Abstract

Photonic hyper-parallel quantum information processing (QIP) can simplify the quantum circuit and improve the information-processing speed, as well as reduce the quantum resource consumption and suppress the photonic dissipation noise. Here, utilizing the singly charged semiconductor quantum dot (QD) inside single-sided optical microcavity as the potentially experimental platform, we propose five schemes for heralded four-qubit hyper-controlled-not (hyper-CNOT) gates, covering all cases of four-qubit hyper-CNOT gates operated on both the polarization and spatial-mode degrees of freedom (DoFs) of a two-photon system. The novel heralding mechanism improves the fidelity of each hyper-CNOT gate to unity in principle without the strict restriction of strong coupling. The adaptability and scalability of the schemes make the hyper-CNOT gates more accessible under current experimental technologies. These heralded high-fidelity photonic hyper-CNOT gates can therefore have immense utilization potentials in high-capacity quantum communication and fast quantum computing, which are of far-reaching significance for QIP.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Based on the principles of quantum mechanics, quantum information processing (QIP) [1] has been shown to exhibit great advantages over classical information processing in secure communication [2,3], algorithm [4,5], simulation [6,7], sensing [810], and machine learning [1113] for its fantastic security and super-fast computing speed. Quantum logic gates are the critical element for QIP. In 1995, Divincenzo et al. [14] showed that one could design all quantum logic circuits with two-qubit gates in principle. Then, Barenco et al. [15] proved that the controlled-not (CNOT) gate is a universal two-qubit quantum logic gate for quantum computation. A set of CNOT and single-qubit gates can complete all unitary operations on arbitrary $n$ qubits [16]. Since then, two-qubit universal quantum CNOT gates have been extensively and in-depth studied on diverse physical platforms, such as linear optics [1720], ion traps [2123], superconducting qubits [2427], nuclear magnetic resonance [2832], nitrogen-vacancy centers [33,34], atoms [35], and quantum dots (QDs) [36,37].

Photons have remarkable transmission speed and multiple degrees of freedom (DoFs) for encoding, which are the perfect candidate for constructing quantum networks. However, complicated photonic quantum circuits will consume lots of quantum resources and introduce noise [3840], which will limit the scale and the performance of the quantum circuits. Different from QIP based on one DoF of photons, hyper-parallel QIP has been proposed in recent years, which encodes quantum information on two or more DoFs of photons, such as polarization, spatial mode, and frequency [4148]. Hyper-parallel quantum gates have some advantages over the ones constructed on one DoF of photons. Utilizing multiple DoFs, they can expand the quantum channel capacity, improve the calculation speed, reduce the noise caused by photonic dissipation, and simplify the quantum circuits. Therefore, the construction of the hyper-CNOT gate, the universal quantum gate in hyper-parallel QIP, is of great significance for high-capacity quantum communication and fast quantum computing.

As a promising solid-state qubit carrier, a semiconductor QD with an injected electron spin (a singly charged QD) can get excellent optical properties due to the optical spin selection rule for exciton transitions and the effect of cavity quantum electrodynamics (QED). In the past decades, the experimental feasibility of this system has been well established. In 2008, Hu et al. [49,50] investigated the optical giant circular birefringence (GCB) induced by a single-electron spin in a QD inside a single-sided optical microcavity. Based on the GCB effect, various quantum interfaces between single electron spins and single photons can be established, and a lot of schemes for scalable and all-optical QIP have been proposed, such as quantum logic gates [5153], cluster-state quantum computing [54], entanglement purification [5558], entanglement concentration [59,60], photonic transistor and router [61,62], and so on. In particular, hyper-parallel quantum gates [41,42,6365] and hyperentangled-Bell-state analysis [66,67] have also been investigated with the QD-cavity platforms in recent years.

Due to the significance of constructing hyper-CNOT gates in hyper-parallel QIP, in this paper, we present a robust interface between a photon and an electron spin confined in a single-sided QD-cavity system, of which the fidelity is unity in principle. Then, we propose five schemes for hyper-CNOT gates utilizing this photon-spin interface, which cover all cases of hyper-CNOT gate by encoding on both the polarization and spatial-mode DoFs of photons, i.e., one photon possesses the control qubits and the other possesses the target qubits, each photon possess both the control qubit and the target qubit, the polarization mode controls the polarization/spatial mode, and the spatial mode controls the spatial/polarization mode. Compared with the combination of cascaded CNOT gates built on one DoF of photons, hyper-CNOT gates can save the quantum resources and suppress the noise caused by the photonic dissipation efficiently. The circuit is simplified and the operation is accelerated. By using the robust photon-spin interface, our schemes improve the fidelities to unity in principle without the restriction of strong coupling. The proposed five hyper-CNOT gates show good adaptability and scalability, which can be used in multi-qubit systems and large-scale QIP, such as high-capacity quantum communication, quantum algorithm, and hyper-entanglement manipulations.

2. Interface between a single photon and a QD-cavity system

The QD-cavity system consists of a singly charged QD embedded in a single-sided optical micro-pillar cavity with a circular cross-section. As shown in Fig. 1(a), we consider a self-assembled In(Ga)As QD or a GaAs interface QD with two distributed Bragg reflectors (DBRs) in the top and bottom, respectively. The bottom DBR is 100% reflective, and the top DBR has partial reflectivity, coupling the light into and out of the cavity in a single side.

 figure: Fig. 1.

Fig. 1. (a) Structure of a singly charged QD coupled to a single-sided micropillar cavity. (b) The energy level of a QD-cavity system and the optical transition rules of negatively charged exciton $X^-$. $|\uparrow \rangle \rightarrow |\uparrow \downarrow \Uparrow \rangle$ and $|\downarrow \rangle \rightarrow |\downarrow \uparrow \Downarrow \rangle$ are driven by the left-circularly photon $|L\rangle$ and the right-circularly photon $|R\rangle$, respectively.

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When an excess electron is injected into the QD, a single incident photon can drive a transition between the ground state of the electron spin and a negatively charged exciton $X^-$. The $X^-$ is composed of two electrons bound to one hole, which exhibits spin-dependent optical transitions due to Pauli’s exclusion principle. As shown in Fig. 1(b), we use $\mid \uparrow \rangle$ and $\mid \downarrow \rangle$ to represent electron-spin states of the excess electron $|\pm \frac {1}{2}\rangle$, $|L\rangle$ and $|R\rangle$ to represent the left-circularly and right-circularly polarized state, respectively. When the electron lies in the spin state $\mid \uparrow \rangle$ ($\mid \downarrow \rangle$), only the $|L\rangle$ ($|R\rangle$) photon couples to the QD to create an $X^-$ in the state $\mid \uparrow \downarrow \Uparrow \rangle$ ($\mid \downarrow \uparrow \Downarrow \rangle$) with two electron spins antiparallel, while the $|R\rangle$ ($|L\rangle$) photon feels an empty cavity. Here, $\mid \Uparrow \rangle$ and $\mid \Downarrow \rangle$ represent heavy-hole spin states $|\pm \frac {3}{2}\rangle$. The spin-quantization axis is along the normal direction of the cavity. Under the Rotating wave approximation (RWA), the Hamiltonian of the QD-cavity system can be written as [68]

$$H=\omega_{X^-}\hat{\sigma}_{+}\hat{\sigma}_-{+}\omega_c\hat{a}^{\dagger}\hat{a}+ig(\hat{\sigma}_+\hat{a}-\hat{a}^{\dagger}\hat{\sigma}_-).$$

Here, $\omega _{X^-}$ and $\omega _c$ represent the frequencies of the $X^-$ transition and the cavity mode, respectively. $\hat {\sigma }_{+}$ and $\hat {\sigma }_{-}$ are the lifting and lowering operators of the exciton $X^-$. $\hat {a}$ and $\hat {a}^{\dagger }$ are the annihilation and creation operators for the cavity mode. $g$ is the coupling strength between $X^-$ and the cavity mode.

We can calculate the optical reflection coefficient of the QD-cavity system by solving the Heisenberg-Langevin equations of the cavity mode operator $\hat {a}$ and the $X^-$ dipole operator $\hat {\sigma }_-$ in the interaction picture [69]

$$\begin{aligned}&\frac{d\hat{a}}{dt}={-}[i(\omega_c-\omega)+\frac{\kappa}{2}+\frac{\kappa_s}{2}]\hat{a}-g\hat{\sigma}_-{-}\sqrt{\kappa}\hat{a}_{in}+\hat{H},\\ &\frac{d\hat{\sigma}_-}{dt}={-}[i(\omega_{X^-}-\omega)+\frac{\gamma}{2}]\hat{\sigma}_-{-}g\hat{\sigma}_z\hat{a}+\hat{G}. \end{aligned}$$

Here, $\omega$ is the frequency of the incident photon. $\hat {\sigma }_z = \hat {\sigma }_+\hat {\sigma }_- - \hat {\sigma }_-\hat {\sigma }_+$ is the population operator. $\gamma$ and $\kappa$ are the decay rates of the $X^-$ dipole and the cavity field mode. $\kappa _s$ is the cavity field leakage rate, including the material background absorption. $\hat {a}_{in}$ and $\hat {a}_{out}$ are the input and output field operators, respectively. $\hat {H}$ and $\hat {G}$ are the noise operators related to reservoirs. In the approximation of the weak excitation, $X^-$ is supposed to be in the ground state most of the time ($\langle \hat {\sigma }_z\rangle = -1$). According to the input-output relation $\hat {a}_{out}=\hat {a}_{in}+\sqrt {\kappa }\hat {a}$, the reflection coefficient of the QD-cavity system is

$$r(\omega,g)=\frac{[i(\omega_{X^-}-\omega)+\frac{\gamma}{2}][i(\omega_c-\omega)-\frac{\kappa}{2}+\frac{\kappa_s}{2}]+g^2}{[i(\omega_{X^-}-\omega)+\frac{\gamma}{2}][i(\omega_c-\omega)+\frac{\kappa}{2}+\frac{\kappa_s}{2}]+g^2}.$$

