## Abstract

Robust universal quantum gates with an extremely high fidelity hold an important position in large-scale quantum computing. Here, we propose a scheme for several robust universal photonic quantum gates on a two-or three-photon system, including the controlled-NOT gate, the Toffoli gate, and the Fredkin gate, assisted by low-Q single-sided cavities. In our scheme, the quantum gates are robust against imperfect process occurring with the photons and the electron spins in diamond nitrogen-vacancy (NV) centers inside low-Q cavities. Errors due to the imperfect process are transferred to some heralding responses, which may lead to a direct recycling procedure to remedy the success probability of the quantum gates. As a result, the adverse impact of the imperfect process on fidelity is eliminated, greatly relaxing the restrictions on implementation of various quantum gates in experiments. Furthermore, the scheme is designed in a compact and heralded style, which can increase the robustness against environmental noise and local fluctuation, thus decreasing the operation time, the error probability, and the quantum resource consumption in a large-scale integrated quantum circuit. The near-unity fidelity and not-too-low efficiency with current achievable experimental techniques guarantees the feasibility of the scheme.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum computation [1] has grown to be one of the most important applications in quantum information processing (QIP) for its excellent performance in certain tasks, including factoring an n-bit integer with the famous Shor algorithm [2] and searching for data in an unsorted database with the Grover algorithm [3] or the optimal Long algorithm [4], for which no traditional counterpart can up to in practical time. The key element in quantum computation is the quantum logic gate, since any quantum task can be decomposed into a series of universal single-qubit unitary gates and two-qubit conditional gates [1, 5–7]. However, if only single- and two-qubit gates are employed, the circuit for certain quantum tasks may be extremely complicated as the number of qubits increases rapidly and therefore is hard to implement in practice [8–12]. Thus the construction of multi-qubit gates such as Toffoli gate [13] and Fredkin gate [14] gains much more attention as it can simplify the design, accelerate the operation, decrease the resource, and lower the decoherence. In addition, these two kinds of multi-qubit gates play a very important role in phase estimation [1], complex quantum algorithms [2–4], quantum error correction [15], and fault tolerant quantum circuit [16].

Tremendous efforts have been devoted to universal quantum gates in both theory and experiment with a variety of special quantum systems, such as the ion trap [17], the nuclear magnetic resonance (NMR) [18,19], quantum dots (QDs) [20–23], nitrogen-vacancy (NV) centers [24–26], superconducting circuits [27–31], photons with only polarization degree of freedom (DOF) [32, 33] and with both polarization and spatial-mode DOFs (hyperparallel photonic quantum computation) [34–40], and hybrid systems [22, 41–45]. Among these systems, the optical one is especially competitive in quantum computation since photons are the ideal quantum information carriers with many advantages, including robustness against the decoherence, easy generation, flexible manipulation and modification, and fast long-distance communication. The main obstacle in scalable quantum computation for optical systems is the lack of strong interaction between individual photons. Some strategies to tackle this problem on different platforms have been developed. A linear quantum computation scheme [46] based on linear optical elements and auxiliary photons is achieved with projective measurements and feedback operations in a probabilistic way. An optical controlled-NOT (CNOT) gate is built by resorting to the nondemolition detection using the cross-kerr nonlinearity in a near-deterministic way [47]. Some proposals [48–52] based on matter-cavity platforms exploiting the cavity quantum electrodynamics (QED) are also put forward.

Among these strategies, an NV center in diamond coupling to a single-sided cavity is attractive because of the flexibility in utilizing the photon-photon interaction, the photon-matter interaction, or the matter-matter interaction in QIP. In recent years, many extraordinary properties in NV centers have been realised in experiment, including ultralong coherence time even at room temperature [45, 53, 54], simple population via highly stable optical transition [55], high-fidelity manipulation [56–58] and read out [59, 60] by using the microwave excitation, efficient information transfer between electron and nuclear spins [61–63], second-scale storage time [64, 65], and good scalability. These properties have enabled rapid progress with NV centers in QIP such as entanglement generation and analysis [39,66–70], hyperentanglement purification and concentration [71–75], and implementation of universal quantum gates and the hyperparallel quantum computation [24, 26, 36, 37, 39, 76–79].

