## Abstract

The two or more degrees of freedoms (DOFs) of photon systems are very useful in hyperparallel photonic quantum computing to accomplish more quantum logic gate operations with less resource, and depress photonic dissipation noise in quantum information processing. We present some flexible and adjustable schemes for hybrid hyper-controlled-not (hyper-CNOT) gates assisted by low-*Q* cavities, on the two-photon systems in both the spatial-mode and the polarization DOFs. These hybrid spatial-polarization hyper-CNOT gates consume less quantum resource and are more robust against photonic dissipation noise, compared with the integration of two cascaded CNOT gates in one DOF. Besides, simultaneous counter-propagation of two photons economize extremely the operation time in the whole process of our schemes. Moreover, these quantum logic gates are more feasible for fast quantum operations in the weak-coupling region of the low-*Q* cavities with current experimental technology, which are much different from strong-coupling cases of the high-*Q* ones.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum information processing (QIP) is powerful with fascinating capability of computing, communication and sensing using quantum mechanical principles [1]. Universal two-qubit operations are the key ingredients in quantum computing. In 1995, Barenco *et al.* proved that any natural building block for quantum computing can be achieved by combining controlled-NOT (CNOT) gates and single-qubit rotations [2], therefore, precise control operations on well-defined quantum systems become extremely important [3]. Deterministic CNOT gate has been implemented both in theory and in experiment by using various physical systems, such as nuclear magnetic resonance [4,5], quantum dots [6–8], diamond nitrogen-vacancy (NV) centers [9–11], superconducting circuits [12, 13], and atoms [14–17]. In 1998, Cerf *et al.* presented a CNOT gate with the spatial-mode of the photon as the control qubit and its polarization as the target one [18]. In 2000, Knill *et al.* put forward a linear optical quantum computing scheme with respect to feedback from photon detectors, which was robust against errors from photon loss and invalid detector [19]. To constitute two-qubit optical gate in a deterministic way, one can resort to Kerr nonlinearities. For instance, Duan and Kimble proposed a scheme for constructing a controlled phase-flip (CPF) gate between an atom trapped in a cavity and a single photon [14]. The strong-coupling interaction between the atom and the cavity can provide the giant Kerr nonlinearity. Quantum logic gates between flying photonic qubits and stationary qubits hold a great promise for quantum communication and computing, especially for quantum repeaters, distributed quantum computing, and blind quantum computing.

In a network of quantum computation [19–21], photons are often regarded as the flying
qubits due to flexible manipulation and robustness against the environment
noise. Increasing the information-carrying capacity by utilizing multiple
degrees of freedoms (DOFs) of a single photon provides us an effective way
to improve the realization of the long-distance quantum computing
[22–25]. Recently, the concept for
hyperparallel photonic quantum computing was proposed with universal
quantum operations performed on two or more DOFs of photon systems
[26–29]. For example, exploiting the giant
optical circular birefringence induced by the double-sided
quantum-dot-cavity system, Wang *et al.* constructed a
deterministic hybrid hyper-CNOT gate, in which the spatial-mode and
polarization states of a photon acted as the two control qubits, whereas
two stationary electron spins in quantum dots confined inside the optical
microcavities served as the two target qubits [27]. With hyperparallel photonic quantum
gates, hyperentanglement generation [30–32], hyperentanglement purification
[33–35], and
hyperentanglement concentration [36, 37] can be
accomplished in the relatively simple way, which will improve the channel
capacity of quantum communication and simplify the process of
operation.

Cavity quantum electrodynamics (QED) is a promising physical platform for constructing
universal quantum logic gates, which can enhance the interaction between a
photon and an atom trapped in a cavity [15–17]. In the past decades, many
theoretical and experimental works have been focused on to realize
large-scale QIP [14–16,38–47]. Most of them
relied on the efficient single-photon input-output process in the
high-*Q* cavity and the strong-coupling interaction between
the confined atoms and the high-*Q* cavity field. However,
the high-*Q* cavity in the strong coupled cases, sensitive
to the ratio *g*/*κ* between the
coupling strength *g* and cavity decay rate
*κ*, does not facilitate to carry out the
input-output process of the photons in experiment. Therefore, it is
significant to realize QIP task in the weak-coupling region of the
low-*Q* cavity. In 1995, Turchette *et al.*
completed the measurement of conditional phase shifts for quantum logic in
the intermediate atom-cavity coupling regime with the
low-*Q* cavity [48]. In [49], Dayan *et al.* demonstrated a robust
and efficient mechanism for the regulated transport of photons under the
same condition [48]. In 2009, An *et al.* proposed a
special scheme with higher fidelity by using low-*Q*
cavities to entangle distant atoms by a single photon, generate entangled
photons, and transfer quantum state to a distant qubit [50]. Besides, there are some new
progress in fabrication of microcavities [51,52], and control of microcavities [53,54]. Microcavities also play an important role in sensing
[55–57].

Motivated by recent works on atom-based quantum logic gates relying on the
low-*Q* cavities, we investigate the possibility to
construct hybrid hyper-CNOT gates on the two-photon system in both the
polarization and spatial-mode DOFs, resorting to the cavity-assisted
interaction, the single-photon input-output process, and the readout on
the auxiliary atom as a result of cavity QED. Our proposals have some
advantages. Firstly, it is relatively easier to perform quantum logical
gates on qubits residing in different DOFs of the same photon, which also
reduces the resources consumed based on the hyperparallel feature.
Secondly, simultaneous counterpropagation of two photons economize
extremely the operation time in the whole process of our schemes. Besides,
our gates work in the weak-coupling region of the low-*Q*
cavities, which further lowers the difficulty of the experimental
operation. Moreover, the relatively long coherence time allows multi-time
operations between the photon and the cavity-atom system. Our calculations
show that the average fidelities and efficiencies of our hybrid hyper-CNOT
gates are higher with current experimental technology. These features
maybe make our proposals have potential applications for high-capacity
quantum computation in the future.

