## Abstract

Under the balance condition of the diamond nitrogen vacancy center embedded in an optical cavity as a result of cavity quantum electrodynamics, we present a robust hyperparallel photonic controlled-phase-flip gate for a two-photon system in both the polarization and spatial-mode degrees of freedom (DOFs), in which the noise caused by the inequality of two reflection coefficients can be depressed efficiently. This gate doubles the quantum entangling operation synchronously on a photon system and can reduce the quantum resources consumed largely and depress the photonic dissipation efficiently, compared with the two cascade quantum entangling gates in one DOF. It has a near unit fidelity. Moreover, we show that the balance condition can be obtained in both the weak coupling regime and the strong coupling regime, and the high-fidelity quantum gate operation is easier to be realized in the balance condition than the ones in the ideal condition in experiment.

© 2017 Optical Society of America

## 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*= ±1) due to spin-spin interaction [52]. There are six excited states according to the Hamilton including spin-orbit and spin-spin interactions and

_{s}*C*

_{3v}symmetry [61]. The specifically excited state $|{A}_{2}\u3009=\frac{1}{\sqrt{2}}\left(|{E}_{-}\u3009|+1\u3009+|{E}_{+}\u3009|-1\u3009\right)$, 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 |

*E*

_{0}〉 is the orbit state of the ground state with the angular momentum projection 0 along the NV axis.

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],

*ω*,

_{e}*ω*, and

_{c}*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*and

_{in}*â*denote the input and output field operators, and they satisfy the boundary relation ${\widehat{a}}_{\mathit{out}}={\widehat{a}}_{\mathit{in}}+\sqrt{\eta}\widehat{a}$. In the weak excitation limit where the electron-spin state in the NV center is dominantly in the ground state with 〈

_{out}*σ*〉 = −1, the reflection coefficient of the one-sided cavity-NV-center system is expressed as [9,64]

_{z}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}*r*

_{0})/(

*F*+ 1). Here,

_{P}*F*= 4

_{P}*g*

^{2}/[

*γ*(

*η*+

*κ*)] is the Purcell factor, and

*r*

_{0}= (

*κ*−

*η*)/(

*κ*+

*η*) is the reflection coefficient for the case

*g*= 0. In the case

*κ*<

*η*and

*F*> |

_{P}*r*

_{0}|, there is a

*π*phase shift between the two reflection coefficients

*r*and

*r*

_{0}, which is very useful in quantum information processing. Usually, the two reflection coefficients |

*r*| and |

*r*

_{0}| are unequal, which may decrease the fidelity of the quantum gate operation. Therefore, the two reflection coefficients |

*r*| and |

*r*

_{0}| 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

*r*

_{0}≃ −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*| ≠ |*r*_{0}|), which has decreased the fidelity of the quantum gate operation. If the reflection coefficients are set to *r*_{0} ≃ −*r*, the infidelity caused by the two unequal reflection coefficients can be efficiently depressed, which requires *r*_{0} = −*r* = −*F _{P}*/(

*F*+ 2) and

_{P}*F*= (1 −

_{P}*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

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 4

*g*

^{2}/[

*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 (

*r*

_{0}≃ −

*r*) to construct the robust universal hyperparallel quantum entangling gate operating on both the polarization and spatial-mode DOFs of photon systems.

## 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.

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 NV_{1} and NV_{2} are both in the balance condition with the reflection coefficient *r* (*r* = −*r*_{0}). The electron-spin states of NV_{1} and NV_{2} are prepared in |+〉_{e1} and |+〉_{e2}, which can be accomplished with the Hadamard operation [|− 1〉 → |+〉, |+ 1〉 → |−〉]. Here,
$|\pm \u3009=\frac{1}{\sqrt{2}}\left(|-1\u3009\pm |+1\u3009\right)$. The initial states of two photons *a* and *b* are prepared in |*ψ _{a}*〉 = (

*α*

_{1}|

*R*〉 +

*β*

_{1}|

*L*〉)

*(*

_{a}*γ*

_{1}|

*a*

_{1}〉 +

*δ*

_{1}|

*a*

_{2}〉) and |

*ψ*〉 = (

_{b}*α*

_{2}|

*R*〉 +

*β*

_{2}|

*L*〉)

