

Optics Express