For simplicity, we tune the frequency $\omega _c$ to match $\omega _{X^-}$, $r(\omega ,g)$ can be expressed as

$$r(\Delta,g)=\frac{(i\Delta+\frac{\gamma}{2})(i\Delta-\frac{\kappa}{2}+\frac{\kappa_s}{2})+g^2}{(i\Delta+\frac{\gamma}{2})(i\Delta+\frac{\kappa}{2}+\frac{\kappa_s}{2})+g^2},$$
where $\Delta =\omega _{X^-}-\omega =\omega _c-\omega$ is the detuning between the incident photon and the cavity mode. When $g=0$, the photon, which uncoupled to the QD, senses an empty cavity with the reflection coefficient
$$r(\Delta,0)=\frac{i\Delta-\frac{\kappa}{2}+\frac{\kappa_s}{2}}{i\Delta+\frac{\kappa}{2}+\frac{\kappa_s}{2}}.$$

Due to cavity QED and the spin selection rule, there is a notable difference between the reflectance coefficients that in coupled case $r(\Delta ,g)$ and uncoupled case $r(\Delta ,0)$. Suppose a circularly polarized photon $|L\rangle$ interacts with a QD-cavity system, in which the QD-spin is in the state $|\psi \rangle = \alpha \mid \uparrow \rangle +\beta \mid \downarrow \rangle$ initially, where $|\alpha |^2+|\beta |^2=1$. The state of the photon-spin system evolves into $|\psi '\rangle = |L\rangle [r(\Delta ,g)\alpha \mid \uparrow \rangle + r(\Delta ,0)\beta \mid \downarrow \rangle ]/\sqrt {p_0}$, where $p_0 = |r(\Delta ,g)\alpha |^2+|r(\Delta ,0)\beta |^2$ is the probability of the photon being reflected by the QD-cavity. When a linearly polarized photon, e.g., the horizontal polarized photon $|H\rangle =(|R\rangle +|L\rangle )/\sqrt {2}$ or the vertical polarized photon $|V\rangle =-i(|R\rangle -|L\rangle )/\sqrt {2}$, enters the QD-cavity system with the QD-spin in the initial state $|\pm \rangle =(\mid \uparrow \rangle \pm \mid \downarrow \rangle )/\sqrt {2}$, the evolution rules of the photon-spin system is as follows

$$\begin{aligned}&|H\rangle|\pm\rangle\rightarrow[r_+(\Delta)|H\rangle|\pm\rangle+ir_-(\Delta)|V\rangle|\mp\rangle]/\sqrt{p_1}, \\ &|V\rangle|\pm\rangle\rightarrow[ir_+(\Delta)|V\rangle|\pm\rangle+r_-(\Delta)|H\rangle|\mp\rangle]/\sqrt{p_1}, \end{aligned}$$
where $r_\pm (\Delta )=[r(\Delta ,0){\pm }r(\Delta ,g)]/2$ and $p_1=[|r(\Delta ,0)|^2+|r(\Delta ,g)|^2]/2$ is the probability the photon interfere in the reflection geometry. After a linearly polarized photon enters the QD-cavity system, the system consisting of the photon and the electron spin evolves into an orthogonal entanglement state with two components: both the photon and electron spin remain unchanged with the probability $|r_+(\Delta )|^2$, and both the photon and electron spin flip with the probability $|r_-(\Delta )|^2$. The first component is due to the imperfect photon scattering process, which is trivial and usually can be filtered out as an error using linear optical elements. The second component is the desired result in QIP for constructing an interface between a single photon and an electron spin. The absolute amplitude $|r_-(\Delta )|$ of $r_-(\Delta )$ versus the frequency detuning $\Delta /\kappa$ are presented in Fig. 2. $|r_-(\Delta )|$ get a maximum of $|r_-(\Delta )|\approx 1$ around the frequency resonance $\Delta =0$ and $\Delta =\pm {g}$ with $\kappa _s\ll \kappa$. Relative high $|r_-(\Delta )|$ can be obtained at several detuning frequency in both strong- and weak-coupling regime. It is favorable for realizing high $|r_-(\Delta )|$ in experiment. The increase of $|r_-(\Delta )|$ is accompanied by a decrease in cavity-decay rate $\kappa _s/\kappa$. After filtering out the error component by linear optical elements, although $|r_-(\Delta )|$ has different values with different parameters, i.e., the coupling rate $g/\kappa$, the cavity-decay rate $\kappa _s/\kappa$, and the frequency detuning $\Delta /\kappa$, as a global coefficient, it only modifies the efficiency of the photon-spin interface.

 figure: Fig. 2.

Fig. 2. The absolute amplitude $|r_-(\Delta )|$ of $r_-(\Delta )$ with $\gamma =0.01\kappa$. (a) is for different coupling rate with $\kappa _s=0$. (b) is for different cavity-decay rate with the coupling rate $g=\kappa$.

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3. Heralded hyper-CNOT gates on two DoFs of a two-photon system

In this section, we propose five schemes for heralded hyper-CNOT gates on the polarization and the spatial-mode DoF of a two-photon system assisted by single-sided QD-cavity systems. We utilize two DoFs of two photons, i.e., the polarization and the spatial mode, to constitute a four-qubit system, with two qubits as control qubits flipping the other two, respectively. The errors caused by the imperfect photon scattering process are passively filtered out by linear optical elements and heralded by single-photon detectors, which lead to a high fidelity. The hyper-CNOT gates on the two-photon system can efficiently save the quantum resource and have better performance compared to the integration of two cascaded CNOT gates in one DOF.

In the schemes, we assume the QD-cavity systems are identical and the electron spin in QD$_j$ is prepared in the state $|\Psi _{e_j}\rangle =|+\rangle _j$ $(j=1, 2)$, initially. The initial states of photon $a$ and $b$ are

$$\begin{aligned}&|\Psi_a\rangle=|\psi^p_a\rangle\otimes|\psi^s_a\rangle=(\alpha_1|H\rangle+\beta_1|V\rangle)_a\otimes(\gamma_1|a_1\rangle+\delta_1|a_2\rangle),\\ &|\Psi_b\rangle=|\psi^p_b\rangle\otimes|\psi^s_b\rangle=(\alpha_2|H\rangle+\beta_2|V\rangle)_b\otimes(\gamma_2|b_1\rangle+\delta_2|b_2\rangle), \end{aligned}$$
where the subscripts $a$ and $b$ are used to distinguish between two photons. The superscripts $p$ and $s$ denote the polarization and the spatial mode. $|a_1\rangle$ $(|b_1\rangle )$ and $|a_2\rangle$ $(|b_2\rangle )$ represent the two spatial modes of photon $a$ $(b)$, respectively. The coefficients satisfy the relation $|\alpha _i|^2+|\beta _i|^2=1$ and $|\gamma _i|^2+|\delta _i|^2=1$ ($i=1, 2$).

3.1 Hyper-CNOT gate type I

The hyper-CNOT gate type I realizes the function that the polarization mode and the spatial mode of photon $a$ simultaneously control the polarization mode and the spatial mode of photon $b$, respectively. The schematic of hyper-CNOT gate type I is depicted in Fig. 3. PBS$_l$ ($l$=1, 2, 3, 4) is a polarization beam splitter, which transmits the photon in the horizontal polarization $|H\rangle$ and reflects the photon in the vertical polarization $|V\rangle$, respectively. H$_m$ ($m$=1, 2, 3, 4) represents a half-wave plate set at $22.5^{\circ }$ to perform a Hadamard operation on the polarization of a photon. BS$_n$ ($n$=1, 2) represents a 50:50 beam splitter used to perform a Hadamard operation on the spatial-mode DoF of a photon. X$_f$ ($f$=1, 2) is a half-wave plate set at $45^{\circ }$ to perform a polarization bit-flip operation $\sigma ^P_X=|H\rangle {\langle }V|+|V\rangle {\langle }H|$. R$_\theta =r_-(\Delta )|c\rangle$ completes a rotation that modifies the shape and intensity of the photons passing through it by the probability amplitude $r_-(\Delta )$, ($c=H, V$). $P_\theta =|H\rangle {\langle }H|+e^{-\frac {\pi }{2}i}|V\rangle {\langle }V|$ denotes a quantum phase gate on the polarization DoF of the photon. VBS$_v$ ($v$=1, 2) denotes an adjustable beam splitter with transmission coefficient $r_-(\Delta )$ and reflection coefficient $\sqrt {1-r_-^2(\Delta )}$. D$_w$ ($w$=1, 2) is a single-photon detector. DLs are delay lines, which make the photon in two spatial modes arrive simultaneously. u and d are used to distinguish between two spatial modes in BLOCKs. BLOCK$_1$ and BLOCK$_2$ represent two different units of interactions. In BLOCK$_1$, only the $|V\rangle$ photon interacts with the electron spin confined in the QD-cavity, while the $|H\rangle$ photon goes through R$_\theta$. After interacting with the QD-cavity system, the photon in spatial mode u goes through $X_1$ to get a polarization bit-flip operation $\sigma ^P_X$. In BLOCK$_2$, both the $|V\rangle$ and the $|H\rangle$ components of the photon interact with the electron spin in the cavity. After the interaction, the photon component in spatial mode u passes through a polarization phase gate $P_{\theta }$. The photon components in both spatial modes u and d pass through $X_2$ to get a bit-flip on polarization. The errors caused by the imperfect photon-spin interaction are filtered by PBSs and heralded by D$_1$ and D$_2$. D$_3$ and D$_4$ are used to announce the transmission errors on spatial modes $a_1$ and $b_1$, respectively. A detailed description of each step can be described as follows.

 figure: Fig. 3.

