1. Introduction

Higher-dimensional photon system is of great interest owing to the outstanding features exhibited in higher capacity [1–3], violations of locality [4–6] and resilience [4, 7] to noise. Its applications in the field of high-dimensional quantum information processing (QIP) have attracted much attention recently, e.g., increasing security against eavesdropping [8, 9] and simplify the implementation of quantum logic [10–12]. Meanwhile, the high-dimensional quantum system [13] permits an exponential computation speedup over classical machines which is another hint at the fundamental relevance of the concept. So far, high-dimensional quantum coding has been implemented in various physical system. In photon systems, the time-energy [14] or position-momentum degrees of freedom (DOF) [15, 16], spatial modes [17–19], continuous variables [20], as well as orbital-angular momentum (OAM) DOF [21–25] are all promising information carriers for high-dimensional QIP. In 2014, Krenn et al. managed to create an entanglement of (100×100) dimensions with only two photons [26]. In solid-state systems, energy levels of atoms [27] and artificial atoms [28] which are generally anharmonically spaced can also offer a promising medium for realizing quantum control in large dimensional Hilbert spaces. In 2015, Anderson et al. design and implement unitary maps in a 16-dimensional Hilbert space associated with the 6S_1/2 ground state of ¹³³ Cs [27].

The information of a high-dimensional photon can be stored into a multimode atomic ensembles [29, 30] while the high-dimensional interaction between two atomic ensembles is difficult to achieve in theory and experiment. Currently, many theoretical and experimental works have been devoted to the small-circuit structure for the few-qubit systems, and it has been well solved. However, the case for high-dimensional few-qudit systems is quite complex. High-dimensional quantum logic gate is a key element in scalable quantum computation [31] and quantum communication. Implementing scalable quantum algorithms requires a high level of control over multiple quantum systems. Although any d-dimensional quantum gates can be achieved by decomposing into sequentially implemented single- and two-qubit gates, it requires much longer time and yields lower overall fidelities for the synthesis procedure for this complex quantum circuit. For example, in a 2ⁿ-dimensional quantum system in which a logic qudit à = A₁ ⊗ A₂ ⊗ ... ⊗ A_n (B̃ = B₁ ⊗ B₂ ⊗ ... ⊗ B_n) is constructed with n 2-dimensional physical qubits A₁(B₁), A₂(B₂),... and A_n(B_n), a 2ⁿ-dimensional controlled-flipped (CF) gate operation [32] can be identified by means of the gate operation sequence as

In practice, the photons are often regarded as the best option for flying qubits due to the fast speed, but photonic qubits are hardly scalable [35] on account of the weak interaction between photons. However, by exploiting a hybrid systems with the interaction between photons and solid-state system, the scalability of the photon system can be improved. The quantized whispering-gallery mode (WGM) resonator is required to be of a high-Q factor and a small mode volume [36, 37] and it provides a good platform for studies of hybrid quantum system. By using WGM resonators, such as microtoroidal resonators [38], WGM bottle microresonators [39], microspheres [40] and diamond-GaP microdisks [41], the strong coupling between a atom or an artificial atom and a photon has been experimentally demonstrated. Interestingly, when a WGM microresonator couples to a fiber, its Q factor is surely degraded [38], allowing the photons to be inserted and transmitted on demand through fibers. In experiment, the single-photon input-output process from a microresonator which is coupled to a single atom has been demonstrated [38,39]. Theoretically, similar physics has been shown useful for entanglement generation [42–45], entanglement concentration for the nitrogen-vacancy (NV) centers in decoherence-free subspace [46], universal quantum computation [47–49] and so on.

In this article, we present some deterministic schemes to construct universal quantum gates between high-dimensional single photons and stationary qubits by employing the three-level Λ-type system and WGM microresonator coupled unit. A Λ system is hereafter referred to as an atom, although it can be implemented by artificial atom systems such as nitrogen-vacancy centers [42,46]. In these gates, the Λ-type atom system which works as a logical qudit can perform the CF operation on a d-dimensional single photon in the spatial-mode DOF. Compared with the traditional method in which a d-dimensional quantum gate is implemented by a synthesis procedures of single- and two-qubit gates, in our scheme, the required interaction times between the photon and solid state system reduce greatly, and it is on a shorter temporal scale for the experimental realization. We show that our scheme can be extended to a 2ⁿ-dimensional case and it provides a simpler construction of key quantum circuits enabling a more feasible optical realization for high-dimensional QIP.

