Robust universal photonic quantum gates operable with imperfect processes involved in diamond nitrogen-vacancy centers inside low-Q single-sided cavities

Ming Li; Mei Zhang

doi:10.1364/OE.26.033129

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_s = ±1〉) due to the spin-spin interaction [45, 86]. The dynamics including the spin-spin interaction, the spin-orbit interaction, and C₃_v symmetry results in six excited states of which the specific state $| A_{2} 〉 = \frac{1}{\sqrt{2}} (| E_{-} 〉 | + 1 〉 + | E_{+} 〉 | - 1 〉)$ is utilized to implement a Λ-type three-level system shown in Fig. 1b [45, 86, 87]. |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 |A₂〉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.

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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]

\hat{H} = ℏ ω_{0} | A 〉 〈 A | + ℏ ω_{c, L} {\hat{a}}_{L}^{†} {\hat{a}}_{L} + ℏ ω_{c, R} {\hat{a}}_{R}^{†} {\hat{a}}_{R} + ℏ g ({\hat{a}}_{L} {\hat{σ}}_{-}^{†} + {\hat{a}}_{L}^{†} {\hat{σ}}_{-} + {\hat{a}}_{R} {\hat{σ}}_{+}^{†} + {\hat{a}}_{R}^{†} {\hat{σ}}_{+}),

where g represents the Rabi frequency. The level transitions of NV centers with

{\hat{σ}}_{\pm} = | \pm 〉 〈 A_{2} |

and

{\hat{σ}}_{\pm}^{†} = | A_{2} 〉 〈 \pm |

follow the selection rules, that is, a right-circular polarized photon |R〉 can only couple the transition between |A₂〉 and |+〉 while a left-circular polarized photon |L〉 can only couple the other. The cavity field creation operator

{\hat{a}}_{k}^{†} (k = R, L)

and the cavity field annihilation operator

{\hat{a}}_{k}^{-}

creates and annihilates a photon in |k〉 state with cavity frequency ω_c_,_k, respectively.

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

\begin{array}{l} \frac{d \hat{a}}{d t} = - [i (ω_{c} - ω_{p}) + \frac{κ}{2} + \frac{κ_{s}}{2}] \hat{a} - g \hat{σ} - \sqrt{κ} {\hat{a}}_{i n} - \sqrt{κ_{s}} {\hat{s}}_{i n}, \\ \frac{d \hat{σ}}{d t} = - [i (ω_{0} - ω_{p}) + \frac{γ}{2}] \hat{σ} (t) - g {\hat{σ}}_{z} \hat{a} + \sqrt{γ} {\hat{σ}}_{z} \hat{N}, \end{array}

where ω_p, ω₀, and ω_c are the frequencies of the cavity, the single photon, and the transition of NV center, respectively.

{\hat{σ}}_{z} = {\hat{σ}}^{†} \hat{σ} - \hat{σ} {\hat{σ}}^{†}

. Operator

\hat{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

{\hat{a}}_{o u t} = {\hat{a}}_{i n} + \sqrt{κ} \hat{a}

and the Fourier transformation, the reflection coefficient can be expressed analytically in the weak excitation limit

(〈 {\hat{σ}}_{z} 〉 = - 1)

as [20, 90]

r (ω_{p}) = \frac{[i (ω_{c} - ω_{p}) - \frac{κ}{2} + \frac{κ_{s}}{2}] [i (ω_{0} - ω_{p}) + \frac{γ}{2}] + g^{2}}{[i (ω_{c} - ω_{p}) + \frac{κ}{2} + \frac{κ_{s}}{2}] [i (ω_{0} - ω_{p}) + \frac{γ}{2}] + g^{2}} .

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 ω₀ = ω_p = ω_c, the reflection coefficients are reduced to

r_{1} = \frac{(κ_{s} - κ) γ + 4 g^{2}}{(κ_{s} + κ) γ + 4 g^{2}}, r_{0} = \frac{(κ_{s} - κ) γ}{(κ_{s} + κ) γ} .