Fig. 3. The schematic diagram of hyper-CNOT gate type I. BLOCK$_1$ and BLOCK$_2$ represent two different units of interactions. In BLOCK$_1$, only the $|V\rangle$ photon interacts with the electron spin confined in the QD-cavity. In BLOCK$_2$, both the $|V\rangle$ and $|H\rangle$ photon interact with the electron spin of the cavity in sequence. PBS$_l$ ($l$=1, 2, 3, 4) is a polarization beam splitter, which transmits the photon in the horizontal polarization $|H\rangle$ and reflects the photon in the vertical polarization $|V\rangle$, respectively. H$_m$ ($m$=1, 2, 3, 4) represents a half-wave plate set at $22.5^{\circ }$ to perform a Hadamard operation on the polarization of a photon. BS$_n$ ($n$=1, 2) represents a 50:50 beam splitter to perform a Hadamard operation on the spatial mode DoF of a photon. X$_f$ ($f$=1, 2) is a half-wave plate set at $45^{\circ }$ to perform a polarization bit-flip operation $\sigma ^P_X=|H\rangle {\langle }V|+|V\rangle {\langle }H|$. R$_\theta =r_-(\Delta )|c\rangle$ complete a rotation that modifies the shape and intensity of the photons passing through it by $r_-(\Delta )$, ($c=H, V$). $P_\theta =|H\rangle {\langle }H|+e^{-\frac {\pi }{2}i}|V\rangle {\langle }V|$ denotes a quantum phase gate on the polarization of the photon. VBS$_v$ ($v$=1, 2) denotes an adjustable beam splitter with transmission coefficient $r_-(\Delta )$ and reflection coefficient $\sqrt {1-r_-^2(\Delta )}$. D$_w$ ($w$=1, 2, 3, 4) is a single-photon detector. DLs are delay lines, which make the photon in two spatial modes arrive simultaneously. u and d denote the spatial mode in BLOCK$_1$ and BLOCK$_2$.

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First, photon $a$ enters the circuit from the left and passes through BLOCK$_1$ in both spatial modes $a_1$ and $a_2$. Before photon $a$ gets PBS$_2$ in BLOCK$_1$, the state of the system composed of photon $a$, photon $b$, QD$_1$ and QD$_2$ is changed from $|\Psi _{abe_1e_2}\rangle _0$ to $|\Psi _{abe_1e_2}\rangle _1$ in unnormalized form

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_0 = &(\alpha_1|H\rangle+\beta_1|V\rangle)_a(\gamma_1|a_1\rangle+\delta_1|a_2\rangle) \\ &\otimes (\alpha_2|H\rangle+\beta_2|V\rangle)_b(\gamma_2|b_1\rangle+\delta_2|b_2\rangle) \otimes |+\rangle_1\otimes|+\rangle_2,\\ |\Psi_{abe_1e_2}\rangle_1 = &[r_-(\Delta)(\alpha_1|H\rangle_d|+\rangle_1+\beta_1|V\rangle_u|-\rangle_1)_a\\ &+ir_+(\Delta)\beta_1|H\rangle_{au}|+\rangle_1](\gamma_1|a_1\rangle+\delta_1|a_2\rangle)\\ &\otimes(\alpha_2|H\rangle+\beta_2|V\rangle)_b(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\otimes|+\rangle_2. \end{aligned}$$

Here, the subscripts $u$ and $d$ denote the spatial modes in BLOCK$_i$ $(i=1, 2)$. The $|H\rangle _{au}$ component of the photon will be transmitted by PBS$_2$ to the single-photon detector D$_1$ and cause a click. The components $|H\rangle _{ad}$ and $|V\rangle _{au}$ will reunite and output from BLOCK$_1$. If the detector D$_1$ clicks, that indicates an error has occurred, the state collapses to the wrong state

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_2 = &ir_+(\Delta)\beta_1|H\rangle_{au}|+\rangle_1(\gamma_1|a_1\rangle+\delta_1|a_2\rangle)\\ &\otimes(\alpha_2|H\rangle+\beta_2|V\rangle)_b(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\otimes|+\rangle_2. \end{aligned}$$

The states of the electron spins do not change. We consider the photon to be damaged, and a new photon $a$ will be put into the circuit to start a recycling procedure. If the detector D$_1$ does not click, the state of the system is

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_3 = &r_-(\Delta)(\alpha_1|H\rangle_d|+\rangle_1+\beta_1|V\rangle_u|-\rangle_1)_a(\gamma_1|a_1\rangle+\delta_1|a_2\rangle)\\ &\otimes(\alpha_2|H\rangle+\beta_2|V\rangle)_b(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\otimes|+\rangle_2. \end{aligned}$$

Then, we let photon $a$ passes through VBS$_1$ and BLOCK$_2$ in spatial modes $a_1$ and $a_2$, respectively. When photon $a$ gets the output port without triggering any photon detector, the state of the system will be

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_4 = &r^2_-(\Delta)(\alpha_1|H\rangle_a|+\rangle_1+\beta_1|V\rangle_a|-\rangle_1)(\gamma_1|a_1\rangle|+\rangle_2+\delta_1|a_2\rangle|-\rangle_2)\\ &\otimes(\alpha_2|H\rangle+\beta_2|V\rangle)_b(\gamma_2|b_1\rangle+\delta_2|b_2\rangle). \end{aligned}$$

For now, the BLOCK$_1$-QD$_1$ interaction gets the polarization of photon $a$ and the electron spin confined in QD$_1$ entangled, and the BLOCK$_2$-QD$_2$ interaction gets the spatial-mode DoF of photon $a$ and the electron spin confined in QD$_2$ entangled, respectively. The qubit which interacts with the QD$_j$ $(j=1, 2)$ first is the control qubit.

After photon $a$ outputs from the circuit, a Hadamard operation H$_e$ is performed on the electron spins in QD$_1$ and QD$_2$, respectively, to accomplish the operation $|+\rangle \rightarrow (|+\rangle +|-\rangle )/\sqrt {2}$ and $|-\rangle \rightarrow (|+\rangle -|-\rangle )/\sqrt {2}$. The system state becomes

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_5 = &r_-^2(\Delta)(\alpha_1|H\rangle_a\frac{|+\rangle_1+|-\rangle_1}{\sqrt{2}}+\beta_1|V\rangle_a\frac{|+\rangle_1-|-\rangle_1}{\sqrt{2}})\\ &\otimes(\gamma_1|a_1\rangle\frac{|+\rangle_2+|-\rangle_2}{\sqrt{2}}+\delta_1|a_2\rangle\frac{|+\rangle_2-|-\rangle_2}{\sqrt{2}})\\ &\otimes(\alpha_2|H\rangle+\beta_2|V\rangle)_b(\gamma_2|b_1\rangle+\delta_2|b_2\rangle). \end{aligned}$$

Then we let photon $b$ enter the circuit. After photon $b$ passes through H$_1$ and H$_2$ in both spatial modes $b_1$ and $b_2$, the state of the system consists of photon $a$, photon $b$, QD$_1$ and QD$_2$ changes into

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_6 = &r_-^2(\Delta)(\alpha_1|H\rangle_a\frac{|+\rangle_1+|-\rangle_1}{\sqrt{2}}+\beta_1|V\rangle_a\frac{|+\rangle_1-|-\rangle_1}{\sqrt{2}})\\ &\otimes(\gamma_1|a_1\rangle\frac{|+\rangle_2+|-\rangle_2}{\sqrt{2}}+\delta_1|a_2\rangle\frac{|+\rangle_2-|-\rangle_2}{\sqrt{2}})\\ &\otimes(\alpha_2'|H\rangle+\beta_2'|V\rangle)_b(\gamma_2|b_1\rangle+\delta_2|b_2\rangle), \end{aligned}$$
where $\alpha _2'=(\alpha _2+\beta _2)/\sqrt {2}$, $\beta _2'=(\alpha _2-\beta _2)/\sqrt {2}$. After photon $b$ passes through BLOCK$_1$ without the detector clicking, Hadamard operations are performed on both the polarization and the spatial-mode DoF of photon $b$ by H$_3$, H$_4$ and BS$_1$, respectively. The state of the system evolves into
$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_7 = &\frac{r_-^3(\Delta)}{\sqrt{2}}\{[\alpha_1|H\rangle_a(\alpha_2|H\rangle_b+\beta_2|V\rangle_b) \\ &+\beta_1|V\rangle_a(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]|+\rangle_1\\ &+[\alpha_1|H\rangle_a(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &-\beta_1|V\rangle_a(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]|-\rangle_1\}\\ &\otimes(\gamma_1|a_1\rangle|+\rangle_2+\delta_1|a_2\rangle|-\rangle_2)(\gamma_2'|b_1\rangle+\delta_2'|b_2\rangle), \end{aligned}$$
where $\gamma _2'=(\gamma _2+\delta _2)/\sqrt {2}$, $\delta _2'=(\gamma _2-\delta _2)/\sqrt {2}$. Then photon $b$ passes through VBS$_2$, BLOCK$_2$, and BS$_2$ in sequence. When no detector clicks, the state of the system will be
$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_8 = &\frac{r_-^4(\Delta)}{2}\{[\alpha_1|H\rangle_a(\alpha_2|H\rangle_b+\beta_2|V\rangle_b) \\ &+\beta_1|V\rangle_a(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]|+\rangle_1\\ &+[\alpha_1|H\rangle_a(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &-\beta_1|V\rangle_a(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]|-\rangle_1\}\\ &{\otimes}\{[\gamma_1|a_1\rangle(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\\ &+\delta_1|a_2\rangle(\delta_2|b_1\rangle+\gamma_2|b_2\rangle)]|+\rangle_2\\ &+[\gamma_1|a_1\rangle(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\\ &-\delta_1|a_2\rangle(\delta_2|b_1\rangle+\gamma_2|b_2\rangle)]|-\rangle_2\}. \end{aligned}$$

If any detector clicks, we need to start a recycling procedure. We reset the spin states in QD$_1$ (QD$_2$) to the $|+\rangle _1$ ($|+\rangle _2$) state before a new photon $a$ enters the circuit. Until there is no detector clicks, we get $|\Psi _{abe_1e_2}\rangle _8$.