The remainder of this article is structured as follows. For the sake of simplicity and clarity, we first explain the physical model of our scheme with the simplest case. In section 2.1, a atom-WGM-waveguide platform is briefly reviewed, and then we exploit this unit to design 4-dimensional and 8-dimensional hybrid CF gate in section 2.2 and 2.3. Subsequently, we show how to extend our scheme to a 2ⁿ-dimensional with the basic Λ-type atom and WGM-waveguide coupled units in section 2.4. Finally, the results and the feasibility are discussed and a summary is given in section 3.

2. The deterministic 2ⁿ-dimensional universal quantum gates using WGM microresonators

Among the multiple DOFs available for high-dimensional quantum information coding, the spatial-mode DOF of photons emerges as a promising alternative as it is more stable against the noise and easily manipulated. The probability that the bit-flip error takes place in the spatial-mode DOF is negligible and it is easy to perform the high-dimensional quantum fast Fourier transform on the single photon in the spatial-mode DOF with only linear optics [50]. Furthermore, based on the OAM-to-polarization coupling effect induced by a modified Mach-Zehnder interferometer [51], the OAM state of a single-photon can be sorted in different paths efficiently in experiment, and with the polarization-independent wavelength division multiplexers (WDMs), photons can be used to guide to different paths according to their frequencies or time-bin states [52]. Therefore, the information encoded in these DOFs can be transformed into the spatial-mode DOF, and one can perform the high-dimensional QIP in these DOFs for photon system with the help of the precise knowledge in high-dimensional spatial-mode DOF QIP.

The basic building block of our high-dimensional hybrid quantum gate is the 4-dimensional (4-d) controlled-X (CX) gate between a 4-dimensional photon in spatial-mode DOF and a atomic qubit of a WGM-waveguide coupled unit. The essence of our architecture is the reflection and the transmission rules of circularly polarized lights interacting with a WGM-waveguide system. Let us first explain our proposal with the basic component step by step.

2.1. Interaction between a polarized photon and a Λ-type atom-WGM resonator coupled system

The basic building block of the present scheme is a Λ-type atom-WGM-waveguide unit, that is, a Λ-type atom fixed on the exterior surface of a WGM resonator side-coupled to four tapered fibers which is shown in Fig. 1. The atom has two degenerate ground states (|± 1〉) and an excited state (|A₂〉). The qubits involved in the atom system are encoded on the states |± 1〉 and the excited one |A₂〉 works as an ancillary state. As demonstrated in Fig. 1, a σ⁻-polarized photon with frequency ω₀ can only induce the transition |+ 1〉 → |A₂〉, where ω₀ is the transition frequency. On the other hand, the σ⁺-polarized photon can only trigger the transition |− 1〉 → |A₂〉. The WGM microresonator interfaced with more than two tapered fibers can be structured as an add-drop system [53, 54], as depicted in Fig. 1. Here, the ports a₀–a₃ are input ports, and ports a′₀–a′₃ are output ports, respectively. In the case where the resonator is strongly coupled with a particle, the resonance frequency of the resonator is modified, thus preventing the buildup of the resonator field. As a result, the photon off-resonant with the resonator is transmitted through the bus waveguide and remains in the initial spatial mode. Otherwise, the input single-photon in port a_j on-resonant with the bare resonator couples inside the resonator and is dropped to the field a′_j+1 (j = 0, 1, 2, 3).

 

Fig. 1 The Λ-type atom is fixed on the exterior surface of the WGM microresonator, which is coupled with four tapered fibers (add-drop structure). The waveguides with the ports a′₀–a′₃ are the bus waveguides and that with ports a₀–a₃ are the drop waveguides, respectively. The effective energy-level diagram of an atom is shown in the dashed frame. The possible cavity-mode-induced transitions are |+ 1〉 → |A₂〉 driven resonantly by absorbing a σ⁻-circular polarized photon and |− 1〉 → |A₂〉 by absorbing a σ⁺-circular polarized photon.