## 2. An atomic-cavity interaction system

Here we consider a system that an atom is trapped in a single-sided optical cavity. The
atom is assumed to be a standard three-level ∧-type system as shown
in Fig. 1. The degenerate ground
states of the atom, i.e., |0〉 and |1〉, are
considered to be the qubit states and the excited level
|*e* 〉 to be the ancillary state. The
optically allowed transitions |1〉 ↔
|*e* 〉 (6*S*_{1/2},
*F* = 4, *m* = 4 →
6*P*_{3/2}, *F′* =
5, *m′* = 5 of cesium atom) can only be
excited by the single left (*L*) polarized photon under the
selection rules, while it decouples the transition |0〉
↔ |*e*〉 due to large detuning. The
top of the cavity is perfectly reflective and the bottom is partially
reflective [50].
Under the Jaynes-Commings model, the Hamiltonian of the whole system
composed of a single cavity mode (*L*-polarized) and an
atom can be given by

*a*and

*a*

^{†}are the annihilation and creation operators of the

*L*-polarized cavity mode with the frequency

*ω*, respectively.

_{c}*σ*,

_{z}*σ*

_{+}, and

*σ*

_{−}are the inversion, raising, and lowering operators of the atom, respectively.

*ω*

_{0}is the frequency difference between the ground level |1〉 and the excited level |

*e*〉 of the atom.

*g*is the atom-cavity coupling strength, which is affected by the trapping position of the atom. The reflection coefficient of a single-photon pulse with the frequency

*ω*injected into the optical cavity can be obtained by solving the Heisenberg-Langevin equations of motion for the internal cavity field and the atomic operator in the interaction picture [58]

_{p}*ω*=

_{c}*ω*−

_{c}*ω*and Δ

_{p}*ω*

_{0}=

*ω*

_{0}−

*ω*.

_{p}*κ*and

*γ*are the cavity damping rate and the atomic decay rate, respectively.

*â*(

_{out}*t*) is the output operator. Here the one-dimensional field operator

*â*(

_{in}*t*) is the cavity input operator which satisfies the commutation relation $[{\widehat{a}}_{\mathit{in}}(t),{\widehat{a}}_{\mathit{in}}^{\u2020}({t}^{\prime})=\delta (t-{t}^{\prime})$. The three-level atom can feel the vacuum input field

*b̂*

_{in(t)}with the commutation relation $[{\widehat{b}}_{\mathit{in}}(t),{\widehat{b}}_{\mathit{in}}^{\u2020}({t}^{\prime})]=\delta (t-{t}^{\prime})$.

In fact, since the atom stays in the ground state at most time, a weak excitation by the single-photon pulse (keeping 〈*σ _{z}*〉 = −1) throughout our operation, one can get the input-output optical property of the cavity field [50]

*g*= 0, one get the amplitude of the output pulse

*r*

_{0}(

*ω*) for an uncoupled cavity (or bare cavity) where the atom does not couple to the input filed, Considering the parameters of the atom-cavity system satisfy the resonant condition

_{p}*ω*

_{0}=

*ω*=

_{c}*ω*, the reflection coefficient can be expressed as

_{p}*e*after reflection. Otherwise, if the photon feels a bare cavity, it will obtain a phase shift

^{iφ}*e*

^{iφ0}. Moreover, when the limitation of a low cavity satisfies

*κ*≫

*g*

^{2}/

*κ*≫

*γ*in the atom-cavity intermediate coupling region, phase shifts

*ϕ*= 0 and

*ϕ*

_{0}=

*π*can be achieved. In detail, when the atom is initially prepared in the state |1〉 with the transition |1〉 ↔ |

*e*〉, the

*L*-polarized photon feels a hot cavity and the corresponding output state of the

*L*-polarized photon can be written as

*L*〉 will only sense a bare cavity. As a result, the corresponding output state can be written as

*R*-polarized pulse which remains unchanged due to having no access to the atom-cavity system, we suppose that the reflection coefficient equals 1. Therefore, the evolution rule of the photon in the polarized state |

*L*〉 (or |

*R*〉) and different atomic states can be summarized as |

*L*〉|1〉 → |

*L*〉|1〉, |

*L*〉|0〉 → −|

*L*〉|0〉, |

*R*〉|1〉 → |

*R*〉|1〉, and |

*R*〉|0〉 → |

*R*〉|0〉. Here the conditional phase shift

*π*allows the construction of a universal quantum gate that can be transformed into any two-qubit gate using rotations of the individual qubits, which are implemented with wave plates for the photon and with Raman transitions for the atom.

## 3. Three types of hybrid hyper-CNOT gates

#### 3.1. Hybrid hyper-CNOT gate *I*

The principle of our photonic four-qubit hybrid hyper-CNOT gate *I* is shown in Fig. 2, in which the quantum circuit is constructed with the reflection rule of *L*-polarized photon interacting with an atom-cavity system. Here *Y*_{1} and *Y*_{2} represent two types of interactions. *Y*_{1} shows the interaction between only the *L* polarization photon in spatial mode |*k*_{1}〉 (|*k*_{2}〉, *k* = *a*, *b*) and the atom trapped in the cavity. For *Y*_{2}, both the *L* and *R* components of the photon interact with the atom trapped in the cavity in sequence, in which before and after the interaction, the *R* component will be flipped with the polarization bit-flip operation X $({\sigma}_{{X}_{K}}^{P}={|L\u3009}_{K}\u3008R|+{|R\u3009}_{K}\u3008L|,K=A,B)$.