Finally, we measure the electron spins in QD$_1$ and QD$_2$ in the orthogonal basis $\{|+\rangle , |-\rangle \}$. If we get the result $|+\rangle _1$ and $|+\rangle _2$, the final state of the two-photon system is expressed as

$$\begin{aligned}|\Psi_{ab}\rangle_1 = &r_-^4(\Delta)[\alpha_1|H\rangle_a(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)+\beta_1|V\rangle_a(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]\\ &{\otimes}[\gamma_1|a_1\rangle(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)+\delta_1|a_2\rangle(\delta_2|b_1\rangle+\gamma_2|b_2\rangle)]\\ = &r_-^4(\Delta)(\alpha_1|H\rangle_a+ \beta_1|V\rangle_a\sigma^P_{X_b})(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &\otimes(\gamma_1|a_1\rangle +\delta_1|a_2\rangle\sigma^S_{X_b})(\gamma_2|b_1\rangle+\delta_2|b_2\rangle). \end{aligned}$$

Here, $\sigma ^P_{X_k}=|H\rangle {\langle }V|+|V\rangle {\langle }H|$ $(k=a, b)$ represents the bit-flip operation on the polarization of photon $k$. $\sigma ^S_{X_k}=|k_1\rangle {\langle }k_2|+|k_2\rangle {\langle }k_1|$ $(k=a, b)$ represents the bit-flip operation on the spatial mode of photon $k$. When the measurement of the electron spin in QD$_1$ is $|-\rangle _1$, a polarization phase-flip operation $\sigma _{Z_k}^P=|H{\rangle \langle }H|-|V{\rangle \langle }V|$ $(k=a, b)$ is performed on the polarization of photon $a$. When the measurement of the electron spin in QD$_2$ is $|-\rangle _1$, a spatial-mode phase-flip operation $\sigma _{Z_k}^S=|k_1{\rangle \langle }k_1|-|k_2{\rangle \langle }k_2|$ $(k=a, b)$ is performed on the spatial-mode of photon $a$. Then we can get the state $|\Psi _{ab}\rangle _1$.

So far, we have completed the hyper-CNOT gate type I operation on the polarization and spatial-mode DoFs of two photons. The polarization (spatial) mode of photon $a$ is the control qubit when the polarization (spatial) mode of photon $b$ is the target qubit.

3.2 Hyper-CNOT gate type II

In the hyper-CNOT gates type I and type II, photon $a$ and photon $b$ enter the circuit sequentially. Photon $a$ possesses two control qubits, photon $b$ possesses two target qubits. The schematic diagram of the hyper-CNOT gate type II is depicted in Fig. 4. Similar to the hyper-CNOT gate type I, both the polarization and the spatial mode of photon $a$ are the control qubits, while both the polarization and the spatial mode of photon $b$ are the target qubits. The difference is that, in this scheme, the polarization of photon $a$ controls the spatial mode of photon $b$, while the spatial mode of photon $a$ controls the polarization of photon $b$, respectively.

 figure: Fig. 4.

Fig. 4. The schematic diagram of the hyper-CNOT gate type II. The polarization and the spatial mode of photon $a$ are the control qubits, while the spatial mode and the polarization of photon $b$ are the target qubits, respectively. The elements have the same function as the ones depicted in Fig. 3.

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Photon $a$ goes the same circuit as it in the hyper-CNOT gate type I. After photon $a$ arrives the output port without any detector clicks, the Hadamard operation H$_e$s are performed on the electron spin confined in QD$_1$ and QD$_2$, respectively. Then we let photon $b$ enters the circuit and passes through BS$_1$, VBS$_2$, BLOCK$_2$, BS$_2$, H$_1$, H$_2$, BLOCK$_1$, H$_3$, H$_4$, sequentially. If any single-photon detector clicks in the process, it indicates an error has occurred. We consider the operation fails, and reset the spin states of QD$_1$ (QD$_2$) to the state $|+\rangle _1$ ($|+\rangle _2$) to start a recycling process. When the two photons are emitted from the output port, and none of the detectors clicks, the hyper-CNOT gate operation succeeds. The state of the system composed of photon $a$, photon $b$, QD$_1$, and QD$_2$ evolves into

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_9 = &\frac{r_-^4(\Delta)}{2}\{[\alpha_1|H\rangle_a(\gamma_2|b_1\rangle+\delta_2|b_2\rangle) \\ &+\beta_1|V\rangle_a(\delta_2|b_1\rangle+\gamma_2|b_2\rangle)]|+\rangle_1\\ &+[\alpha_1|H\rangle_a(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\\ &-\beta_1|V\rangle_a(\delta_2|b_1\rangle+\gamma_2|b_2\rangle)]|-\rangle_1\}\\ &\otimes\{[\gamma_1|a_1\rangle(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &+\delta_1|a_2\rangle(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]|+\rangle_2\\ &+[\gamma_1|a_1\rangle(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &-\delta_1|a_2\rangle(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]|-\rangle_2\}. \end{aligned}$$

Then, we measure the electron spins in QD$_1$ and QD$_2$ in the orthogonal basis $\{|+\rangle , |-\rangle \}$. If the measurement is $|+\rangle _1$ and $|+\rangle _2$, the final state of the two-photon system is

$$\begin{aligned}|\Psi_{ab}\rangle_2 = &r_-^4(\Delta)(\alpha_1|H\rangle_a+\beta_1|V\rangle_a\sigma^S_{X_b})(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\\ &\otimes(\gamma_1|a_1\rangle+\delta_1|a_2\rangle\sigma^P_{X_b})(\alpha_2|H\rangle_b+\beta_2|V\rangle_b). \end{aligned}$$

When the measurement outcomes of the electron spin in QD$_1$ is $|-\rangle _1$, a polarization phase-flip operation $\sigma _{Z_a}^P$ is performed on photon $a$. When the measurement of the electron spin in QD$_2$ is $|-\rangle _1$, a spatial-mode phase-flip operation $\sigma _{Z_a}^S$ is performed on photon $a$. Then we can also get the state $|\Psi _{ab}\rangle _2$.

In a word, we have completed the type II hyper-CNOT gate operation on the polarization and spatial-mode DoFs of two photons. The polarization and the spatial mode of photon $a$ simultaneously control the spatial mode and the polarization of photon $b$, respectively.

3.3 Hyper-CNOT gate type III

The schematic diagram of the hyper-CNOT gate type III is illustrated in Fig. 5. Different from the type I and II hyper-CNOT gate, the two photons enter the circuit from opposite directions simultaneously. In this scheme, the spatial mode of photon $a$ $(b)$ is the control qubit, when the polarization of photon $b$ $(a)$ is the target qubit. Photon $a$ passes through BLOCK$_2$, VBS$_1$, H$_1$, H$_3$, BLOCK$_1$, H$_2$, and H$_4$ in sequence. Simultaneously, photon $b$ enters the circuit from the opposite side and passes through BLOCK$_2$, VBS$_2$, H$_6$, H$_8$, BLOCK$_1$, H$_5$, and H$_7$, sequentially. Before the interaction between QD$_1$ (QD$_2$) and photon $b$ (photon $a$), a Hadamard operation H$_e$ is performed on the electron spin confined in QD$_1$ (QD$_2$), respectively. Without any detector clicks, the state of the system becomes

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_{10} = &\frac{r_-^4(\Delta)}{2}\{[\gamma_1|a_1\rangle(\alpha_2|H\rangle_b+\beta_2|V\rangle_b) \\ &+\delta_1|a_2\rangle(\beta_2|H\rangle_b+\alpha_2|V\rangle_a)]|+\rangle_1\\ &+[\gamma_1|a_1\rangle(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &-\delta_1|a_2\rangle(\beta_2|H\rangle_b+\alpha_2|V\rangle_b)]|-\rangle_1\}\\ &\otimes\{[\gamma_2|b_1\rangle(\alpha_1|H\rangle_a+\beta_1|V\rangle_a)\\ &+\delta_2|b_2\rangle(\beta_1|H\rangle_a+\alpha_1|V\rangle_a)]|+\rangle_2\\ &+[\gamma_2|b_1\rangle(\alpha_1|H\rangle_a+\beta_1|V\rangle_a)\\ &-\delta_2|b_2\rangle(\beta_1|H\rangle_a+\alpha_1|V\rangle_a)]|-\rangle_2\}. \end{aligned}$$

After that, we measure the electron spins in QD$_1$ and QD$_2$ in the orthogonal basis $\{|+\rangle , |-\rangle \}$. When the result is $|+\rangle _1$ and $|+\rangle _2$, the state of the system is

$$\begin{aligned}|\Psi_{ab}\rangle_3 = &r_-^4(\Delta)(\gamma_1|a_1\rangle+\delta_1|a_2\rangle\sigma^P_{X_b})(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &\otimes(\gamma_2|b_1\rangle+\delta_2|b_2\rangle\sigma^P_{X_a})(\alpha_1|H\rangle_a+\beta_1|V\rangle_a). \end{aligned}$$

When the measurement outcomes of the electron spin in QD$_1$ (QD$_2$) is $|-\rangle _1$ ($|-\rangle _2$), a spatial phase-flip operation $\sigma _{Z_a}^S$ ($\sigma _{Z_b}^S$) is performed on photon $a$ ($b$). Then we can get the state $|\Psi _{ab}\rangle _3$.

 figure: Fig. 5.