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The Λ-type atom-WGM resonator coupled system exhibits similar features with the Jaynes-Cummings model [55]. Considering an incident single-photon pulse with frequency ω_p from the input port a₁, the Heisenberg equations of motion for the cavity-field annihilation operator a_k driven by the input field a_in and the lowering operator σ₋ of the atom can be described as [39, 42]

To make sure a weak excitation by the single-photon pulse on the Λ-type atom initially prepared in the ground state, we have 〈σ_z〉 = −1 through the whole processing. Based on the input and output relation $a_{\mathit{out}}=a_{\mathit{in}}-\sqrt{k}a$, r(ω_p) = |a_out/a_in| and t(ω_p) = 1 + r(ω_p), where a_out is the output field operator, we can solve the steady-state solution of the reflection and transmission coefficients by setting the time varying part to zero and adiabatically eliminating the cavity mode, expressed as

Under the resonance condition with ω₀ = ω_c = ω_p, the transmission and reflection coefficients t(ω_p) and r(ω_p) for the coupled cavity system can be written as

One of the physical requirements is the access to the WGM microresonator interfaced with tapered fibers. Fortunately, the high Q microtoroid resonators have been demonstrated by Armani et al. in 2003 [57]. Photonic qubits could be coupled into the resonators through the tapered fiber waveguide. Here the tapered fiber waveguides could be fabricated by placing the single-mode fiber on the moveable brackets and heated by the flame of a hydrogen torch. Note that it has been demonstrated that waveguide-to-waveguide power transfer efficiency of 93% is measured and the insertion loss remains less than 0.02% in the optical four port resonant coupler (add-drop geometry), using ultrahigh Q (> 10⁸) toroidal microcavities [53]. Motivated by this technology, our proposed system could be realized by using several tapered fibers coupled with the resonator on different sides, and it has been demonstrated experimentally [39, 53, 58].

2.2. Implementation of the CF gate operations on the 4-dimensional photon and the Λ system

In this section, we show how to build a quantum circuit for 4-dimensional CF gate, in which the stationary qubits in the Λ-type atom 1 (Λ₁) and the Λ-type atom (Λ₂) work as the control qubits, while the 4-dimensional spatial-mode state of the photon A serve as the target qubit, based on the spin-selective optical transition rules of the atom summarized above. Here, |ĩ〉 (i = 0, 1, 2, 3) denotes the 4-dimensional logical quantum state for the 2-electron-spin system composed of Λ₁ and Λ₂, which is correspondingly given by

The 2-dimensional qubit states of the atom controlling the states of a 4-dimensional photon in the spatial-mode DOF flipped operations, such as

The construction of our hybrid 4-dimensional CF gate is shown in Fig. 2. Suppose the flying photon A is prepared in the state |φ_A〉₀ = |σ₋〉(α|a₀〉 + β|a₁〉 + γ|a₂〉 + ξ|a₃〉), where the arbitrary complex parameters α, β, γ and ξ satisfy the relations |α|² + |β|² + |γ|² + |ξ|² = 1. The state of 2 atoms (Λ₁ and Λ₂) is initially prepared in the state |φ_e〉₀ = (α̃|0̃〉 + β̃|1̃〉 + γ̃|2̃〉 + ξ̃|3̃〉)Λ₂Λ₁ and |α̃|² + |β̃|² + |γ̃|² + |ξ̃|² = 1. As illustrated in Fig. 2(a), the incident photon A first flies through a set of half-wave plates(HWPs), which serves as a π-phase-add operation on the |σ₋〉 photon. Then the photon A passes through the 4-dimensional CX gate, and the state of the photon A-Λ₁-Λ₂ system is changed as

 

Fig. 2 Schematic diagram for a 4-dimensional controlled-flip gate (a 4-dimensional CF gate) operating on the 2 atomic qubits and a 4-dimensional single-photon system. The HWP represents a half-wave plate which serves as a π-phase-add operation on the |σ₋〉 photon. c₁, c₂, c₃, and c₄ are four circulators, and the transmission direction of the c₁ and c₄ is clockwise while the transmission direction of the c₂ and c₃ is counter clockwise.