*(*

_{b}*γ*

_{2}|

*b*

_{1}〉 +

*δ*

_{2}|

*b*

_{2}〉). Here,

*i*

_{1}and

*i*

_{2}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 *a*_{1} and *a*_{2} pass through NV_{1} and waveform corrector (WFC), the state of the system consists of the electron spin *e*_{1} (in NV_{1}) and the photon *a* is changed from |*ψ _{a}*〉 ⊗ |+〉

_{e1}to |Ψ

_{ae1}〉. Here,

*i*

_{2}is used to map |

*i*

_{2}〉 to

*r*|

*i*

_{2}〉 [65], which can leave the fidelity of quantum gate operation intact. Then we put the wavepackets from two spatial modes

*a*

_{1}and

*a*

_{2}into X

_{1}, CPBS

_{1}, CPBS

_{2}, DL, NV

_{2}, WFC, CPBS

_{3}, CPBS

_{4}, and X

_{2}in sequence as shown in Fig. 3. The state of the system consists of the electron spins

*e*

_{1},

*e*

_{2}(in NV

_{1}and NV

_{2}) and the photon

*a*is changed from |Ψ

_{ae1}〉 ⊗ |+〉

_{e2}to |Ψ

_{ae1e2}〉. Here,

*e*

_{1}and

*e*

_{2}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 *e*_{1} and *e*_{2}. Then we put photon *b* into the quantum circuit shown in Fig. 3. After the wavepackets from two spatial modes *b*_{1} and *b*_{2} pass through the quantum circuit, we can also obtain the result of the robust hybrid CPF gate for the electron spins *e*_{1}, *e*_{2} and the spatial-mode and polarization DOFs of photon *b*. The state of the system consists of the electron spins *e*_{1}, *e*_{2} and the photons *a*, *b* is changed from |Ψ_{ae1e2}〉 ⊗ |*ψ _{b}*〉 to

At last, we perform Hadamard operations on two electron spins *e*_{1} and *e*_{2} 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

*e*

_{1}is |+ 1〉

_{e1}, a spatial-mode phase-flip operation ${\sigma}_{z}^{S}=|{i}_{1}\u3009\u3008{i}_{1}|-|{i}_{2}\u3009\u3008{i}_{2}|$ is required to perform on photon

*a*to obtain the state |

*ψ*〉. If the state of electron spin

_{ab}*e*

_{2}is |+ 1〉

_{e2}, a polarization phase-flip operation ${\sigma}_{z}^{P}=-|R\u3009\u3008R|+|L\u3009\u3008L|$ is required to perform on photon

*a*to obtain the state |

*ψ*〉. 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.

_{ab}## 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 (∼ 100*μ*s) [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*| ≠ |*r*_{0}|) 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, |*r*_{0}| ∼ 0.7) in their experiment. Here, we investigate the balance condition (*r* ≃ −*r*_{0}) 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*| ≠ |

*r*

_{0}|, after photon

*a*in the state |

*ψ*〉 interacts with the one-sided cavity-NV-center system NV

_{a}_{1}in the state |+〉 as shown in Fig. 3, the state of the system

*ae*

_{1}is transformed to

*a*are changed. In the balance condition, the state of the system

*ae*

_{1}is expressed as

*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

*ae*

_{1}is transformed to

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*+ 2), which is increased with the raise of the Purcell factor

_{P}*F*as shown in Fig. 5. The efficiency of the robust quantum gate operation is dependent on the reflection coefficient

_{P}*r*. In Fig. 6, the efficiency of the robust hyper-CPF gate is calculated vs Purcell factor

*F*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

_{P}*F*. 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

_{P}*F*= 9 (e.g.,

_{P}*g*∼ 2

*π*× 0.5GHZ,

*η*∼ 2

*π*× 7.77GHZ,

*γ*∼ 2

*π*× 13MHZ,

*λ*∼ 0.1 [75]), the efficiency is

*E*= 66.9% for the robust hyper-CPF gate. In the strong coupling regime with

*F*= 400 (e.g.,

_{P}*g*∼ 2

*π*× 2GHZ,

*η*∼ 2

*π*× 3GHZ,

*γ*∼ 2

*π*× 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.

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* ≃ −*r*_{0}), 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).

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