Fig. 5. The schematic diagram of hyper-CNOT gate type III. The spatial mode of photon $a$ $(b)$ is the control qubit, when the polarization mode of photon $b$ $(a)$ is the target qubit. The elements have the same function as the ones depicted in Fig. 3.

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3.4 Hyper-CNOT gate type IV

As depicted in Fig. 6, the schematic diagram of the hyper-CNOT gate type IV is similar to the hyper-CNOT gate type III. The two photons enter the circuit simultaneously and propagate in the opposite direction. In this scheme, the polarization of photon $a$ $(b)$ controls the spatial mode of photon $b$ $(a)$. Photon $a$ passes through BLOCK$_1$, BS$_1$, VBS$_1$, BLOCK$_2$, and BS$_2$ in sequence, simultaneously photon $b$ enters the circuit from the right side and passes through BLOCK$_1$, BS$_4$, BLOCK$_2$, VBS$_2$, and BS$_3$ sequentially. Before photon $a$ (photon $b$) interacts with QD$_2$ (QD$_1$), a Hadamard operation H$_e$ is performed on the electron spin confined in QD$_2$ (QD$_1$), respectively. Without any detector clicks, the state of the system composed of photon $a$, photon $b$, QD$_1$, and QD$_2$ is

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_{11} = &\frac{r_-^4(\Delta)}{2}\{[\alpha_1|H\rangle_a(\gamma_2|b_1\rangle+\delta_2|b_2\rangle) \\ &+\beta_1|V\rangle_a(\delta_2|b_1\rangle+\gamma_2|b_2\rangle)]|+\rangle_1\\ &+[\alpha_1|H\rangle_a(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\\ &-\beta_1|V\rangle_a(\delta_2|b_1\rangle+\gamma_2|b_2\rangle)]|-\rangle_1\}\\ &\otimes\{[\alpha_2|H\rangle_b([\gamma_1|a_1\rangle+\delta_1|a_2\rangle)\\ &+\beta_2|V\rangle_b(\delta_1|a_1\rangle+\gamma_1|a_2\rangle)]|+\rangle_2\\ &+[\alpha_2|H\rangle_b(\gamma_1|a_1\rangle+\delta_1|a_2\rangle)\\ &-\beta_2|V\rangle_b(\delta_1|a_1\rangle+\gamma_1|a_2\rangle)]|-\rangle_1\}. \end{aligned}$$

Then we measure the electron spins in QD$_1$ and QD$_2$ in the orthogonal basis $\{|+\rangle , |-\rangle \}$. When the measurement is $|+\rangle _1$ and $|+\rangle _2$, the state of the system is

$$\begin{aligned}|\Psi_{ab}\rangle_4 = &r_-^4(\Delta)(\alpha_1|H\rangle_a+\beta_1|V\rangle_a\sigma^S_{X_b})(\gamma_2|b_1\rangle+\delta_2|b_2\rangle)\\ &\otimes(\alpha_2|H\rangle_b+\beta_2|V\rangle_b\sigma^S_{X_a})(\gamma_1|a_1\rangle+\delta_1|a_2\rangle). \end{aligned}$$

When the measurement of the electron spin in QD$_1$ (QD$_2$) is $|-\rangle _1$ ($|-\rangle _2$), a polarization phase-flip operation $\sigma _{Z_a}^P$ ($\sigma _{Z_b}^P$) is performed on photon $a$ ($b$). Then the state $|\Psi _{ab}\rangle _4$ is obtained. In this scheme, the polarization of photon $b$ $(a)$ is the control qubit, when the spatial mode of photon $a$ $(b)$ is the target qubit.

 figure: Fig. 6.

Fig. 6. The schematic diagram of hyper-CNOT gate type IV. The polarization mode of photon $a$ $(b)$ is the control qubit, when the spatial mode of photon $b$ $(a)$ is the target qubit. The elements have the same function as the ones depicted in Fig. 3.

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3.5 Hyper-CNOT Gate Type V

The schematic diagram of the hyper-CNOT gate type V is shown in Fig. 7. The two photons enter the circuit simultaneously and propagate oppositely similar to the hyper-CNOT gate type III and IV. In this scheme, the polarization of photon $a$ is the control qubit, when that of photon $b$ is the target qubit. And the spatial mode of photon $b$ is the control qubit when that of photon $a$ is the target qubit. Photon $a$ passes the same circuit as it in the hyper-CNOT gate type IV. Simultaneously, photon $b$ enters the circuit from the right side and passes through BLOCK$_2$, VBS$_2$, H$_3$, H$_4$, BLOCK$_1$, H$_1$, and H$_4$ in sequence. Before photon $a$ (photon $b$) interacts with QD$_2$ (QD$_1$), a Hadamard operation H$_e$ is performed on the electron spin confined in QD$_2$ (QD$_1$), respectively. Without any detector clicks, we can get the state of the system

$$\begin{aligned}|\Psi_{abe_1e_2}\rangle_{12} = &\frac{r_-^4(\Delta)}{2}\{[\alpha_1|H\rangle_a(\alpha_2|H\rangle_b+\beta_2|V\rangle_b) \\ &+\beta_1|V\rangle_a(\beta_2|H\rangle_b+\alpha_2|V\rangle_2)]|+\rangle_1\\ &+[\alpha_1|H\rangle_a(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &-\beta_1|V\rangle_a(\beta_2|H\rangle_b+\alpha_2|V\rangle_2)]|-\rangle_1\}\\ &\otimes\{[\gamma_2|b_1\rangle(\gamma_1|a_1\rangle+\delta_1|a_2\rangle)\\ &+\delta_2|b_2\rangle(\delta_1|a_1\rangle+\gamma_1|a_2\rangle)]|+\rangle_2\\ &+[\gamma_2|b_1\rangle(\gamma_1|a_1\rangle+\delta_1|a_2\rangle)\\ &-\delta_2|b_2\rangle(\delta_1|a_1\rangle+\gamma_1|a_2\rangle)]|-\rangle_1\}. \end{aligned}$$

Then, we measure the electron spins in QD$_1$ and QD$_2$ in the orthogonal basis $\{|+\rangle , |-\rangle \}$. When we get $|+\rangle _1$ and $|+\rangle _2$, the final state of the two-photon system is

$$\begin{aligned}|\Psi_{ab}\rangle_5 = &r_-^4(\Delta)(\alpha_1|H\rangle_a+\beta_1|V\rangle_a\sigma^P_{X_b})(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &\otimes(\gamma_2|b_1\rangle+\delta_2|b_2\rangle\sigma^S_{X_a})(\delta_1|a_1\rangle+\gamma_1|a_2\rangle). \end{aligned}$$

When the measurement of the electron spin in QD$_1$ is $|-\rangle _1$, a polarization phase-flip operation $\sigma _{Z_a}^P$ is performed on photon $a$. When the measurement of the electron spin in QD$_2$ is $|-\rangle _2$, a spatial phase-flip operation $\sigma _{Z_b}^S$ is performed on photon $b$. Then we get the state $|\Psi _{ab}\rangle _5$. In this scheme, the polarization of photon $a$ controls the polarization of photon $b$, while the spatial mode of photon $b$ controls the spatial mode of photon $a$.

 figure: Fig. 7.

Fig. 7. The schematic diagram of hyper-CNOT gate type V. The polarization of photon $a$ is the control qubit, when that of photon $b$ is the target qubit. The spatial mode of photon $b$ is the control qubit when that of photon $a$ is the target qubit. The elements have the same function as the ones depicted in Fig. 3.

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In the schemes of hyper-CNOT gate type III, IV, and V, photon $a$ and photon $b$ possess both the control qubit and the target qubit. Two photons simultaneously enter the circuit in opposite directions, which can save the time consumption of the operations. If any detector clicks, which indicates the failure of the operation, we reset the spin state of QD$_1$ (QD$_2$) to the state $|+\rangle _1$ ($|+\rangle _1$), and start a recycling process. The hyper-CNOT gate operation will be completed when two photons output the circuit with no detector clicks.