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The quantum circuit shown in Fig. 2(b) can be used to construct the other controlled quantum gate, the 4-d CX² gate, between a 2-dimensional atom (Λ₂) and a 4-dimensional single-photon system. As shown in Fig. 2(b), c₁, c₂, c₃, and c₄ are four circulators, and the transmission direction of the c₁ and c₄ is clockwise while the transmission direction of the c₂ and c₃ is counter clockwise. When a σ⁻ photon is injected into the input-port a′₁ (a′₀, a′₂, or a′₃), if the state of the Λ₂ is |− 1〉, the input σ⁻ photon on-resonant with the bare resonator passes through the circulator c₂ (c₄, c₁, or c₃), couples inside the resonator twice and is dropped to the field a″₃ (a″₂, a″₀, or a″₁) with an acquired phase shift 2ϕ₀; otherwise, if the Λ₂ is |+ 1〉 (in the case where the resonator is strongly coupled with a particle, the resonance frequency of the resonator is modified, thus preventing the buildup of the resonator field), the σ⁻ photon remains in the initial spatial mode with an acquired phase shift ϕ.

Therefore, under the resonance condition and strong-coupling regime where we approximately have ϕ = 0, ϕ₀ = π, the spin-selective optical transition rule of the 4-d CX² quantum gate performed on a NV center and a WGM resonator [shown in Fig. 2(b)] can be summarized as

According to the input-output relation shown in Eq. (11), after the interaction with the Λ₂ atom-WGM resonator coupled system [the 4-d CX² gate shown in Fig. 2(b)], the state of the complicated system A-Λ₁-Λ₂ transforms to

2.3. Implementation of the controlled-flip gate operations on the 8-dimensional photon and the Λ system

In this section, we show how to build a quantum circuit for 8-dimensional CF gate, in which the stationary qubits in the atom 1 (Λ₁), the atom 2 (Λ₂), and the atom 3 (Λ₃) work as the control qubits, while the 8-dimensional spatial-mode state of the photon A serves as the target qudit. Here, |ĩ〉 (i = 0, 1, 2, 3, 4, 5, 6, 7) denotes the 8-dimensional logical quantum state for the 3-qubit system composed of the Λ₁, Λ₂, and Λ₃, correspondingly given by

The architecture of our hybrid 8-dimensional CF gate is demonstrated in Fig. 3. We assume that the flying photon A is prepared in the state |φ_A〉₈ = |σ₋〉 (α₀|a₀〉 + α₁|a₁〉 + α₂|a₂〉 + α₃|a₃〉 + α₄|a₄〉 + α₅|a₅〉 + α₆|a₆〉 + α₇|a₇〉), where the arbitrary complex parameters α_j satisfy the relations ∑_j |α_j|² = 1. The state of 3 atoms is initially prepared in the state |φ_e〉₃ = (β̃₀|0̃〉 + β̃₁|1̃〉 + β̃₂|2̃〉 + β̃₃|3̃〉 + β̃₄|4̃〉 + β̃₅|5̃〉 + β̃₆|6̃〉 + β̃₇|7̃〉)_Λ3Λ2Λ1 and ∑_i |β̃_i|² = 1.

 

Fig. 3 Schematic diagram for a 8-dimensional controlled-flip gate (a 8-d CF gate) operating on the 3-qubit atoms and a 8-dimensional single-photon system. The 50:50 beam splitter (BS) works as the Hadamard gate on the 2-dimensional spatial-mode photonic state.