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 = α₁|R〉₁ + β₁|L〉₁ and |ϕ^p〉₂ = α₂|R〉₂+ β₂|L〉₂, and the NV center is prepared in $| ϕ^{e} 〉 = \frac{1}{\sqrt{2}} (| + 〉 + | - 〉)$ . H_i(i = 1, 2, 3) and H′ are the half wave plates which achieve the rotation, $| R 〉 \to \frac{1}{\sqrt{2}} (| R 〉 + | L 〉)$ , $| L 〉 \to \frac{1}{\sqrt{2}} (| R 〉 - | L 〉)$ and the rotation, $| R 〉 \to \frac{1}{\sqrt{2}} (| R 〉 + | L 〉)$ , $| L 〉 \to \frac{1}{\sqrt{2}} (| L 〉 - | R 〉)$ , respectively. R_θ modifies the shape and the amplitude of the photon passing through. PBS_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 H₂, the NV center, H′ and combines with the modified mode reflected by PBS₁ at PBS₂. If the detector doesn’t click, the system gets into

{| ψ 〉}_{1} = \frac{α_{1}}{\sqrt{2}} {| R 〉}_{1} (| + 〉 - | - 〉) + \frac{β_{1}}{\sqrt{2}} {| L 〉}_{1} (| + 〉 + | - 〉) .

Then, a Hadamard operation H_e, $| + 〉 \to | x_{+} 〉 = \frac{1}{\sqrt{2}} (| + 〉 + | - 〉)$ and $| - 〉 \to | x_{-} 〉 = \frac{1}{\sqrt{2}} (| + 〉 - | - 〉)$ , 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

{| ψ 〉}_{2} = (α_{1} {| R 〉}_{1} | - 〉 + β_{1} {| L 〉}_{1} | + 〉) {| ϕ^{p} 〉}_{2} .

Third, the second photon passes through H₁, PBS₁, PBS₂, and H₃. Provided that the detector does not click, the system becomes

{| ψ 〉}_{3} = - α_{1} {| R 〉}_{1} (α_{2} {| L 〉}_{2} + β_{21} {| R 〉}_{2}) | - 〉 + β_{1} {| L 〉}_{1} (α_{2} {| R 〉}_{2} + β_{2} {| L 〉}_{2}) | + 〉 .

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

\begin{matrix} {| ψ 〉}_{4} = | - 〉 [α_{1} {| R 〉}_{1} (α_{2} {| L 〉}_{2} + β_{2} {| R 〉}_{2}) + β_{1} {| L 〉}_{1} (α_{2} {| R 〉}_{2} + β_{2} {| L 〉}_{2})] \\ + | + 〉 [- α_{1} {| R 〉}_{1} (α_{2} {| L 〉}_{2} + β_{2} {| R 〉}_{2}) + β_{1} {| L 〉}_{1} (α_{2} {| R 〉}_{2} + β_{2} {| L 〉}_{2})] . \end{matrix}

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.

Fig. 2 Schematic of the robust photonic CNOT gate completing a bit-flip operation on photon 2 (the target qubit) when photon 1 (the control qubit) is in the right-circular polarization |R〉. PBS is the circular polarization beam splitter which transmits the right-circular polarized photon and reflects the left-circular polarized photon. H and H′ are half-wave plates which achieve the rotations, $| R 〉 \to \frac{1}{\sqrt{2}} (| R 〉 + | L 〉)$ , $| L 〉 \to \frac{1}{\sqrt{2}} (| R 〉 - | L 〉)$ and the rotations, $| R 〉 \to \frac{1}{\sqrt{2}} (| R 〉 + | L 〉)$ , $| L 〉 \to \frac{1}{\sqrt{2}} (| L 〉 - | R 〉)$ , respectively. R_θ modifies the shape and the intensity of the photon passing through. D is a single-photon detector. SW is an optical switch which controls photons 1 and 2 passing through the system successively.

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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 $| ϕ^{e} 〉 = 1 / \sqrt{2} ({| + 〉}_{i} + {| - 〉}_{i}) (i = 1, 2)$ . First, the control qubits, photon 1 and photon 2, pass through the system, i.e. photon 1 passes through PBS₁, H₁(R_θ₁), NV₁, $H_{1}^{'}$ and PBS₂ and photon 2 passes through PBS₃, H₂ (R_θ₂), NV₂, $H_{2}^{'}$ and PBS₄. If the detectors don’t click, the system, composed of the two control qubits and the two NV centers, evolves into

\begin{matrix} {| φ 〉}_{1} = [α_{1} {| R 〉}_{1} ({| + 〉}_{1} - {| - 〉}_{1}) + β_{1} {| L 〉}_{1} ({| + 〉}_{1} + {| - 〉}_{1})] \\ \otimes [α_{2} {| R 〉}_{2} ({| + 〉}_{2} - {| - 〉}_{2}) + β_{2} {| L 〉}_{2} ({| + 〉}_{2} + {| - 〉}_{2})] . \end{matrix}

Then, a Hadamard operation H_e is performed on the two NV centers respectively. Photon 1 and photon 2 get entangled with NV₁ and NV₂, respectively. It can be detailed as

{| φ 〉}_{2} = (α_{1} {| R 〉}_{1} {| - 〉}_{1} + β_{1} {| L 〉}_{1} {| + 〉}_{1}) (α_{2} {| R 〉}_{2} {| - 〉}_{2} + β_{2} {| L 〉}_{2} {| + 〉}_{2}) .