4. Discussion and summary

Our schemes are based on the interaction between the polarization of the photon and the electron spin embedded in the QD-cavity system. The performance of the hyper-CNOT gate is described by fidelity and efficiency. We define the fidelity as ${F}={|\langle \Psi _{r}|\Psi _{i}\rangle |^2}$, where $|\Psi _{r}\rangle$ represents the final state of the system composed of excess electron spins and photons with realistic QD-cavities. $|\Psi _{i}\rangle$ is the final state of the system in ideal conditions. In realistic conditions, we only consider the influence of system parameters. The effects of photon dissipation loss, the imperfections of single-photon source and linear optical elements, and the dark counts of single-photon detectors on the fidelity are discussed later. Take the hyper-CNOT gate type I for example. The ideal final state of the system should be

$$\begin{aligned}|\Psi_{i}\rangle = &(\alpha_1|H\rangle_a+ \beta_1|V\rangle_a\sigma^P_{X_b})(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &\otimes(\gamma_1|a_1\rangle +\delta_1|a_2\rangle\sigma^S_{X_b})(\gamma_2|b_1\rangle+\delta_2|b_2\rangle). \end{aligned}$$

According to Fig. 3, the final state in the realistic condition in unnormalized form is

$$\begin{aligned}|\Psi_{r}\rangle = &r_-^4(\Delta)(\alpha_1|H\rangle_a+ \beta_1|V\rangle_a\sigma^P_{X_b})(\alpha_2|H\rangle_b+\beta_2|V\rangle_b)\\ &\otimes(\gamma_1|a_1\rangle +\delta_1|a_2\rangle\sigma^S_{X_b})(\gamma_2|b_1\rangle+\delta_2|b_2\rangle). \end{aligned}$$

The average fidelity of the scheme is

$$\overline{F}=\overline{|\langle\Psi_{r}|\Psi_{i}\rangle|^2}= \frac{1}{4{\pi}^4}\int_{0}^{2\pi}d{\theta_p}\int_{0}^{2\pi} d{\phi_p}\int_{0}^{2\pi}d{\theta_s}\int_{0}^{2\pi} d{\phi_s}\frac{{|\langle\Psi_{r}|\Psi_{i}\rangle|^2}}{{\langle}\Psi_r|\Psi_r\rangle} =1.$$

Where $cos\theta _p=\alpha _1$, $\sin \theta _p=\beta _1$, $cos\phi _p=\alpha _2$, $sin\phi _p=\beta _2$, $cos\theta _s=\gamma _1$, $\sin \theta _s=\delta _1$, $cos\phi _s=\gamma _2$, $sin\phi _s=\delta _2$. Due to the heralding mechanism, errors have been filtered out and heralded by single-photon detectors. The fidelity of each hyper-CNOT gate is unity in principle, which is robust to the coupling rate $g/\kappa$ and the cavity-decay rate $\kappa _s/\kappa$.

We define the efficiency of the hyper-CNOT gate as the probability that the photons output the circuit successfully. Due to the similar construction of the schemes, the efficiencies of the hyper-CNOT gate type I-V are

$$\begin{aligned}{\eta} = &|r_-(\Delta)|^8=\left|\frac{r(\Delta,0)-r(\Delta,g)}{2}\right|^8\\ = &\left|\frac{-4g^2/\kappa^2}{(2i\Delta/\kappa+1+\kappa_s/\kappa)[(2i\Delta/\kappa+\gamma/\kappa)(2i\Delta/\kappa+1+\kappa_s/\kappa)+4g^2/\kappa^2]}\right|^8. \end{aligned}$$

According to Fig. 2, $|r_-(\Delta )|$ get relevant high values around $\Delta =0$ and $\Delta ={\pm }g$. Since $\gamma =0.3$ $\mu$eV can be achieved, we take $\gamma =0.01\kappa$ here [70]. The efficiencies $\eta$ is the function of the coupling strength $g/\kappa$ and the cavity-decay rate $\kappa _s/\kappa$ when $\Delta =0$ and $\Delta ={\pm }g$. The relation between $g/\kappa$ and $r_-(\Delta )$ is shown in Fig. 8(a) and (b), and the relation between $g/\kappa$ and ${\eta }$ is shown in Fig. 8(c) and (d). According to Fig. 8(a) and (b), although relatively high $|r_-(\Delta )|$ can be obtained at both resonant frequency $\Delta =0$ and detuning frequency $\Delta ={\pm }g$, frequency detuning still reduces $|r_-(\Delta )|$ to some extent. The frequency detuning also affects the efficiencies. According to Fig. 8(c), at resonant frequency $\Delta =0$, in the case $\kappa _s=0.01\kappa$, the efficiency is 85.21% at $g=0.5\kappa$, 90.5% at $g=\kappa$, and 92.03% at $g=2.4\kappa$. When the side leakage is negligible, the efficiency is 92.35% at $g=0.5\kappa$, 98.02% at $g=\kappa$, and 99.65% at $g=2.4\kappa$. According to Fig. 8(d), when the frequency detuning is $\Delta =g$, in the case $\kappa _s=0.01\kappa$, the efficiency is 34.4% at $g=\kappa$, and 71.71% at $g=2.4\kappa$. When the side leakage is negligible, the efficiency is 37.82% at $g=\kappa$, and 77.91% at $g=2.4\kappa$. It can be seen that under the frequency detuning condition $\Delta ={\pm }g$, the efficiency of the scheme is lower than that under the resonance condition $\Delta =0$, but it still works. At the same time, to get a high efficiency, $\kappa _s/\kappa$ should be as small as possible, which can be realized via reducing the number of the top mirror pairs in the DBRs [71]. By improving the design and manufacturing process of micro-cavities, $\kappa _s=0.01\kappa$ could possibly be achieved in a micro-pillar cavity [7274].

 figure: Fig. 8.

Fig. 8. (a) and (b) are the relation between the coupling rate $g/\kappa$ and the probability amplitude $|r_-(\Delta )|$ at the frequency detuning $\Delta =0$ and $\Delta =g$, respectively. (c) and (d) are the relation between $g/\kappa$ and the efficiency $\eta$ at the frequency detuning $\Delta =0$ and $\Delta =g$, respectively. We take $\gamma =0.01\kappa$ here.

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For In(Ga)As or GaAs charged QDs, to maintain the weak excitation approximation valid, the interval between two photons enter the cavity could be in nanosecond level [75]. The electron spin coherence time $T^e$ can reach $T^e>3{\mu }$s by spin echo techniques [76], which is sufficient for multi-qubit operations. The electron spin state can be prepared by optical pumping or optical cooling [77,78], and nanosecond microwave pulses or picosecond/femtosecond optical pulses [79,80]. The photon can be produced by nanosecond laser pulses and QD single-photon sources [8183]. Some imperfections would affect the qualities of the hyper-CNOT gates. Such as the imperfect single-photon source, linear elements, and the balancing of PBSs and BSs, which can be improved by the development of manufacturing process. Fortunately, due to the heralding mechanism, errors caused by the realistic interaction between photons and electron spins will be filtered out and heralded by the detectors, which would not affect the fidelities of the hyper-CNOT gates. The dark count of the single-photon detectors may decrease the efficiencies of the hyper-CNOT gates slightly [84], which will be improved by choosing the appropriate temperature and parameters for the single-photon detectors [85]. Moreover, recent research has shown that the microcavity made of LiN can overcome the high loss of traditional silicon cavities [86], and microcavities can be used in nonreciprocal response and conversion [87], producing non-classical states for quantum teleportation [88], working as interface [89], and many other applications [90,91]. With the continuous improvement of microcavity manufacturing processes, the QD-cavity system is more and more realistic in manufacturing and becomes a promising candidate for QIP.

Different from the cascaded CNOT gate on only one DoF of photons, the proposed hyper-CNOT gates, which using two DoFs of photons for encoding, reduce the overhead of photons, save quantum resources efficiently, and reduce the photon dissipation noise. The circuit is simplified, which can turn the existing QIP schemes based on one DoF more compact. Compared with the previously hyper-CNOT gate schemes [47], we utilize the heralded mechanism for announcing errors, so that the fidelities of the hyper-CNOT gates can achieve unity in principle, which are robust to the coupling rate $g/\kappa$ and the cavity decay rate $\kappa _s/\kappa$. The hyper-CNOT gates can work without the restriction of strong coupling. In scheme III-V, two photons enter the circuit simultaneously, which can further shorter the operation time to improving operating speed.

In summary, hyper-CNOT gates play an essential role in fast QIP. Based on single-sided QD-cavity systems, we have proposed five high-fidelity heralded hyper-CNOT gates operated on both the polarization and spatial mode DoFs of a two-photon system. Each hyper-CNOT gate has a high fidelity of unity by utilizing the heralding mechanism, which can filter the imperfect photon scattering errors. They can work faithfully and efficiently without the restriction of strong coupling. Because of less rigid experimental conditions, the hyper-CNOT gates have good practicality. Since the two DoFs of each photon can be encoded as different qubits independently, the scalability and feasibility of the schemes make them promising applications, such as expanding the quantum channel and simplifying the existing circuit using one DoF. It can also be used in large-scaled QIP networks, such as quantum super-dense coding, quantum teleportation, and quantum computation. In multi-qubit systems, they can be applied in the generation, analysis, and purification of hyper-entanglement.