Suppose that the auxiliary atom *j* in the cavity *j* is initially prepared in the state ${|\phi \u3009}_{j}=\frac{1}{\sqrt{2}}{\left(|0\u3009+|1\u3009\right)}_{j}$ (*j* = 1, 2) and the initial states of the photons *A* and *B* are

*A*in spatial mode |

*a*

_{1}〉 or |

*a*

_{2}〉 propagates from left to right (passing through

*Y*

_{1}-atom 1, BS

_{1},

*Y*

_{2}-atom 2, and BS

_{2}sequentially), while the photon

*B*in spatial mode |

*b*

_{1}〉 or |

*b*

_{2}〉 propagates from right to left (passing through

*Y*

_{1}-atom 2, BS

_{3},

*Y*

_{2}-atom 1, and BS

_{4}sequentially). Due to the interaction between

*Y*

_{1}-atom 1 and the photon

*A*is the same as the interaction between

*Y*

_{1}-atom 2 and the photon

*B*, we use the former as an example to discuss in detail. After the

*Y*

_{1}-atom 1 interacts with the photon

*A*, the joint state |

*φ*〉

_{1}⊗ |

*φ*〉

*of the system composed of the atom 1 and the photon*

_{A}*A*is evolved into

*Y*

_{1}-atom

*j*(

*j*= 1, 2) with the photon

*A*(

*B*) can be viewed as the polarization-controlled-Z gate (P-CZ gate). After performing the Hadamard operation on the atom 1 applying a

*π*/2-pulse (that is, $|1\u3009\to \frac{1}{\sqrt{2}}(|1\u3009+|0\u3009)$ and $|0\u3009\to \frac{1}{\sqrt{2}}(|1\u3009-|0\u3009)$ in the basis {|1〉, |0〉}), and the Hadamard operation on the spatial-mode DOF of the photon

*A*(namely, $|{a}_{1}\u3009\to \frac{1}{\sqrt{2}}(|{a}_{1}\u3009+|{a}_{2}\u3009)$ and $|{a}_{2}\u3009\to \frac{1}{\sqrt{2}}(|{a}_{1}\u3009-|{a}_{2}\u3009)$) using BS

_{1}, the state |Ψ〉

_{1A}in Eq. (9) becomes

*B*interacts with

*Y*

_{1}-atom 2, and immediately the Hadamard operations are performed on the atom 2 and the spatial-mode DOF of the photon

*B*, the complex state |

*ϕ*〉

*⊗ |*

_{B}*ϕ*〉

_{2}becomes

Subsequently, the photon *A* interacts with the *Y*_{2}-atom 2, meanwhile the photon *B* interacts with the *Y*_{2}-atom 1. The complex state |Ψ′〉_{1A} ⊗ |Ψ′〉_{2B} of the system composed of the two photons (*A* and *B*) and two atoms (1 and 2) is evolved into

*Y*

_{2}-atom 2 and the photon

*A*(

*Y*

_{2}-atom 1 and the photon

*B*) can be viewed as the spatial-mode controlled-Z gate (S-CZ gate). Here DL makes the photon

*A*(

*B*) in spatial mode

*a*

_{1}(

*b*

_{1}) and

*a*

_{2}(

*b*

_{2}) reach the second BS

_{2}(BS

_{4}) simultaneously. That is, fiber loops to storage the photon for enough long time needed by the interaction between the single photon and the atom. After performing the Hadamard operations on the atom

*j*in the cavity

*j*(

*j*= 1, 2) and the Hadamard operations on the spatial-mode DOF of the photon

*A*(

*B*) using BS

_{2}(BS

_{4}), the state |Ψ〉

_{12AB}becomes

Finally, after measuring the auxiliary atoms 1 and 2 under the orthogonal basis {|1〉, |0〉}, if two auxiliary atoms are both in the basis |1〉, the joint system of the two photons *A* and *B* collapses into

*A*(

*B*) when the outcome |0〉 of the auxiliary atom 1 (atom 2), the hybrid hyper-CNOT gate

*I*is completed.

In a word, the quantum circuit shown in Fig. 2 can be used to accomplish the deterministic hybrid hyper-CNOT gate *I* on the two photons *A* and *B*. In detail, the polarization state of the photon *A* is the control qubit when the spatial-mode state of the photon *B* is the target qubit, meanwhile, the polarization state of the photon *B* is the control qubit when the spatial-mode state of the photon *A* is the target qubit. That is, the polarization states of the photons *A* and *B* simultaneously control the spatial-mode states of the photons *B* and *A*, respectively.

#### 3.2. Hybrid hyper-CNOT gate *II*

The quantum circuit of the hybrid hyper-CNOT gate *II* in Fig. 3(a) is similar to the one of the hybrid hyper-CNOT gate *I* in Fig. 2 in many ways. In Fig. 3(a), the Hadamard operation is performed on the polarization DOF rather than the spatial-mode DOF of the photons *A* and *B* by *BS*, which is substituted by *H*, that is, $|R\u3009\to \frac{1}{\sqrt{2}}(|R\u3009+|L\u3009)$, and $|L\u3009\to \frac{1}{\sqrt{2}}(|R\u3009-|L\u3009)$. Besides, the interaction between *Y*_{2}-atom 1 (*Y*_{2}-atom 2) and the photon *A* (*B*) and the interaction between *Y*_{1}-atom 2 (*Y*_{1}-atom 1) and the photon *A* (*B*) occur in turn, which leads to realizing S-CZ gate and the P-CZ gate sequentially.

Suppose that the initial states of the photons *A* and *B* are shown in Eq. (7). Moreover, |*φ*〉* _{j}* is the initial state of the auxiliary atom

*j*in the cavity

*j*(

*j*= 1, 2). In Fig. 3(a), the photon

*A*in two spatial modes |

*a*

_{1}〉 and |

*a*

_{2}〉 passes through

*Y*

_{2}-atom 1,

*H*

_{1},

*Y*

_{1}-atom 2, and

*H*

_{2}sequentially, while the photon

*B*in the two spatial modes |

*b*

_{1}〉 and |

*b*

_{2}〉 passes through

*Y*

_{2}-atom 2,

*H*

_{3},

*Y*

_{1}-atom 1, and

*H*

_{4}sequentially. After the interaction

*Y*(

_{i}*i*= 1, 2)-atom

*j*(

*j*= 1, 2) with the photon

*A*(

*B*), one can perform the Hadamard operation on the atom

*j*in the cavity

*j*(

*j*= 1, 2) every time. The DL guarantees the photon

*A*(

*B*) in its own two spatial modes |

*a*

_{1}〉 (|

*b*

_{1}〉) and |

*a*

_{2}〉 (|

*b*

_{2}〉) to simultaneously reach

*H*

_{2}(

*H*

_{4}).