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The incident photon A will fly through the Λ₁-WGM resonator unit twice as one can see in Fig. 3(a). Here, the a_j is input port and the a′_j is output port for the spatial-mode state |a_j〉 of the photon, respectively. First, after passing through a set of circulators [the transmission direction of the circulators are illustrated in the Fig. 3(a)] and HWPs, the photon A interacts with the Λ₁-WGM resonator unit, and the state of the photon A-Λ₁-Λ₂-Λ₃ system evolves as

After performing the hybrid 8-d CX gate on the photon A and the Λ₁, the spatial-mode component |a_j′〉 of the photon A is injected into the Λ₂-WGM resonator unit from the input port a_j′ as shown in Fig. 3(b), and according to the Eq. (5), when the electron-spin state of Λ₂ is in the state |− 1〉, the spatial-mode states of the photon will evolve as

Finally, after performing the hybrid 8-d CX² gate on the photon A and the Λ₂, the spatial-mode component |a_j″〉 of the photon A is injected into the Λ₃-WGM resonator unit from the input port a_j″ as shown in Fig. 3(c), and according to the Eq. (12), when the state of atom Λ₃ is in the state |− 1〉, the spatial-mode states of the photon will evolve as

2.4. Extending to 2ⁿ-dimensional CF gate operation on the photon and the Λ system

With a good knowledge of 4-dimensional and 8-dimensional CF gate operations in the hybrid system, we are able to extend our scheme to a 2ⁿ-dimensional case with the basic atom and WGM resonator coupled units. A hybrid 2ⁿ-dimensional CF gate operation can be identified by means of the gate operation sequence as

Any 2ⁿ-dimensional ${\text{CX}}_{2^n}^k$ operation optical circuit can easily be constructed by the 8-d CX gate, the 8-d CX² gate or 4-d CX² gate. For example, according to Eq. (12), one can directly exchange the spatial-mode states of a single photon as |a_j〉 ↔ |a_j+2ⁿ⁻¹〉 when the state of the n-th atom is |− 1〉 by repeating 4-d CX² gate for 2ⁿ⁻² times, thus a ${\text{CX}}_{2^n}^{2^{n-1}}$ operation between the n-th atom and the 2ⁿ-dimensional photon can be achieved. Moreover, according to Eq. (20), one can directly exchange the spatial-mode states of a single photon as |a_j〉 → |a_j+2ⁿ⁻²〉 → |a_j+2n−1〉 → |a_{j+3×2ⁿ⁻²}〉 → |a_j〉 when the state of the (n − 1)-th atom is |− 1〉 by repeating 8-d CX gate for 2ⁿ⁻³ times, implementing a ${\text{CX}}_{2^n}^{2^{n-2}}$ operation between an atom and a 2ⁿ-dimensional photon. As for other ${\text{CX}}_{2^n}^k$ operations, the spatial-mode states of a single photon will change as |a_j〉 → |a_j+k〉 → |a_j+2k〉 → .... → |a_{2ⁿ+ j−k}〉 → |a_j〉 when the electron-spin state of Λ_i center is in the state | − 1〉, and the corresponding ${\text{CX}}_{2^n}^k$ operation optical circuit can be recursively assembled out of 8-d CX gates and 8-d CX² gates. For instance, one should repeat the 8-d CX² gate operation for 4 times and perform a 8-d CX gate operation to achieve a ${\text{CX}}_{32}^1$ operation on the atom and a 32-dimensional photon.