Funding

National Natural Science Foundation of China (61671085, 61701035).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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14. D. P. DiVincenzo, “Two-bit gates are universal for quantum computation,” Phys. Rev. A 51(2), 1015–1022 (1995). [CrossRef]  

15. A. Barenco, “A universal two-bit gate for quantum computation,” Proc. R. Soc. London, Ser. A 449(1937), 679–683 (1995). [CrossRef]  

16. 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(5), 3457–3467 (1995). [CrossRef]  

17. E. Knill, R. Laflamme, and G. J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409(6816), 46–52 (2001). [CrossRef]  

18. K. Nemoto and W. J. Munro, “Nearly deterministic linear optical controlled-not gate,” Phys. Rev. Lett. 93(25), 250502 (2004). [CrossRef]  

19. M. A. Nielsen, “Optical quantum computation using cluster states,” Phys. Rev. Lett. 93(4), 040503 (2004). [CrossRef]  

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(6964), 264–267 (2003). [CrossRef]  

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26. W. Qin, X. Wang, A. Miranowicz, Z. Zhong, and F. Nori, “Heralded quantum controlled-phase gates with dissipative dynamics in macroscopically distant resonators,” Phys. Rev. A 96(1), 012315 (2017). [CrossRef]  

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29. M. Nakahara, Y. Kondo, K. Hata, and S. Tanimura, “Demonstrating quantum algorithm acceleration with NMR quantum computer,” Phys. Rev. A 70(5), 052319 (2004). [CrossRef]  

30. Y. Mori, R. Sawae, M. Kawamura, T. Sakata, and K. Takarabe, “Quantum circuits for an effective pure state in NMR quantum computer,” Int. J. Quantum Chem. 105(6), 758–761 (2005). [CrossRef]  

31. G. R. Feng, G. F. Xu, and G. L. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110(19), 190501 (2013). [CrossRef]  

32. M. Jiang, T. Wu, J. W. Blanchard, G. R. Feng, X. H. Peng, and D. Budker, “Experimental benchmarking of quantum control in zero-field nuclear magnetic resonance,” Sci. Adv. 4(6), eaar6327 (2018). [CrossRef]  

33. 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(4), 042323 (2013). [CrossRef]  

34. M. Li, J. Y. Lin, and M. Zhang, “High-fidelity hybrid quantum gates between a flying photon and diamond nitrogen-vacancy centers assisted by low-Q single-sided cavities,” Ann. Phys. 531(1), 1800312 (2019). [CrossRef]  

35. L. M. Duan and H. J. Kimble, “Scalable photonic quantum computation through cavity-assisted interactions,” Phys. Rev. Lett. 92(12), 127902 (2004). [CrossRef]  

36. H. R. Wei and F. G. Deng, “Scalable photonic quantum computing assisted by quantum-dot spin in double-sided optical microcavity,” Opt. Express 21(15), 17671–17685 (2013). [CrossRef]  

37. C. Cao, Y. H. Han, L. Zhang, L. Fan, Y. W. Duan, and R. Zhang, “High fidelity universal quantum controlled gates on electron-spin qubits in quantum dots inside single-sided optical microcavities,” Adv. Quantum Technol. 2(10), 1900081 (2019). [CrossRef]  

38. Y. Shi, “Both toffoli and controlled-NOT need little help to do universal quantum computing,” Quantum Inf. Comput. 3, 84–92 (2003).

39. 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. 06(03), 447–462 (2008). [CrossRef]  

40. V. V. Shende and I. L. Markov, “On the CNOT-cost of toffoli gates,” Quantum Inf. Comput. 9, 461–486 (2009).

41. B. C. Ren, H. R. Wei, and F. G. Deng, “Deterministic photonic spatial-polarization hyper-controlled-not gate assisted by a quantum dot inside a one-side optical microcavity,” Laser Phys. Lett. 10(9), 095202 (2013). [CrossRef]  

42. B. C. Ren and F. G. Deng, “Hyper-parallel photonic quantum computation with coupled quantum dots,” Sci. Rep. 4(1), 4623 (2015). [CrossRef]  

43. B. C. Ren, G. Y. Wang, and F. G. Deng, “Universal hyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime,” Phys. Rev. A 91(3), 032328 (2015). [CrossRef]  

44. T. Li and G. L. Long, “Hyperparallel optical quantum computation assisted by atomic ensembles embedded in double-sided optical cavities,” Phys. Rev. A 94(2), 022343 (2016). [CrossRef]  

45. B. C. Ren and F. G. Deng, “Robust hyperparallel photonic quantum entangling gate with cavity QED,” Opt. Express 25(10), 10863–10873 (2017). [CrossRef]  

46. B. C. Ren, A. H. Wang, A. Alsaedi, T. Hayat, and F. G. Deng, “Three-photon polarization-spatial hyperparallel quantum fredkin gate assisted by diamond nitrogen vacancy center in optical cavity,” Ann. Phys. (Berlin, Ger.) 530(5), 1800043 (2018). [CrossRef]  

47. F. F. Du and Z. R. Shi, “Robust hybrid hyper-controlled-not gates assisted by an input-output process of low-Q cavities,” Opt. Express 27(13), 17493 (2019). [CrossRef]  

48. H. R. Wei, W. Q. Liu, and N. Y. Chen, “Implementing a two-photon three-degrees-of-freedom hyper-parallel controlled phase flip gate through cavity-assisted interactions,” Ann. Phys. 532(4), 1900578 (2020). [CrossRef]  

49. C. Hu, A. Young, J. O’Brien, W. Munro, and J. Rarity, “Giant optical faraday rotation induced by a single electron spin in a quantum dot: Applications to entangling remote spins via a single photon,” Phys. Rev. B 78(8), 085307 (2008). [CrossRef]  

50. C. Hu, W. Munro, and J. Rarity, “Deterministic photon entangler using a charged quantum dot inside a microcavity,” Phys. Rev. B 78(12), 125318 (2008). [CrossRef]  

51. C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80(20), 205326 (2009). [CrossRef]  

52. 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(16), 160503 (2010). [CrossRef]  

53. L. Fan and C. Cao, “Deterministic CNOT gate and complete Bell-state analyzer on quantum-dot-confined electron spins based on faithful quantum nondemolition parity detection,” J. Opt. Soc. Am. B 38(5), 1593–1603 (2021). [CrossRef]  

54. T. Li and F. G. Deng, “Error-rejecting quantum computing with solid-state spins assisted by low-Q optical microcavities,” Phys. Rev. A 94(6), 062310 (2016). [CrossRef]  

55. C. Wang, R. Zhang, Y. Zhang, and H. Q. Ma, “Multipartite electronic entanglement purification using quantum-dot spin and microcavity system,” Quantum Inf. Process. 12(1), 525–536 (2013). [CrossRef]  

56. C. Cao, C. Wang, L. Y. He, and R. Zhang, “Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime,” Opt. Express 21(4), 4093–4105 (2013). [CrossRef]  

57. G. Y. Wang, Q. Ai, F. G. Deng, and B. C. Ren, “Imperfect-interaction-free entanglement purification on stationary systems for solid quantum repeaters,” Opt. Express 28(13), 18693–18706 (2020). [CrossRef]  

58. L. C. Lu, B. C. Ren, X. Wang, M. Zhang, and F. G. Deng, “General quantum entanglement purification protocol using a controlled-phase-flip gate,” Ann. Phys. (Berlin, Ger.) 532(4), 2000011 (2020). [CrossRef]  

59. C. Wang, “Efficient entanglement concentration for partially entangled electrons using a quantum-dot and microcavity coupled system,” Phys. Rev. A 86(1), 012323 (2012). [CrossRef]  

60. C. Cao, L. Fan, X. Chen, Y. W. Duan, T. J. Wang, R. Zhang, and C. Wang, “Efficient entanglement concentration of arbitrary unknown less-entangled three-atom W states via photonic faraday rotation in cavity QED,” Quantum Inf. Process. 16(4), 98 (2017). [CrossRef]  

61. C. Y. Hu, “Photonic transistor and router using a single quantum-dot-confined spin in a single-sided optical microcavity,” Sci. Rep. 7(1), 45582 (2017). [CrossRef]  

62. C. Cao, Y. W. Duan, X. Chen, R. Zhang, T. J. Wang, and C. Wang, “Implementation of single-photon quantum routing and decoupling using a nitrogen-vacancy center and a whispering-gallery-mode resonator-waveguide system,” Opt. Express 25(15), 16931–16946 (2017). [CrossRef]  

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2021 (3)

L. Fan and C. Cao, “Deterministic CNOT gate and complete Bell-state analyzer on quantum-dot-confined electron spins based on faithful quantum nondemolition parity detection,” J. Opt. Soc. Am. B 38(5), 1593–1603 (2021).
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Z. H. Yan, J. L. Qin, Z. Z. Qin, X. L. Su, X. J. Jia, C. D. Xie, and K. C. Peng, “Generation of non-classical states of light and their application in deterministic quantum teleportation,” Fundam. Res. 1(1), 43–49 (2021).
[Crossref]

B. Shi and Z. Zhou, “Quantum interface for high-dimensional quantum states encoded in an orbital angular momentum space,” Fundam. Res. 1(1), 88–90 (2021).
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2020 (8)

Y. Jiao, C. H. Bai, D. Y. Wang, S. Zhang, and H. F. Wang, “Optical nonreciprocal response and conversion in a tavis-cummings coupling optomechanical system,” Quant. Eng. 2(2), e39 (2020).
[Crossref]

C. R. Chang, Y. C. Lin, K. L. Chiu, and T. W. Huang, “The second quantum revolution with quantum computers,” AAPPS Bull. 30, 9–22 (2020).