Usually the matter-cavity platforms are expected to work in strong coupling regime where the Rabi frequency exceeds the decay rates of both the cavity and the involved matter spin, leading to a near-deterministic input-output process which in turn results in high performance for many proposals [24, 26, 36, 37, 39, 39, 66, 69–74, 76, 77]. The decrease in coupling strength aggravates the input-output process and leads to an imperfect one, which may further deteriorate the fidelity, the scalability, or the feasibility of the schemes. Several tactics have been introduced to mitigate this problem and it turns out that the performances of the QIP in both weak and strong coupling regimes have been improved greatly. A maximally entangling gate between the atom and the photon is achieved with an error-heralding mechanism [80]. A near-deterministic controlled phase gate in optical cavity is achieved in a heralded way [81]. Some error-detected blocks [82–84] are built on quantum dot-spins assisted by single-or double-sided cavities. A quadratic fidelity improvement in controlled gates is achieved in superconducting system [85]. A robust hyperparallel quantum entangling gate [39] for a two-photon system assisted by a one-sided cavity NV-center system with the balance condition is proposed.

In this paper, we present a scheme for robust photonic quantum computation, including the construction of the CNOT gate, the Toffoli gate, and the Fredkin gate on photon qubits, workable with imperfect process in low-Q single-sided cavities. Usually, the imperfect process inevitably introduces some errors in the quantum gates. Here, some detectors are employed to herald the errors, which, together with subsequent repeat-until-success remedies, guarantees an extremely high fidelity of the quantum gates. As a result, the adverse impact of the imperfect process on fidelity is eliminated, greatly relaxing the restrictions on experiment.

## 2. The NV-cavity platform

The negatively charged NV center, a defect in a diamond, shown in Fig. 1a, consists of a substitutional nitrogen atom replacing a carbon atom, an adjacent vacancy and six electrons from the nitrogen atom and the three carbons that surrounds the vacancy. The ground triple state splits with 2.88 GHz between the magnetic sublevels |0〉 (|*m _{s}* = 0〉) and |±〉 (|

*m*= ±1〉) due to the spin-spin interaction [45, 86]. The dynamics including the spin-spin interaction, the spin-orbit interaction, and

_{s}*C*

_{3}

*symmetry results in six excited states of which the specific state $|{A}_{2}\u3009=\frac{1}{\sqrt{2}}\left(|{E}_{-}\u3009|+1\u3009+|{E}_{+}\u3009|-1\u3009\right)$ is utilized to implement a Λ-type three-level system shown in Fig. 1b [45, 86, 87]. |*

_{v}*E*

_{±}〉 are orbital states with angular momentum projection ±1 along the NV axis. According to spin-conserving optical transitions shown in Fig. 1b, the excited state |

*A*

_{2}〉decays with an equal probability to the two ground states |+〉 and |−〉 radiating a right-circular polarized photon |

*R*〉 and a left-circular polarized photon |

*L*〉, respectively.

An NV-cavity unit, where the optical transition process of the NV center can be enhanced by coupling it to an single-sided optical cavity, can supply nonlinearity interaction between a circularly polarized light and a diamond NV center. The cavity have two frequency-degenerate but polarization-nondegenerate modes [88]. The dynamics of the system can be described by the Hamiltonian [89]

*g*represents the Rabi frequency. The level transitions of NV centers with ${\widehat{\sigma}}_{\pm}=|\pm \u3009\u3008{A}_{2}|$ and ${\widehat{\sigma}}_{\pm}^{\u2020}=|{A}_{2}\u3009\u3008\pm |$ follow the selection rules, that is, a right-circular polarized photon |

*R*〉 can only couple the transition between |

*A*

_{2}〉 and |+〉 while a left-circular polarized photon |

*L*〉 can only couple the other. The cavity field creation operator ${\widehat{a}}_{k}^{\u2020}(k=R,L)$ and the cavity field annihilation operator ${\widehat{a}}_{k}^{-}$ creates and annihilates a photon in |