Similar to the hybrid hyper-CNOT gate *I*, the hybrid hyper-CNOT gate *II* can be achieved by measuring the states of auxiliary atoms 1 and 2 in the orthogonal basis {|1〉, |0〉} and making some feed-forward operations. In detail, if both the auxiliary atoms 1 and 2 are in the state |0〉, the spatial-mode phase-flip operation ${\sigma}_{{Z}_{K}}^{S}=|{k}_{1}\u3009\u3008{k}_{1}|-|{k}_{2}\u3009\u3008{k}_{2}|(k=a,b;\hspace{0.17em}K=A,B)$ on the spatial-mode DOF of the photons *A* and *B* are performed, respectively. Otherwise, if two auxiliary atoms are both in the basis |1〉, the result of the hybrid hyper-CNOT gate *II* is obtained as

In a word, the spatial-mode states of the photons *A* and *B* simultaneously control the polarization states of the photons *B* and *A*, respectively. It is self-evident that the control qubits swap with the target qubits in both types *I* and *II*.

#### 3.3. Hybrid hyper-CNOT gate *III* and hybrid hyper-CNOT^{N} gate

^{N}

The hybrid hyper-CNOT gate *III* in Fig. 3(b) is used to complete the task in which the polarization state and the spatial-mode state of the photon *A* simultaneously control the spatial-mode state and the polarization state of the photon *B*, respectively. The initial states of the two photons *A* and *B* and two auxiliary atoms *j* in the cavity *j*(*j* = 1, 2) are the same as the one of the hybrid hyper-CNOT gate *I* and *II*. In the first step, let the photon *A* get through *Y*_{1}-atom 1 and *Y*_{2}-atom 2 sequentially. After the *Y*_{1}-atom 1 and *Y*_{2}-atom 2 interact with the photon *A*, the atoms 1 and 2 are performed the Hadamard operation. Subsequently, let the photon *B* get through *Y*_{2}-atom 2, BS_{5}, *H*_{5}, *Y*_{1}-atom 1, *Y*_{2}, *H*_{5}, and BS_{6} sequentially. Then one continues to perform the Hadamard operation on the atoms 1 and 2 again after the interactions *Y*_{2}-atom 2 and *Y*_{1}-atom 1 with the photon *B*. Eventually, one measures the states of auxiliary atoms 1 and 2 in the orthogonal basis {|1〉, |0〉}, and makes the feed-forward operations, similar to the discussion of the hybrid hyper-CNOT gate *I* and *II*. If the auxiliary atom 1 (atom 2) is in the state |0〉, the polarization (spatial-mode) phase-flip operation ${\sigma}_{{Z}_{A}}^{P}({\sigma}_{{Z}_{A}}^{S})$ is performed on the polarization (spatial-mode) DOF of the photon *A*. Therefore, the hybrid hyper-CNOT gate *III* is obtained as

The hybrid hyper-CNOT gate *III* is different from the hybrid hyper-CNOT gate *I* and *II*, because the photons *A* and *B* do not simultaneously interact with *Y*_{1}-atom 1 and *Y*_{2}-atom 2, which means the photon *B* goes through the quantum circuit after the photon *A* interacts with the atom-cavity (1 or 2). Moreover, our hybrid hyper-CNOT gate *III* can be easily extended to establish the multi-qubit hybrid hyper-CNOT* ^{N}* gate, in which one photon as control qubit and the rest

*N*-photon as target qubits. The stationary-qubit systems [59,60] have been used widely to study the CNOT

*gate. We will describe the procedure of the multi-qubit hybrid hyper-CNOT*

^{N}*gate below, which useful to realize quantum algorithms [61], correct errors [62], and prepare the hyperentanglement for*

^{N}*N*photons [63].

Suppose that the initial states of the control photon *A* and two auxiliary atoms *j* in the cavity *j* (*j* = 1, 2) remain |*φ*〉* _{A}* and |

*φ*〉

*(*

_{j}*j*= 1, 2), respectively, and the

*N*target photons

*B*

_{1},

*B*

_{2}, · · · , and

*B*are in the superposition state of the form ${|\varphi \u3009}_{{B}_{m}}={|\varphi \u3009}_{{B}_{m}}^{P}\otimes {|\varphi \u3009}_{{B}_{m}}^{S}(m=1,2\dots )$, where ${|\varphi \u3009}_{{B}_{m}}^{P}=\text{sin}{\theta}_{1}^{m}{|R\u3009}_{{B}_{m}}+\text{cos}{\theta}_{1}^{m}{|L\u3009}_{{B}_{m}}$ and ${|\varphi \u3009}_{{B}_{m}}^{S}=\text{sin}{\theta}_{2}^{m}|{b}_{1}^{m}\u3009+\text{cos}{\theta}_{2}^{m}|{b}_{2}^{m}\u3009$. To implement the hybrid hyper-CNOT

_{N}*gate, we could use the quantum circuit depicted by Fig. 3(b) by substituting the target photon*

^{N}*B*in the original hybrid hyper-CNOT gate

*III*with the photon string

*B*

_{1},

*B*

_{2}, · · ·, and

*B*. Besides, the target photon

_{N}*B*

_{m+1}should be input into the cavity in turn after

*B*goes away the cavity. Meanwhile, the time delay between each two target photons

_{m}*B*and

_{m}*B*

_{m+1}could be utilized to discriminate their spatial modes. To be detail, after the control photon

*A*interacts with the

*Y*

_{1}-atom 1 and

*Y*

_{2}-atom 2 in turn, the atoms 1 and 2 are performed the Hadamard operations. Subsequently, let the first target photon

*B*

_{1}with its spatial modes ${b}_{1}^{1}$ and ${b}_{2}^{1}$ get through

*Y*

_{2}-atom 2, BS

_{5}, H

_{5},

*Y*

_{1}-atom 1,

*Y*

_{2}, H

_{5}, and BS

_{6}sequentially, similar to the target photon

*B*of the hybrid hyper-CNOT gate

*III*. After the photon

*B*

_{1}passes though the setup and propagates into the output modes ${b}_{1}^{1}$ and ${b}_{2}^{1}$, let the target photon

*B*after the photon

_{m}*B*

_{m−1}be emitted by the cavity with the similar procedure. After all the

*N*target photons pass though the setup, the Hadamard operations on the two atoms 1 and 2 are performed. To decouple the two atoms, the measurements on the atoms are performed. When the outcomes are both the states |0〉, the (

*N*+ 1)-photon system is projected into the state ${|\mathrm{\Psi}\u3009}_{\mathit{III}}^{N}$. Here

*B*involved, respectively. That is, the multi-qubit hybrid hyper-CNOT

_{m}*gates in double DOFs on the*

^{N}*N*+ 1 photons are achievable after successively operating the control photon

*A*and target photon

*B*

_{1},

*B*

_{2}, · · · , and

*B*.