Third, photon 3 is injected into the system and experiences a Hadamard operation at $H_{3}^{'}$ , the system becomes

\begin{matrix} {| φ 〉}_{3} = \frac{1}{\sqrt{2}} (α_{3} - β_{3}) {| R 〉}_{3} (α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{1} \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) \\ + \frac{1}{\sqrt{2}} (α_{3} + β_{3}) {| L 〉}_{3} (α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{1} \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) . \end{matrix}

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 PBS₅ to the export directly. For the present, we only consider the evolution of the component |τ〉₀, where

\begin{matrix} {| τ 〉}_{0} = {| R 〉}_{3} (α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{2} \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) . \end{matrix}

Fourth, the photon passes through H₃, PBS₆, H₄ (R_θ₃), NV₁, $H_{4}^{'}$ and PBS₇. If the detector doesn’t click, the state evolves into

\begin{matrix} {| τ 〉}_{1} = \frac{1}{\sqrt{2}} (α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{2}) (- {| R 〉}_{3} + {| L 〉}_{3}) \\ + \frac{1}{\sqrt{2}} (β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) ({| R 〉}_{3} + {| L 〉}_{3}) . \end{matrix}

After the photon experiences a Hadamard operation at H₅, the state becomes

\begin{matrix} {| τ 〉}_{2} = - (α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{2}) {| L 〉}_{3} \\ + (β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) {| R 〉}_{3} . \end{matrix}

Fifth, the photon passes through PBS₈, R_θ₄(H₆, NV₂, $H_{5}^{'}$ ), and PBS₉. If the detector does not click, the state becomes

\begin{matrix} {| τ 〉}_{3} = (α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} - α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{2}) {| L 〉}_{3} \\ + (β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) {| R 〉}_{3} . \end{matrix}

Then, after the photon passes through the H₁₀, PBS₁₂, H₉ (R_θ₆), NV₁, $H_{6}^{'}$ , PBS₁₁ and H₈, the state changes into

\begin{matrix} {| τ 〉}_{4} = {| R 〉}_{3} (- α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{2} \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) . \end{matrix}

Seventh, substituting the component |τ〉₄ into Eq. (11). After the reflected mode from PBS₅ is modified by R_θ₅ and combines with the component |τ〉₄ at PBS₁₀ with no detector clicks, the system evolves into

\begin{matrix} {| φ 〉}_{5} = \frac{1}{\sqrt{2}} (α_{3} + β_{3}) {| L 〉}_{3} (α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{1} \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) \\ + \frac{1}{\sqrt{2}} (α_{3} - β_{3}) {| R 〉}_{3} (- α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{1} \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2}) . \end{matrix}

A bit-flip operation at X and a Hadamard operation at H₇ transform the system into

\begin{matrix} {| φ 〉}_{4} = α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} {| - 〉}_{1} {| - 〉}_{2} (α_{3} {| L 〉}_{3} + β_{3} {| R 〉}_{3}) \\ + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} {| - 〉}_{1} {| + 〉}_{2} (α_{3} {| R 〉}_{3} + β_{3} {| L 〉}_{3}) \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} {| + 〉}_{1} {| - 〉}_{2} (α_{3} {| R 〉}_{3} + β_{3} {| L 〉}_{3}) \\ + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} {| + 〉}_{1} {| + 〉}_{2} (α_{3} {| R 〉}_{3} + β_{3} {| L 〉}_{3}) . \end{matrix}

Finally, a Hadamard operation is performed on the two NV centers respectively and a ${\hat{σ}}_{z}$ 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,

\begin{matrix} {| φ 〉}_{5} = α_{1} α_{2} {| R 〉}_{1} {| R 〉}_{2} (α_{3} {| L 〉}_{3} + β_{3} {| R 〉}_{3}) + α_{1} β_{2} {| R 〉}_{1} {| L 〉}_{2} (α_{3} {| R 〉}_{3} + β_{3} {| L 〉}_{3}) \\ + β_{1} α_{2} {| L 〉}_{1} {| R 〉}_{2} (α_{3} {| R 〉}_{3} + β_{3} {| L 〉}_{3}) + β_{1} β_{2} {| L 〉}_{1} {| L 〉}_{2} (α_{3} {| R 〉}_{3} + β_{3} {| L 〉}_{3}) . \end{matrix}

Fig. 3 (a) Schematic of the robust photonic quantum Toffoli gate completing 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〉. X achieves a bit-flip operation on the polarization of a photon passing through. (b) The order of the operations that photons interact with NV centers for Toffoli gate.