*k*〉 state with cavity frequency

*ω*

_{c}_{,}

*respectively.*

_{k,}In the case when the input photon is coupled with the transition, the Heisenberg-Langevin equations for the NV transition operator $\widehat{\sigma}$ (where the subscripts ± are neglected) with spontaneous emission rate *γ* and for the cavity field annihilation operator $\widehat{a}$ driven by the single photon input field ${\widehat{a}}_{in}$ and subject to the side-leakage bath ${\widehat{s}}_{in}$ with strength *κ* and *κ _{s}* respectively are [20, 90]

*ω*,

_{p}*ω*

_{0}, and

*ω*are the frequencies of the cavity, the single photon, and the transition of NV center, respectively. ${\widehat{\sigma}}_{z}={\widehat{\sigma}}^{\u2020}\widehat{\sigma}-\widehat{\sigma}{\widehat{\sigma}}^{\u2020}$. Operator $\widehat{N}$ presents the noise bath related to the spontaneous emission of the NV center. The reflection coefficient is determined by the ratio between the effective output field and the input field. Using the standard cavity input-output relationship ${\widehat{a}}_{out}={\widehat{a}}_{in}+\sqrt{\kappa}\widehat{a}$ and the Fourier transformation, the reflection coefficient can be expressed analytically in the weak excitation limit $(\u3008{\widehat{\sigma}}_{z}\u3009=-1)$ as [20, 90]

_{c}In the case when the input photon is uncoupled with the transition, the coupling strength *g* vanishes in the reflection coefficient in Eq. (3). In the following, we add subscript *j* to distinguish the case where the input photon uncouples with the transition (*j* = 0) from the other one where the input photon couples with the transition (*j* = 1). Considering the system in the resonant condition *ω*_{0} = *ω _{p}* =

*ω*, the reflection coefficients are reduced to

_{c}## 3. Robust photonic CNOT gate

The photonic CNOT gate completes a bit-flip operation on photon 2 (the target qubit) when photon 1 (the control qubit) is in the right-circular polarization |*R*〉; Otherwise, nothing is done on the target qubit. The quantum circuit for implementing a robust photonic CNOT gate is shown in Fig. 2. The two photons are prepared in the polarization state |*ϕ ^{p}*〉

_{1}=

*α*

_{1}|

*R*〉

_{1}+

*β*

_{1}|

*L*〉

_{1}and |

*ϕ*〉

^{p}_{2}=

*α*

_{2}|

*R*〉

_{2}+

*β*

_{2}|

*L*〉

_{2}, and the NV center is prepared in $|{\varphi}^{e}\u3009=\frac{1}{\sqrt{2}}(|+\u3009+|-\u3009)$.

*H*(

_{i}*i*= 1, 2, 3) and

*H′*are the half wave plates which achieve the rotation, $|R\u3009\to \frac{1}{\sqrt{2}}(|R\u3009+|L\u3009)$, $|L\u3009\to \frac{1}{\sqrt{2}}(|R\u3009-|L\u3009)$ and the rotation, $|R\u3009\to \frac{1}{\sqrt{2}}(|R\u3009+|L\u3009)$, $|L\u3009\to \frac{1}{\sqrt{2}}(|L\u3009-|R\u3009)$, respectively.

*R*modifies the shape and the amplitude of the photon passing through. PBS

_{θ}*(*

_{i}*i*= 1, 2) is the polarized beam splitter which transmits the right-circular polarized photon and reflects the left-circular polarized photon.