_{N}## 4. Discussion and conclusion

Our hybrid hyper-CNOT gates work in an intermediate coupling region, which is considerably different from the strong coupling cases [14–17]. There are some technical challenges to realize the strong coupling strength in experiment, compared with intermediate coupling strength of the atom-cavity system [48, 49, 64]. Moreover, in order to obtain shorter operation time, it is necessary to achieve the atom-cavity photon scattering with a bad cavity in experiment. In [48], Turchette *et al.* made a measurement on the conditional phase shifts for quantum logic with the experimental parameters (*g*, *κ*, *γ*)/2*π* ∼ (20, 75, 2.5) MHz, which satisfy the limitation of a bad cavity *κ* ≫ *g*^{2}/*κ* ≫ *γ* exactly in an intermediate coupling region (*g* = 0.27*κ*). In [49], Dayan *et al.* presented an intermediate atom-cavity coupling in experiment, where a Cs atom is trapped in a microtoroidal resonator. They provided a set of experimental parameters (*g*, *κ*, *γ*)/2*π* ∼ (70, 165 ± 15, 2.6) MHz with the atom-cavity detuning *ω*_{0} = *ω _{c}*, which still satisfied the requirements of a bad cavity. The

*Q*value in [49] is more than 10

^{4}in the low-

*Q*cavity limit, which is also very suitable to our protocol requiring the

*Q*value of larger than 10

^{2}. Besides, it should be noted that that although the

*Q*can be low, a high

*Q*/

*V*is usually required in cavity QED systems, in which

*V*is the mode volume. In [64], Tiecke

*et al.*experimentally demonstrated a quantum optical switch, in which a single atom switched the phase of a photon and a single photon modified the atom’s phase, by coupling a photon to a single atom trapped in the near field of a nanoscale photonic crystal cavity attached to an optical fibre taper. The atom is trapped about 200 nm from the surface in an optical lattice formed by the interference of an optical tweezer and its reflection from the side of the cavity. The atom utilizing a short (3 ns) pulse of light was excited in the optical trap and resonant with the |5

*S*

_{1/2},

*F*= 2〉 → |5

*P*

_{2/3},

*F′*= 3〉 transition (near 780 nm) accompanied by the experimental parameters (2

*g*,

*κ*,

*γ*)/2

*π*∼ (1.096 ± 0.03, 25, 0.006) GHz under the limitation of a low cavity.

When the photon interacts with the atom-cavity system, the incident photon may be inevitably lost. Therefore, it becomes particularly important to consider the efficiencies of the hybrid hyper-CNOT gates, *η* = *η _{output}*/

*η*, which is defined as the yield of the incident photon, that is, the ratio that takes into account these gates’ output photon number

_{input}*η*and input photon number

_{output}*η*. Besides, all the discussions for the constructions of the hybrid hyper-CNOT gates are in the ideal case

_{input}*r*(

*ω*) ∼ 1 and

_{p}*r*

_{0}(

*ω*) = −1. The imperfection in phase and amplitude of the reflection photons reduces the performance of our gates, so it is necessary to consider the feasibilities of our gates, which can be evaluated by the fidelity defined as the overlap of the output states of the system in the ideal case |

_{p}*ψ*〉 and the real case |

_{ideal}*ψ*〉,

_{real}*F*= |〈

*ψ*|

_{ideal}*ψ*〉|

_{real}^{2}. Taking the hybrid hyper-CNOT gate

*I*as an example, where |

*ψ*〉 can be described by in Eq. (14), and the corresponding output state |

_{ideal}*ψ*〉 can be obtain by substituting the real optical transition rules |

_{real}*L*〉|1〉 →

*r*(

*ω*) |

_{p}*L*〉|1〉, and |

*L*〉|0〉 →

*r*

_{0}(

*ω*)|

_{p}*L*〉|0〉 for the ideal case |

*ψ*〉 during the evolution of the whole system. Here,

_{ideal}*ξ*

_{1}=

*ξ*

_{4}= 2+2

*r*,

*ξ*

_{2}=

*ξ*

_{3}= 2−2

*r*, ${\xi}_{5}={\xi}_{8}=2r+2r{r}_{0}+{r}^{2}-{r}_{0}^{2}$, ${\xi}_{6}={\xi}_{7}=2r-2r{r}_{0}-{r}^{2}+{r}_{0}^{2}$,

*λ*

_{1}=

*λ*

_{2}= sin

*α*,

_{i}*λ*

_{3}=

*λ*

_{4}= cos

*α*,

_{i}*ζ*

_{1}= sin

*β*sin

_{j}*α*,

_{i}*ζ*

_{2}= cos

*β*sin

_{j}*α*,

_{i}*ζ*

_{3}= cos

*β*cos

_{j}*α*, and

_{i}*ζ*

_{4}= sin

*β*cos

_{j}*α*(

_{i}*i*≠

*j*∈ {1, 2}). In order to make our discussion more practical and general, we use the average efficiency defined as ${\overline{\eta}}_{C1}=\frac{1}{{\left(2\pi \right)}^{4}}{\int}_{0}^{2\pi}d{\alpha}_{1}{\int}_{0}^{2\pi}d{\beta}_{1}{\int}_{0}^{2\pi}d{\alpha}_{2}{\int}_{0}^{2\pi}d{\beta}_{2}{\eta}_{C1}$ and the average fidelity defined as ${\overline{F}}_{C1}=\frac{1}{{\left(2\pi \right)}^{4}}{\int}_{0}^{2\pi}d{\alpha}_{1}{\int}_{0}^{2\pi}d{\beta}_{1}{\int}_{0}^{2\pi}d{\alpha}_{2}{\int}_{0}^{2\pi}d{\beta}_{2}{F}_{C1}$ to characterize the performance of our hybrid hyper-CNOT gate