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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 NV₁ after photon 3 interacts with it. Then, photon 2 interacts with NV₁, if the detectors don’t click, the system evolves into

\begin{matrix} {| ϑ 〉}_{1} = \frac{1}{\sqrt{2}} {| x_{+} 〉}_{1} [- α_{3} {| R 〉}_{3} (α_{2} {| L 〉}_{2} + β_{2} {| R 〉}_{2}) + β_{3} {| L 〉}_{3} (α_{2} {| R 〉}_{2} + β_{2} {| L 〉}_{2})] {| ϕ^{p} 〉}_{1} \\ + \frac{1}{\sqrt{2}} {| x_{-} 〉}_{1} [α_{3} {| R 〉}_{3} (α_{2} {| L 〉}_{2} + β_{2} {| R 〉}_{2}) + β_{3} {| L 〉}_{3} (α_{2} {| R 〉}_{2} + β_{2} {| L 〉}_{2})] {| ϕ^{p} 〉}_{1} . \end{matrix}

Fig. 4 (a) Schematic of the robust photonic quantum Fredkin gate completing a polarization swap operation between photon 2 and photon 3 (the target qubits) if photon 1 (the control qubit) is in right-circular polarization |R〉. X achieves a bit-flip operation on the polarization of a photon passing through. T is the robust photonic Toffoli gate depicted in Section 4. (b) The order of operation that photons interact with NV centers for Fredkin gate.

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Third, NV₁ is measured in the basis |x_±〉. If NV₁ is measured in |x₋〉, nothing is done on the photon, else, photon 3 should experience a phase operation |R〉 →−|R〉. Here, we assume NV₁ 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

\begin{matrix} | ϑ 〉 = α_{1} {| R 〉}_{1} (α_{2} α_{3} {| L 〉}_{2} {| R 〉}_{3} + β_{2} α_{3} {| R 〉}_{2} {| L 〉}_{3} + α_{2} β_{3} {| R 〉}_{2} {| R 〉}_{3} + β_{2} β_{3} {| L 〉}_{2} {| L 〉}_{3}) | x_{+} 〉 \\ + β_{1} {| L 〉}_{1} (α_{2} α_{3} {| L 〉}_{2} {| R 〉}_{3} + β_{2} α_{3} {| R 〉}_{2} {| R 〉}_{3} + α_{2} β_{3} {| R 〉}_{2} {| L 〉}_{3} + β_{2} β_{3} {| L 〉}_{2} {| L 〉}_{3}) | x_{+} 〉 . \end{matrix}

Fifth, photon 3 and photon 2 pass through NV₁ successively. Before and after photon 2 passes through NV₁, a Hadamard operation is performed on NV₁. If the detectors do not click, the system evolves into

\begin{matrix} | ϑ 〉 = {| - 〉}_{1} {α_{1} {| R 〉}_{1} (α_{2} α_{3} {| R 〉}_{2} {| R 〉}_{3} + β_{2} α_{3} {| R 〉}_{2} {| L 〉}_{3} + α_{2} β_{3} {| L 〉}_{2} {| R 〉}_{3} + β_{2} β_{3} {| L 〉}_{2} {| L 〉}_{3}) \\ + β_{1} {| L 〉}_{1} (α_{2} α_{3} {| R 〉}_{2} {| R 〉}_{3} + β_{2} α_{3} {| L 〉}_{2} {| R 〉}_{3} + α_{2} β_{3} {| R 〉}_{2} {| L 〉}_{3} + β_{2} β_{3} {| L 〉}_{2} {| L 〉}_{3})} \\ + {| + 〉}_{1} {α_{1} {| R 〉}_{1} (- α_{2} α_{3} {| R 〉}_{2} {| R 〉}_{3} + β_{2} α_{3} {| R 〉}_{2} {| L 〉}_{3} - α_{2} β_{3} {| L 〉}_{2} {| R 〉}_{3} + β_{2} β_{3} {| L 〉}_{2} {| L 〉}_{3}) \\ + β_{1} {| L 〉}_{1} (- α_{2} α_{3} {| R 〉}_{2} {| R 〉}_{3} - β_{2} α_{3} {| L 〉}_{2} {| R 〉}_{3} + α_{2} β_{3} {| R 〉}_{2} {| L 〉}_{3} + β_{2} β_{3} {| L 〉}_{2} {| L 〉}_{3})} . \end{matrix}