3. Discussion and conclusion

As described in Eq. (1), a 2ⁿ-dimensional CF gate can be achieved by decomposing into sequentially implemented single- and two-qubit gates, and the number of two-qubit gates required is ${\sum }_{i=0}^{n-1}(n-i)(2i-1)$. For the case n=2, $U_{\tilde{A}}^{\tilde{B}}(4\times 4)=C_{B_1}^{A_1}\cdot C_{B_2}^{A_2}\cdot T_{B_1}^{A_2{B}_2}$, while in the situation of n = 3, $U_{\tilde{A}}^{\tilde{B}}(8\times 8)=C_{B_1}^{A_1}{C}_{B_2}^{A_2}{C}_{B_3}^{A_3}\cdot T_{B_1}^{A_2{B}_2}{T}_{B_2}^{A_3{B}_3}\cdot T_{B_1}^{A_3{B}_3{B}_2}$. That is, the fundamental requirements for constructing a 4-dimensional CF gate and a 8-dimensional CF gate are 5 times’ 2-qubit interaction and 14 times’ 2-qubit interaction, respectively, by decomposing 4- and 8-dimensional Hilbert space into 2-dimensional quantum space. However, in our scheme, the single photons only need to interact with atoms 2 times or 5 times for separately implementing a 4-dimensional CF gate or a 8-dimensional CF gate operation. To vividly demonstrate the merits of our scheme, we calculate the required interaction times of the present schemes as pictured in Fig. 4. The black curve S1 stands for the interaction time between the high-dimensional photon and atom by decomposing d-dimensional Hilbert space into 2-dimensional quantum space, whereas the line S2 represents that of the our scheme. Apparently, in our scheme, the interaction times of a d-dimensional (d < 128) two-qudit CF quantum gate can be remarkably reduced, especially when d < 64, to less than half the interaction times required in a conventional way. Meanwhile, it is on a shorter temporal scale for the experimental realization. Furthermore, compared with the Eq. (1), the form of Eq. (25) is more simple and easily to extend. In order to improve the dimension of QIP, one only needs to add an atom-WGM resonator coupled unit shown in Fig. 3(b) rather than a complex quantum circuit composed of (n + 1) multi-qubit gating operations. In addition, the order of the interaction between the photon and the atoms can be exchanged arbitrarily in our scheme, but in traditional scheme described in Eq. (1), it must to be implemented in a certain sequence. Obviously, our scheme provides a simpler construction of key quantum circuits enabling a more feasible optical realization for high-dimensional QIP.

 

Fig. 4 The required interaction times between the photon and atoms of the present schemes for constructing a d-dimensional CF gate. The black curve S1 stands for the traditional method in which a d-dimensional quantum gates can be achieved by decomposing into sequentially implemented single- and two-qubit gates, whereas the line S2 represents that of the our scheme.

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The level configuration under our consideration in Fig. 1 can be found in ⁸⁷Rb; for example, the level with F = 1 (e.g., 5²S_1/2 of Rb) acts as the ground state and the excited state could be 5²P_3/2. And the high efficiency switching of optical signals between two optical fibers controlled by a single Rb atom trapped at the center of a bottle cavity has been demonstrated experimentally [39]. Besides the atom, the artificial atoms such as NV centers can also be considered as a good carrier of stationary qubits in our scheme. The NV centers are negatively charged with six electrons from the nitrogen and three carbons surrounding the vacancy. The ground state is a spin triplet with the zero field splitting at 2.87 GHz between levels |0〉 (m_s = 0) and |± 1〉 (m_s = ±1) owing to spin-spin interactions [59]. The six excited states are defined by group theory [60] as |A₁〉 = |E₋〉| + 1〉 − |E₊〉| − 1〉, |A₂〉 = |E₋〉| + 1〉 + |E₊〉| − 1〉, |E₁〉 = |E₋〉| − 1〉 − |E₊〉| + 1〉, |E₂〉 = |E₋〉| − 1〉 + |E₊〉| + 1〉, |E_x〉 = |X〉|0〉, |E_y〉 = |Y〉|0〉, with |E₋〉, |E+〉, |X〉, and |Y〉 being orbital states. The six excited states are eigenstates of the full Hamiltonian including spin-orbit and spin-spin interactions in the absence of any perturbation, such as by an external magnetic field or crystal strain. The spin-orbit interaction splits the pair (A₁, A₂) from the others by at least about 5.5 GHz. The spin-spin interaction increases the energy gap and produces a gap of 3.3 GHz between |A₂〉 and |A₁〉 [42, 60]. Thus, in the limit of low strain, the |A₂〉 state is robust with the stable symmetric properties, and decays to the ground state sublevels |− 1〉 and |+ 1〉, respectively, and, with a low probability, decays nonradiatively to |0〉.