*I*. The fidelities

*F*

_{C2}(

*F*

_{C3}) of the hybrid hyper-CNOT gates

*II*(

*III*) is similar to the one

*F*

_{C1}of the hybrid hyper-CNOT gate

*I*, and can be obtained by transforming the positions of

*α*

_{1},

*α*

_{2},

*β*

_{1}, and

*β*

_{2}in Eq. (18), so they possess the same average fidelity. Therefore, we adopt the average fidelity

*F̄*

_{C1}to display the feature of three kinds of hybrid hyper-CNOT gates. So does the following discussion of the average efficiency of the hybrid hyper-CNOT gates. Obviously, the average fidelity and efficiency of our hybrid hyper-CNOT gates are assisted by the nonlinear interaction between the photon and the auxiliary atom in the cavity shown in Figs. 4(a) and 4(b), respectively. Based on the experimental parameters mentioned in [48], the average fidelity and efficiency of our hybrid hyper-CNOT gate are

*F̄*

_{C1}= 0.9943, and

*η̄*

_{C1}= 0.9061, respectively. The average fidelity and efficiency of our hybrid hyper-CNOT gate are improved to

*F̄*

_{C1}= 0.9989 and

*η̄*

_{C1}= 0.9478 under the condition [49], respectively. The average fidelity and efficiency of our hybrid hyper-CNOT gate become

*F̄*

_{C1}= 0.9999 and

*η̄*

_{C1}= 0.9898 in the case of [64], respectively. The results above show that the average fidelities and efficiencies of our hybrid hyper-CNOT gates can acquire higher values with low-

*Q*cavities.

The mechanism of our hybrid hyper-CNOT gates are in principle determinated in experiment if the unavoidable effects of photon loss, non-ideal single-photon sources, imperfect linear-optical elements (BSs and CPBSs), and invalid dark detectors atomic detections are not taken into account. These imperfections reduce the efficiencies and fidelities of our schemes. An ideal single-photon source should be work efficiently, controlled, exactly, but no single-photon sources can meet these requirements with the current technology. However, these technical imperfections will be improved with the further development of linear optical elements.

In this paper, the hybrid hyper-CNOT gate *I* (*II*) shows that the polarization (spatial-mode) states of the photons *A* and *B* simultaneously control the spatial-mode (polarization) states of the photons *B* and *A*, respectively. It is self-evident that the control qubits of the hybrid hyper-CNOT gate *I* are changed into the target qubits of the hybrid hyper-CNOT gate *II* and vice versa. Two photons *A* and *B* simultaneously interact with atom-cavity system in the opposite direction, which means shorter operation time required in the whole process of our schemes. Moreover, the hybrid hyper-CNOT gate *III* shows that the polarization state and the spatial-mode state of the photon *A* simultaneously control the spatial-mode state and the polarization state of the photon *B*, respectively. Meanwhile, it could be directly generalized into the multi-qubit hyper-CNOT* ^{N}* gate, which is of great importance when performing the hyperentanglement preparation and redundant hyperencoding procedure [65,66], and could find its potential application in the memory-less quantum communication [67, 68] in the hyper-parallel quantum network. In addition, our multi-qubit hybrid hyper-CNOT

*gate needs only two auxiliary atoms, which makes our scheme easier, compared with the multi-qubit gate constituted with two-qubit gates and single-qubit gates [2]. These hybrid hyper-CNOT gates can be used to set up some interesting devices for high-capacity direct transmission of the information from one quantum communication node to another with a polynomial gain for the efficiency of distributed QIP. Compared with other protocols with input-output process [14–17], this protocol does not need the confined atom strong coupled to a high-*

^{N}*Q*cavity, and extends the earlier protocols with single-photon to continuous variable regime, which could greatly relax the experimental requirement. Therefore, these hybrid spatial-polarization hyper-CNOT gates are more robust against photonic dissipation noise.

In conclusion, we have constructed three types of flexible hybrid hyper-CNOT gates assisted by low-*Q* cavities and investigated the possibility of parallel quantum computation without utilizing extra spatial modes or polarization modes. The hybrid spatial-polarization hyper-CNOT gates consume less quantum resource and are more robust against photonic dissipation noise, compared with the integration of two cascaded CNOT gates in one DOF. In contrast to the hybrid hyper-CNOT gate on the one photon and the stationary electron spins in quantum dots [27], our hybrid hyper-CNOT gates are ultimately realized on the two photon systems, and the atoms of the low-*Q* cavities are only the auxiliary resource. Besides, the operation time of our hybrid hyper-CNOT gates are economized extremely in the whole process of our schemes due to simultaneous counter-propagation of two photons, which are much different from previous hyper-CNOT gate [26]. Moreover, these gates are performed with cavity-assisted photon scattering in the intermediate coupling region, possessing relatively long coherence time, which are much different from strong-coupling cases of the high-*Q* ones. Therefore, it is more feasible to realize not only fast quantum operations, but also multi-time operations between the photon and the cavity-atom system. In addition, our calculations show that the average fidelities and efficiencies of our gates are high with current experimental technology.

## Funding

National Natural Science Foundation of China under Grant (11747024, 51635011, 61571406, 61704158); in part by the Shanxi “1331 Project ” Key Subjects Construction.