*D*is a single-photon detector. The optical switch

*SW*(

_{i}*i*= 1, 2) makes photons 1 and 2 pass through the device successively. The time interval between the two photons should be less than the decoherence time of the NV center. First, photon 1 splits into two modes at PBS1, the transmitted component of which interacts with

*H*

_{2}, the NV center,

*H′*and combines with the modified mode reflected by PBS

_{1}at PBS

_{2}. If the detector doesn’t click, the system gets into

Then, a Hadamard operation *H _{e}*, $|+\u3009\to |{x}_{+}\u3009=\frac{1}{\sqrt{2}}(|+\u3009+|-\u3009)$ and $|-\u3009\to |{x}_{-}\u3009=\frac{1}{\sqrt{2}}(|+\u3009-|-\u3009)$, is performed on the NV center before the optical switch allows photon 2 to enter. Upon photon 2’s presence, the system is in the state

Third, the second photon passes through *H*_{1}, PBS_{1}, PBS_{2}, and H_{3}. Provided that the detector does not click, the system becomes

Fourth, a Hadamard operation is performed on the NV center, and the state evolves into

If the outcome of the measurement on the NV center is |−〉, the robust CNOT gate on the polarization of photons is directly achieved. Otherwise, one has to implement a phase-flip operation *σ _{z}* = |

*L*〉〈

*L*|−|

*R*〉〈

*R*| on the first photon.

## 4. Robust photonic Toffoli gate

The photonic Toffoli gate completes a bit-flip operation on photon 3 (the target qubit) when both photon 1 and photon 2 (the control qubits) are in the right-circular polarization |*R*〉; Otherwise, nothing is done on the target qubit. The quantum circuit for implementing a robust photonic Toffoli gate is shown in Fig. 3. The initial state of the photons are prepared in |*ϕ ^{p}* 〉

*=*

_{i}*α*〉

_{i}|R*+*

_{i}*β*|

_{i}*L*〉

*(*

_{I}*i*= 1, 2, 3) and the initial state of the two NV centers are prepared in $|{\varphi}^{e}\u3009=1/\sqrt{2}({|+\u3009}_{i}+{|-\u3009}_{i})(i=1,2)$. First, the control qubits, photon 1 and photon 2, pass through the system, i.e. photon 1 passes through PBS

_{1},

*H*

_{1}(

*R*

_{θ}_{1}), NV

_{1}, ${H}_{1}^{\prime}$ and PBS

_{2}and photon 2 passes through PBS

_{3},

*H*

_{2}(

*R*

_{θ}_{2}), NV

_{2}, ${H}_{2}^{\prime}$ and PBS

_{4}. If the detectors don’t click, the system, composed of the two control qubits and the two NV centers, evolves into

Then, a Hadamard operation *H _{e}* is performed on the two NV centers respectively. Photon 1 and photon 2 get entangled with NV

_{1}and NV

_{2}, respectively. It can be detailed as

Third, photon 3 is injected into the system and experiences a Hadamard operation at ${H}_{3}^{\prime}$, the system becomes

Clearly, the first term in Eq. (11) will be transmitted to interact the optical elements and the NV centers while the second term in Eq. (11) will be reflected by PBS_{5} to the export directly. For the present, we only consider the evolution of the component |*τ*〉_{0}, where

Fourth, the photon passes through *H*_{3}, PBS_{6}, *H*_{4} (*R _{θ}*

_{3}), NV

_{1}, ${H}_{4}^{\prime}$ and PBS

_{7}. If the detector doesn’t click, the state evolves into

After the photon experiences a Hadamard operation at *H*_{5}, the state becomes

Fifth, the photon passes through PBS_{8}, *R _{θ}*

_{4}(

*H*

_{6}, NV

_{2}, ${H}_{5}^{\prime}$), and PBS

_{9}. If the detector does not click, the state becomes

Then, after the photon passes through the *H*_{10}, PBS_{12}, *H*_{9} (*R _{θ}*

_{6}), NV

_{1}, ${H}_{6}^{\prime}$, PBS

_{11}and

*H*

_{8}, the state changes into

Seventh, substituting the component |*τ*〉_{4} into Eq. (11). After the reflected mode from PBS_{5} is modified by *R _{θ}*

_{5}and combines with the component |

*τ*〉

_{4}at PBS

_{10}with no detector clicks, the system evolves into

A bit-flip operation at *X* and a Hadamard operation at *H*_{7} transform the system into