If the NV₁ is measured on the state |−〉, the Fredkin gate is achieved directly. If NV₁ is measured on the state |+〉, a phase flip operation performed on photon 3 completes the Fredkin gate. Besides, if NV₁ 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 ${\bar{n}}_{output}$ to the input photon number ${\bar{n}}_{input}$ , i.e., $η = {\bar{n}}_{output} / {\bar{n}}_{input}$ . 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 $| T 〉 = (\frac{α}{\sqrt{2}} (| R 〉 + | L 〉) + β | L 〉) | N V 〉$ . If the NV center is in |NV〉 = a|+〉 + b|−〉, the state evolves into $| T 〉 = \frac{α}{\sqrt{2}} [(a r | + 〉 + b r_{0} | - 〉) | R 〉 + (a r_{0} | + 〉 + b r | - 〉) | L 〉] + β | L 〉 | N V 〉$ . Fourth, the photon experiences a Hadamard-like operation at H′, the state becomes $| T 〉 = \frac{α}{2} [(r - r_{0}) (a | + 〉 - b | - 〉) | R 〉 + (r_{0} + r) (a | + 〉 + b | - 〉) | L 〉] + β | L 〉 | N V 〉$ . Another channel, β|L〉 encounters a R_θ, then the state of the system can be rewrote as $| T 〉 = \frac{1}{2} (r - r_{0}) [a | + 〉 (α | R 〉 + β | L 〉) - b | - 〉 (α | R 〉 - β | L 〉)] + \frac{1}{2} (r_{0} + r) β | R 〉 | N V 〉$ . The second term in |T〉 represents the error. The first term in |T〉 is the successful case with a probability of $T = {| \frac{r + r_{0}}{2} |}^{2}$ . Then, the efficiencies η_C, η_T and η_F corresponding to those of our CNOT gate, Toffoli gate and Fredkin gate, respectively, can be derived as

\begin{array}{l} η_{C} = T_{1}, \\ η_{T} = \frac{T_{1} + T_{2} + T_{1}^{2} T_{2}}{3}, \\ η_{F} = \frac{T_{2} + T_{1}^{2} T_{3} + T_{1}^{2} T_{2}^{2} T_{3}}{3}, \end{array}

where

T_{k} = {| \frac{r_{1 k} - r_{0 k}}{2} |}^{2}

represents the survival probability after the photon interacts with the k-th NV center. We evaluate the efficiencies changing vs. the cooperativity

C = \frac{g}{\sqrt{(κ_{s} + κ) γ}}

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_s 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 C and a small leakage rate κ_s/κ are available. With parameters g_ZPL = 2π × 0.3 GHz, κ = 2π × 26 GHz, γ_total = 2π × 13 MHz, γ_ZPL = 2π × 0.4 MHz [92], the cooperativity C can approximately reach 3. With this achievable cooperativity, the efficiencies of these gates can reach η_C = 0.95, η_T = 0.91, and η_F = 0.85 when no side-leakage presents, and η_C = 0.78, η_T = 0.68, and η_F = 0.52 with a leakage rate 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.

Fig. 5 The efficiencies vs. the cooperativity $C = \frac{g}{\sqrt{(κ_{s} - κ) γ}}$ with leak rates κ_s = 0 (the red solid line), κ_s = 0.01κ (the blue dashed line), κ_s = 0.05κ (the brown dash-dot line), and κ_s = 0.10κ (the orange dotted line):(a) the efficiency η_C of the CNOT gate, (b) the efficiency η_T of the Toffoli gate, and (c) the efficiency η_F of the Fredkin gate. The pink solid vertical line indicates the achievable cooperativity C = 3 with current technology.

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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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Robust universal photonic quantum gates operable with imperfect processes involved in diamond nitrogen-vacancy centers inside low-Q single-sided cavities

Abstract

1. Introduction

2. The NV-cavity platform

3. Robust photonic CNOT gate

4. Robust photonic Toffoli gate

5. Robust photonic Fredkin gate

6. Discussion and summary

Funding

Acknowledgments

References

Cited By

Figures (5)

Equations (23)

Optics Express