## References

**1. **M. A. Nielsen and I. Chuang, “Quantum computation and quantum information,” (2002).

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

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

**4. **G. R. Feng, G. F. Xu, and G. L. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. **110**, 190501 (2013). [CrossRef] [PubMed]

**5. **T. Xin, J. S. Pedernales, E. Solano, and G. L. Long, “Entanglement measures in embedding quantum simulators with nuclear spins,” Phys. Rev. A **97**, 022322 (2018). [CrossRef]

**6. **X. Q. Li, Y. W. Wu, D. C. Steel, D. Gammon, T. Stievater, D. Katzer, D. Park, C. Piermarocchi, and L. Sham, “An all-optical quantum gate in a semiconductor quantum dot,” Science **301**, 809–811 (2003). [CrossRef] [PubMed]

**7. **C. Bonato, F. Haupt, S. S. Oemrawsingh, J. Gudat, D. Ding, M. P. van Exter, and D. Bouwmeester, “Cnot and bell-state analysis in the weak-coupling cavity qed regime,” Phys. Rev. Lett. **104**, 160503 (2010). [CrossRef] [PubMed]

**8. **H. R. Wei and F. G. Deng, “Universal quantum gates for hybrid systems assisted by quantum dots inside double-sided optical microcavities,” Phys. Rev. A **87**, 022305 (2013). [CrossRef]

**9. **C. Wang, Y. Zhang, R. Z. Jiao, and G. S. Jin, “Universal quantum controlled phase gate on photonic qubits based on nitrogen vacancy centers and microcavity resonators,” Opt. Express **21**, 19252–19260 (2013). [CrossRef] [PubMed]

**10. **H. R. Wei and F. G. Deng, “Compact quantum gates on electron-spin qubits assisted by diamond nitrogen-vacancy centers inside cavities,” Phys. Rev. A **88**, 042323 (2013). [CrossRef]

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

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

**13. **J. M. Chow, J. M. Gambetta, A. Córcoles, S. T. Merkel, J. A. Smolin, C. Rigetti, S. Poletto, G. A. Keefe, M. B. Rothwell, and J. Rozen, “Universal quantum gate set approaching fault-tolerant thresholds with superconducting qubits,” Phys. Rev. Lett. **109**, 060501 (2012). [CrossRef] [PubMed]

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

**15. **L. Isenhower, E. Urban, X. Zhang, A. Gill, T. Henage, T. A. Johnson, T. Walker, and M. Saffman, “Demonstration of a neutral atom controlled-not quantum gate,” Phys. Rev. Lett. **104**, 010503 (2010). [CrossRef] [PubMed]

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

**17. **B. Hacker, S. Welte, G. Rempe, and S. Ritter, “A photon-photon quantum gate based on a single atom in an optical resonator,” Nature **536**, 193 (2016). [CrossRef] [PubMed]

**18. **N. J. Cerf, C. Adami, and P. G. Kwiat, “Optical simulation of quantum logic,” Phys. Rev. A **57**, R1477 (1998). [CrossRef]

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

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

**21. **Y. B. Sheng and L. Zhou, “Blind quantum computation with a noise channel,” Phys. Rev. A **98**, 052343 (2018). [CrossRef]

**22. **E. T. Campbell, “Enhanced fault-tolerant quantum computing in d-level systems,” Phys. Rev. Lett. **113**, 230501 (2014). [CrossRef] [PubMed]

**23. **D. Zhou, B. Zeng, Z. Xu, and C. Sun, “Quantum computation based on d-level cluster state,” Phys. Rev. A **68**, 062303 (2003). [CrossRef]

**24. **C. Wang, F. G. Deng, Y. S. Li, X. S. Liu, and G. L. Long, “Quantum secure direct communication with high-dimension quantum superdense coding,” Phys. Rev. A **71**, 044305 (2005). [CrossRef]

**25. **Z. Yang, O. S. Magaña Loaiza, M. Mirhosseini, Y. Zhou, B. Gao, L. Gao, S. M. H. Rafsanjani, G. L. Long, and R. W. Boyd, “Digital spiral object identification using random light,” Light. Sci. & Appl. **6**, e17013 (2017). [CrossRef]

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

**27. **T. J. Wang, Y. Zhang, and C. Wang, “Universal hybrid hyper-controlled quantum gates assisted by quantum dots in optical double-sided microcavities,” Laser Phys. Lett. **11**, 025203 (2014). [CrossRef]

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

**29. **G. Y. Wang, T. Li, Q. Ai, and F. G. Deng, “Self-error-corrected hyperparallel photonic quantum computation working with both the polarization and the spatial-mode degrees of freedom,” Opt. Express **26**, 23333–23346 (2018). [CrossRef] [PubMed]

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

**31. **G. Y. Wang, Q. Ai, B. C. Ren, T. Li, and F. G. Deng, “Error-detected generation and complete analysis of hyperentangled bell states for photons assisted by quantum-dot spins in double-sided optical microcavities,” Opt. Express **24**, 28444–28458 (2016). [CrossRef] [PubMed]

**32. **G. Y. Wang, B. C. Ren, F. G. Deng, and G. L. Long, “Complete analysis of hyperentangled bell states assisted with auxiliary hyperentanglement,” Opt. Express **27**, 8994–9003 (2019). [CrossRef] [PubMed]

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

**34. **F. F. Du, T. Li, and G. L. Long, “Refined hyperentanglement purification of two-photon systems for high-capacity quantum communication with cavity-assisted interaction,” Annals Phys. **375**, 105–118 (2016). [CrossRef]

**35. **G. Y. Wang, T. Li, Q. Ai, A. Alsaedi, T. Hayat, and F. G. Deng, “Faithful entanglement purification for high-capacity quantum communication with two-photon four-qubit systems,” Phys. Rev. Appl. **10**, 054058 (2018). [CrossRef]

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

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

**38. **J. M. Raimond, M. Brune, and S. Haroche, “Manipulating quantum entanglement with atoms and photons in a cavity,” Rev. Mod. Phys. **73**, 565 (2001). [CrossRef]

**39. **J. Beugnon, M. P. Jones, J. Dingjan, B. Darquié, G. Messin, A. Browaeys, and P. Grangier, “Quantum interference between two single photons emitted by independently trapped atoms,” Nature **440**, 779 (2006). [CrossRef] [PubMed]

**40. **M. Hijlkema, B. Weber, H. P. Specht, S. C. Webster, A. Kuhn, and G. Rempe, “A single-photon server with just one atom,” Nat. Phys. **3**, 253 (2007). [CrossRef]