As mentioned above, the key elements of our high-dimensional controlled-flip operation scheme are the 4-dimensional logic gates. To demonstrate our theoretical results, we now propose an experimentally accessible quantum device which uses a diamond NV center coupling to a resonator which have been studied in 2-dimensional universal quantum computation [47]. In experiment, by using microspheres [40] and diamond-GaP microdisks [41], strong coupling between the NV center and WGM resonator has been demonstrated. With its exceeded second-scale storage time [61] and nanosecond-scale manipulating time [62], NV center exhibits good properties as a promising candidate for QIP. By defining the gate fidelity as F = 〈φ_f|φ₀〉, we can obtain the fidelities of the two gate operations. Here |φ₀〉 is the ideal final state, and |φ_f〉 is the finial state in realistic by considering experimental factors. In the resonant case, the evolution of an initial state $|\varphi 〉=\frac{1}{2}{\left(|+1〉+|-1〉\right)}_{N_1}{\left(|+1〉+|-1〉\right)}_{N_2}{|{\sigma }^{-}〉}_A{|a_0〉}_A$ is $|\varphi 〉\to \frac{1}{2}{|{\sigma }^{-}〉}_A\left\{{|-1〉}_{N_1}\left[{|-1〉}_{N_2}{|a_3^{″}〉}_A+{|+1〉}_{N_2}\otimes {\left(t|a_1^{″}〉+r^2|a_3^{″}〉\right)}_A\right]+{|+1〉}_{N_1}\left[{{|-1〉}_{N_2}{\left(t|a_2^{″}〉-r|a_3^{″}〉\right)}_A+{|+1〉}_{N_2}\left(t^2|a_0^{″}〉+t{r}^2|a_2^{″}〉-tr|a_1^{″}〉\right)-r^3|a_3^{″}〉)}_A\right]\right\}$, Thus, the fidelity of the 4-dimensional CF gate operation is

 

Fig. 5 The fidelity and efficiency of our scheme in 4-dimensional case as a function of the parameter $g/\sqrt{\kappa \gamma }$. The blue solid curve represents the fidelity of the 4-dimensional CF gate while the red dashed line corresponds to the efficiency. Here, $g/\sqrt{\kappa \gamma }\ge 0.5$.

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Meanwhile, since the mixing among the excited states of the NV center caused by the strain effect in the diamond [60], we briefly discuss our scheme by considering a negligible energy shift due to a relatively low strain. We suppose that for the N–V center prepared in |+ 1〉, the strain induces an imperfect dipolar transition between the excited state and the state |+ 1〉, and σ⁻ would mix in the transition with ξ² probability due to the involvement of an σ⁻-polarized component [42, 60]. As a result, the fidelity of our scheme in this case solves as $F=\frac{{\left(2-{\xi }^2\right)}^2}{4}$, and we have F = 0.985(0.993) in the case of ξ = 0.12(0.08). The above results manifest that the higher fidelity depends on the smaller spontaneous emission rate of NV center and the higher NV center-resonator coupling strength. The result of our simulation is in good qualitative consistency with the theoretical speculations.

In summary, we present and demonstrate a general technique that harnesses high-dimensional information carriers to implement the deterministic quantum logic gates. The technique that we describe is independent of the physical encoding of quantum information and the way in which the elemental gates have been constructed in Section II. Consequently, it has the potential to be integrated in conjunction with existing technology. Clearly the tools are available to exploit our technique, the benefits of which lie at the increased capacity of single photons for carrying information, enabling a simpler construction of key quantum circuits and a more feasible optical realization for high-dimensional QIP. The performance of our scheme was analyzed in this hybrid system. Our scheme has the ability to map from the atomic state to photonic system for communication across a quantum network, as well as the ability to map from the photonic system to atomic state (atomic systems offer very long coherence times) for information storage after quantum communication [68]. Furthermore, it can also be exploited to implement some essential two-photon quantum operation to connect the two neighboring nodes in a network, such as high-dimensional Bell-state analysis. Thus, the hybird physical system proposed in this paper is a good candidate for realizing high-capacity long-distance quantum communication. Moreover, Our proposed scheme is an efficient way to implement quantum gates allowing high-dimensional quantum computing tasks to be demonstrated on a shorter time scale. It provides a platform for further studies of hybrid quantum information processing and high-capacity quantum communication.