Finally, a Hadamard operation is performed on the two NV centers respectively and a ${\widehat{\sigma}}_{z}$ operation is performed on photon *i* (*i* = 1, 2) if the outcome of the measurement on the *i*-th NV is |−〉. The robust photonic Toffoli gate is achieved,

## 5. Robust photonic Fredkin gate

The photonic Fredkin gate completes a polarization swap operation between photon 2 and photon 3 (the target qubits) if photon 1 (the control qubit) is in the right-circular polarization |*R*〉; If the photon is in the left-circular polarization |*L*〉, nothing is done on the target qubits. The quantum circuit for implementing a robust photonic Fredkin gate is shown in Fig. 4. The photonic Fredkin gate can be easily achieved [91] by performing a CNOT operation with photon 3 as the control qubit and photon 2 as the target qubit before and after the photons are injected into the import of the Toffoli gate as the one depicted in Fig. 3. Photon 1, photon 3 and photon 2 are injected into the system subsequently. First, a Hadamard operation is performed on NV_{1} after photon 3 interacts with it. Then, photon 2 interacts with NV_{1}, if the detectors don’t click, the system evolves into

Third, NV_{1} is measured in the basis |*x*_{±}〉. If NV_{1} is measured in |*x*_{−}〉, nothing is done on the photon, else, photon 3 should experience a phase operation |*R*〉 →−|*R*〉. Here, we assume NV_{1} is in |*x*_{+}〉. Forth, the photons enter the T and experience a Toffoli gate operation, if the detectors don’t click, the state evolves into

Fifth, photon 3 and photon 2 pass through NV_{1} successively. Before and after photon 2 passes through NV_{1}, a Hadamard operation is performed on NV_{1}. If the detectors do not click, the system evolves into

If the NV_{1} is measured on the state |−〉, the Fredkin gate is achieved directly. If NV_{1} is measured on the state |+〉, a phase flip operation performed on photon 3 completes the Fredkin gate. Besides, if NV_{1} is measured on the |*x*_{−}〉 in the third step, a phase operation to achieve |−〉 → −|−〉 should be performed before photon 2 and photon 3 interact with it for the second time.

## 6. Discussion and summary

The efficiency of the gate can be characterized by the ratio of the output photon number ${\overline{n}}_{\text{output}}$ to the input photon number ${\overline{n}}_{\text{input}}$, i.e., $\eta ={\overline{n}}_{\text{output}}/{\overline{n}}_{\text{input}}$. If a photon, |*T*〉 = (*α*|*R*〉 + *β*|*L*〉)|*NV*〉, flies to interact with an NV center unit including PBS, *H*(*R _{θ}*),

*H*′, and PBS. It first splits into two channels at PBS. Then one of the channels, here we use

*α*|

*R*〉 to do this demonstration, encounters a Hadamard operation at

*H*. The state becomes $|T\u3009=(\frac{\alpha}{\sqrt{2}}(|R\u3009+|L\u3009)+\beta |L\u3009)|NV\u3009$. If the NV center is in |

*NV*〉 =

*a*|+〉 +

*b*|−〉, the state evolves into $|T\u3009=\frac{\alpha}{\sqrt{2}}[(ar|+\u3009+b{r}_{0}|-\u3009)|R\u3009+(a{r}_{0}|+\u3009+br|-\u3009)|L\u3009]+\beta |L\u3009|NV\u3009$. Fourth, the photon experiences a Hadamard-like operation at

*H′*, the state becomes $|T\u3009=\frac{\alpha}{2}[(r-{r}_{0})(a|+\u3009-b|-\u3009)|R\u3009+({r}_{0}+r)(a|+\u3009+b|-\u3009)|L\u3009]+\beta |L\u3009|NV\u3009$. Another channel,

*β*|

*L*〉 encounters a

*R*, then the state of the system can be rewrote as $|T\u3009=\frac{1}{2}(r-{r}_{0})[a|+\u3009(\alpha |R\u3009+\beta |L\u3009)-b|-\u3009(\alpha |R\u3009-\beta |L\u3009)]+\frac{1}{2}({r}_{0}+r)\beta |R\u3009|NV\u3009$. The second term in |