**41. **K. M. Fortier, S. Y. Kim, M. J. Gibbons, P. Ahmadi, and M. S. Chapman, “Deterministic loading of individual atoms to a high-finesse optical cavity,” Phys. Rev. Lett. **98**, 233601 (2007). [CrossRef] [PubMed]

**42. **A. D. Boozer, A. Boca, R. Miller, T. E. Northup, and H. J. Kimble, “Reversible state transfer between light and a single trapped atom,” Phys. Rev. Lett. **98**, 193601 (2007). [CrossRef] [PubMed]

**43. **R. Gehr, J. Volz, G. Dubois, T. Steinmetz, Y. Colombe, B. L. Lev, R. Long, J. Esteve, and J. Reichel, “Cavity-based single atom preparation and high-fidelity hyperfine state readout,” Phys. Rev. Lett. **104**, 203602 (2010). [CrossRef] [PubMed]

**44. **N. Kalb, A. Reiserer, S. Ritter, and G. Rempe, “Heralded storage of a photonic quantum bit in a single atom,” Phys. Rev. Lett. **114**, 220501 (2015). [CrossRef] [PubMed]

**45. **X. F. Zhou, Y. S. Zhang, and G. C. Guo, “Nonlocal gate of quantum network via cavity quantum electrodynamics,” Phys. Rev. A **71**, 064302 (2005). [CrossRef]

**46. **L. M. Duan, B. Wang, and H. Kimble, “Robust quantum gates on neutral atoms with cavity-assisted photon scattering,” Phys. Rev. A **72**, 032333 (2005). [CrossRef]

**47. **X. M. Lin, P. Xue, M. Y. Chen, Z. H. Chen, and X. H. Li, “Scalable preparation of multiple-particle entangled states via the cavity input-output process,” Phys. Rev. A **74**, 052339 (2006). [CrossRef]

**48. **Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. **75**, 4710 (1995). [CrossRef] [PubMed]

**49. **B. Dayan, A. Parkins, T. Aoki, E. Ostby, K. Vahala, and H. Kimble, “A photon turnstile dynamically regulated by one atom,” Science **319**, 1062–1065 (2008). [CrossRef] [PubMed]

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

**51. **M. Wang, R. B. Wu, J. T. Lin, J. H. Zhang, Z. W. Fang, Z. F. Chai, and Y. Cheng, “Chemo-mechanical polish lithography: a pathway to low loss large scale photonic integration on lithium niobate on insulator (lnoi),” Quantum Eng. **1**, e9 (2019). [CrossRef]

**52. **X. F. Liu, F. Lei, M. Gao, X. Yang, G. Q. Qin, and G. L. Long, “Fabrication of a microtoroidal resonator with picometer precise resonant wavelength,” Opt. Lett. **41**, 3603–3606 (2016). [CrossRef] [PubMed]

**53. **M. Wang, Y. Z. Wang, X. S. Xu, Y. Q. Hu, and G. L. Long, “Characterization of microresonator-geometry-deformation for cavity optomechanics,” Opt. Express **27**, 63–73 (2019). [CrossRef] [PubMed]

**54. **X. F. Liu, F. C. Lei, T. J. Wang, G. L. Long, and C. Wang, “Gain lifetime characterization through time-resolved stimulated emission in a whispering-gallery mode microresonator,” Nanophotonics **8**, 127–134 (2018). [CrossRef]

**55. **T. Wang, X. F. Liu, Y. Q. Hu, G. Q. Qin, D. Ruan, and G. L. Long, “Rapid and high precision measurement of opto-thermal relaxation with pump-probe method,” Sci. Bull. **63**, 287–292 (2018). [CrossRef]

**56. **D. E. Liu, “Sensing kondo correlations in a suspended carbon nanotube mechanical resonator with spin-orbit coupling,” Quantum Eng. **1**, e10 (2019). [CrossRef]

**57. **W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, “Exceptional points enhance sensing in an optical microcavity,” Nature **548**, 192 (2017). [CrossRef] [PubMed]

**58. **M. O. Scully and M. S. Zubairy, “Quantum optics,” (1999).

**59. **G. W. Lin, X. B. Zou, X. M. Lin, and G. C. Guo, “Robust and fast geometric quantum computation with multiqubit gates in cavity qed,” Phys. Rev. A **79**, 064303 (2009). [CrossRef]

**60. **H. F. Wang, A. D. Zhu, and S. Zhang, “One-step implementation of a multiqubit phase gate with one control qubit and multiple target qubits in coupled cavities,” Opt. Lett. **39**, 1489–1492 (2014). [CrossRef] [PubMed]

**61. **T. Beth and M. Rötteler, “Quantum algorithms: applicable algebra and quantum physics,” in *Quantum information*, (Springer, 2001), pp. 96–150. [CrossRef]

**62. **T. H. Taminiau, J. Cramer, T. van der Sar, V. V. Dobrovitski, and R. Hanson, “Universal control and error correction in multi-qubit spin registers in diamond,” Nat. nanotechnology **9**, 171 (2014). [CrossRef]

**63. **M. Šašura and V. Bužek, “Multiparticle entanglement with quantum logic networks: Application to cold trapped ions,” Phys. Rev. A **64**, 012305 (2001). [CrossRef]

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

**65. **L. Zhou and Y. B. Sheng, “Complete logic bell-state analysis assisted with photonic faraday rotation,” Phys. Rev. A **92**, 042314 (2015). [CrossRef]

**66. **Y. B. Sheng and L. Zhou, “Distributed secure quantum machine learning,” Sci. Bull. **62**, 1025–1029 (2017). [CrossRef]

**67. **W. Munro, A. Stephens, S. Devitt, K. Harrison, and K. Nemoto, “Quantum communication without the necessity of quantum memories,” Nat. Photonics **6**, 777 (2012). [CrossRef]

**68. **S. Muralidharan, J. Kim, N. Lütkenhaus, M. D. Lukin, and L. Jiang, “Ultrafast and fault-tolerant quantum communication across long distances,” Phys. Rev. Lett. **112**, 250501 (2014). [CrossRef] [PubMed]