_{θ}*T*〉 represents the error. The first term in |

*T*〉 is the successful case with a probability of $T={\left|\frac{r+{r}_{0}}{2}\right|}^{2}$. Then, the efficiencies

*η*and

_{C}, η_{T}*η*corresponding to those of our CNOT gate, Toffoli gate and Fredkin gate, respectively, can be derived as

_{F}*k*-th NV center. We evaluate the efficiencies changing vs. the cooperativity $C=\frac{g}{\sqrt{({\kappa}_{s}+\kappa )\gamma}}$ with different leak rates

*κ*/

_{s}*κ*in Fig. 5, which shows that the efficiencies of all three gates asymptotically approach to unity with large cooperativity

*C*and small leak rates

*κ*/

_{s}*κ*. However, the increase in the side-leakage

*κ*does lead to some inefficiency. A comparison among the three gates indicates that the fewer times the photon interacts with the NV center, the less adverse effect the side-leakage imposes and the higher the efficiency is achieved. Therefore, one can complete these gates with much better performance when a large cooperativity

_{s}*C*and a small leakage rate

*κ*/

_{s}*κ*are available. With parameters

*g*

_{ZPL}= 2

*π*× 0.3 GHz,

*κ*= 2

*π*× 26 GHz,

*γ*

_{total}= 2

*π*× 13 MHz,

*γ*

_{ZPL}= 2

*π*× 0.4 MHz [92], the cooperativity

*C*can approximately reach 3. With this achievable cooperativity, the efficiencies of these gates can reach

*η*= 0.95,

_{C}*η*= 0.91, and

_{T}*η*= 0.85 when no side-leakage presents, and

_{F}*η*= 0.78,

_{C}*η*= 0.68, and

_{T}*η*= 0.52 with a leakage rate

_{F}*κ*/

_{s}*κ*= 0.1 (pink solid vertical lines shown in the figure). Besides, these quantum gates can be successfully achieved in the end by some recycling procedures heralded by the clicks on the single-photon detectors, even in a low efficiency situation where the leakage rate is large and the cooperativity is not strong enough.

The fidelities of these gates can in principle reach unity if both the imperfection of the linear optical elements and the dephasing of the NV centers are ignored. In our proposed scheme, all parameters characterizing the imperfect NV centers and the finite line-width of single photons have been incorporated into the global coefficient of the final state, exhibiting good robustness to the local noise and the local fluctuation. Thus, the infidelity due to the imperfect input-output procedure or the not strong enough coupling strength has been transformed into the inefficiency, which can be ultimately controlled. Moreover, the dephasing of the NV centers hardly affects the fidelities of these quantum gates since the short timescale of the input-output process (about 2 ns) [45], the subnanosecond electron-spin manipulation control [57], and the photon coherence time (about 10 ns) are much shorter than the electron-spin coherence time which exceeds 10 ms [54].

In summary, we have proposed a scheme for the robust photonic quantum computation including the CNOT gate, the Toffoli gate, and the Fredkin gate assisted by the NV-cavity platform with the practical input-output procedure. The scheme improves the fidelity of these quantum gates to unity at the cost of nominal inefficiency, showing great robustness to the imperfect input-output process and various sources of practical noise. The coupling strength does not impact the fidelity of the gates, which means the scheme can work well in all coupling strength regimes. The scheme is designed in a compact and heralded style, which greatly lessens the adverse impact of the environmental noise, the local fluctuation, and the finite line-width of the single photon on the fidelity and the efficiency. Thus, the scheme is expected to decrease the operation time, the error probability, and the quantum resource consumption in a large scale integrated quantum circuit. The near-unity fidelity and not-too-low efficiency promises that the scheme is realizable with current technology.

## Funding

National Natural Science Foundation of China (11475021); National Key Basic Research Program of China (2013CB922000).

## Acknowledgments

We thank H. R. Wei, B. C. Ren, and F. G. Deng for the discussion about the calculations in our scheme.

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