G. Y. Wang, Q. Ai, F. G. Deng, and B. C. Ren, “Imperfect-interaction-free entanglement purification on stationary systems for solid quantum repeaters,” Opt. Express 28(13), 18693–18706 (2020).
[Crossref]

L. C. Lu, B. C. Ren, X. Wang, M. Zhang, and F. G. Deng, “General quantum entanglement purification protocol using a controlled-phase-flip gate,” Ann. Phys. (Berlin, Ger.) 532(4), 2000011 (2020).
[Crossref]

C. Cao, L. Zhang, Y. H. Han, P. P. Yin, L. Fan, Y. W. Duan, and R. Zhang, “Complete and faithful hyperentangled-bell-state analysis of photon systems using a failure-heralded and fidelity-robust quantum gate,” Opt. Express 28(3), 2857–2872 (2020).
[Crossref]

H. R. Wei, W. Q. Liu, and N. Y. Chen, “Implementing a two-photon three-degrees-of-freedom hyper-parallel controlled phase flip gate through cavity-assisted interactions,” Ann. Phys. 532(4), 1900578 (2020).
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T. Li and G. L. Long, “Quantum secure direct communication based on single-photon bell-state measurement,” New J. Phys. 22(6), 063017 (2020).
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X. L. Ouyang, X. Z. Huang, Y. K. Wu, W. G. Zhang, X. Wang, H. L. Zhang, L. He, X. Y. Chang, and L. M. Duan, “Experimental demonstration of quantum-enhanced machine learning in a nitrogen-vacancy-center system,” Phys. Rev. A 101(1), 012307 (2020).
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2019 (7)

J. Allcock and S. Zhang, “Quantum machine learning,” Natl. Sci. Rev. 6(1), 26–28 (2019).
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M. Li, J. Y. Lin, and M. Zhang, “High-fidelity hybrid quantum gates between a flying photon and diamond nitrogen-vacancy centers assisted by low-Q single-sided cavities,” Ann. Phys. 531(1), 1800312 (2019).
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F. F. Du and Z. R. Shi, “Robust hybrid hyper-controlled-not gates assisted by an input-output process of low-Q cavities,” Opt. Express 27(13), 17493 (2019).
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C. Cao, Y. H. Han, L. Zhang, L. Fan, Y. W. Duan, and R. Zhang, “High fidelity universal quantum controlled gates on electron-spin qubits in quantum dots inside single-sided optical microcavities,” Adv. Quantum Technol. 2(10), 1900081 (2019).
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M. Wang, R. Wu, J. Lin, J. Zhang, Z. Fang, Z. Chai, and Y. Cheng, “Chemo-mechanical polish lithography: A pathway to low loss large-scale photonic integration on lithium niobate on insulator,” Quant. Eng. 1(1), e9 (2019).
[Crossref]

J. Romero, “Shaping up high-dimensional quantum information,” AAPPS Bull. 29, 2–4 (2019).

X. W. He, Y. F. Song, Y. Yu, B. Ma, Z. S. Chen, X. J. Shang, H. Q. Ni, B. Q. Sun, X. M. Dou, H. Chen, H. Y. Hao, T. T. Qi, S. S. Huang, H. Q. Liu, X. B. Su, X. L. Su, Y. J. Shi, and Z. C. Niu, “Quantum light source devices of In(Ga)As semiconductor self-assembled quantum dots,” J. Semicond. 40(7), 071902 (2019).
[Crossref]

2018 (6)

B. C. Ren, A. H. Wang, A. Alsaedi, T. Hayat, and F. G. Deng, “Three-photon polarization-spatial hyperparallel quantum fredkin gate assisted by diamond nitrogen vacancy center in optical cavity,” Ann. Phys. (Berlin, Ger.) 530(5), 1800043 (2018).
[Crossref]

J. Z. Liu, H. R. Wei, and N. Y. Chen, “A heralded and error-rejecting three-photon hyper-parallel quantum gate through cavity-assisted interactions,” Sci. Rep. 8(1), 1885 (2018).
[Crossref]

B. Y. Xia, C. Cao, Y. H. Han, and R. Zhang, “Universal photonic three-qubit quantum gates with two degrees of freedom assisted by charged quantum dots inside single-sided optical microcavities,” Laser Phys. 28(9), 095201 (2018).
[Crossref]

Z. Zeng, “Self-assisted complete hyperentangled Bell state analysis using quantum-dot spins in optical microcavities,” Laser Phys. Lett. 15(5), 055204 (2018).
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M. Jiang, T. Wu, J. W. Blanchard, G. R. Feng, X. H. Peng, and D. Budker, “Experimental benchmarking of quantum control in zero-field nuclear magnetic resonance,” Sci. Adv. 4(6), eaar6327 (2018).
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S. Rosenblum, Y. Y. Gao, P. Reinhold, C. Wang, C. J. Axline, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. H. Devoret, and R. J. Schoelkopf, “A cnot gate between multiphoton qubits encoded in two cavities,” Nat. Commun. 9(1), 652 (2018).
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2017 (6)

W. Qin, X. Wang, A. Miranowicz, Z. Zhong, and F. Nori, “Heralded quantum controlled-phase gates with dissipative dynamics in macroscopically distant resonators,” Phys. Rev. A 96(1), 012315 (2017).
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C. Cao, L. Fan, X. Chen, Y. W. Duan, T. J. Wang, R. Zhang, and C. Wang, “Efficient entanglement concentration of arbitrary unknown less-entangled three-atom W states via photonic faraday rotation in cavity QED,” Quantum Inf. Process. 16(4), 98 (2017).
[Crossref]

C. Y. Hu, “Photonic transistor and router using a single quantum-dot-confined spin in a single-sided optical microcavity,” Sci. Rep. 7(1), 45582 (2017).
[Crossref]

C. Cao, Y. W. Duan, X. Chen, R. Zhang, T. J. Wang, and C. Wang, “Implementation of single-photon quantum routing and decoupling using a nitrogen-vacancy center and a whispering-gallery-mode resonator-waveguide system,” Opt. Express 25(15), 16931–16946 (2017).
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B. C. Ren and F. G. Deng, “Robust hyperparallel photonic quantum entangling gate with cavity QED,” Opt. Express 25(10), 10863–10873 (2017).
[Crossref]

X. Xie, Q. Xu, B. Shen, J. Chen, Q. Dai, Z. Shi, L. Yu, Z. Wang, and S. Hai-Zhi, “InGaAsP/InP micropillar cavities for 1.55 μm quantum-dot single photon sources,” J. Phys.: Conf. Ser. 844, 012002 (2017).
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2016 (6)

G. Jin, B. Liu, J. He, and J. Wang, “High on/off ratio nanosecond laser pulses for a triggered single-photon source,” Appl. Phys. Express 9(7), 072702 (2016).
[Crossref]

C. Schneider, P. Gold, S. Reitzenstein, S. Hoefling, and M. Kamp, “Quantum dot micropillar cavities with quality factors exceeding 250.000,” Appl. Phys. B 122(1), 19 (2016).
[Crossref]

T. Li and G. L. Long, “Hyperparallel optical quantum computation assisted by atomic ensembles embedded in double-sided optical cavities,” Phys. Rev. A 94(2), 022343 (2016).
[Crossref]

T. Li and F. G. Deng, “Error-rejecting quantum computing with solid-state spins assisted by low-Q optical microcavities,” Phys. Rev. A 94(6), 062310 (2016).
[Crossref]

V. Giesz, N. Somaschi, G. Hornecker, T. Grange, B. Reznychenko, L. De Santis, J. Demory, C. Gomez, I. Sagnes, A. Lemaitre, O. Krebs, N. D. Lanzillotti-Kimura, L. Lanco, A. Auffeves, and P. Senellart, “Coherent manipulation of a solid-state artificial atom with few photons,” Nat. Commun. 7(1), 11986 (2016).
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S. J. Wei and G. L. Long, “Duality quantum computer and the efficient quantum simulations,” Quantum Inf. Process. 15(3), 1189–1212 (2016).
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2015 (4)

X. Cai, D. Wu, Z. Su, M. Chen, X. Wang, L. Li, N. Liu, C. Lu, and J. W. Pan, “Entanglement-based machine learning on a quantum computer,” Phys. Rev. Lett. 114(11), 110504 (2015).
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T. R. Tan, J. P. Gaebler, Y. Lin, Y. Wan, R. Bowler, D. Leibfried, and D. J. Wineland, “Multi-element logic gates for trapped-ion qubits,” Nature 528(7582), 380–383 (2015).
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B. C. Ren and F. G. Deng, “Hyper-parallel photonic quantum computation with coupled quantum dots,” Sci. Rep. 4(1), 4623 (2015).
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B. C. Ren, G. Y. Wang, and F. G. Deng, “Universal hyperparallel hybrid photonic quantum gates with dipole-induced transparency in the weak-coupling regime,” Phys. Rev. A 91(3), 032328 (2015).
[Crossref]

2014 (1)

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(2), 025203 (2014).
[Crossref]

2013 (7)

C. Wang, R. Zhang, Y. Zhang, and H. Q. Ma, “Multipartite electronic entanglement purification using quantum-dot spin and microcavity system,” Quantum Inf. Process. 12(1), 525–536 (2013).
[Crossref]

C. Cao, C. Wang, L. Y. He, and R. Zhang, “Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime,” Opt. Express 21(4), 4093–4105 (2013).
[Crossref]

B. C. Ren, H. R. Wei, and F. G. Deng, “Deterministic photonic spatial-polarization hyper-controlled-not gate assisted by a quantum dot inside a one-side optical microcavity,” Laser Phys. Lett. 10(9), 095202 (2013).
[Crossref]

H. R. Wei and F. G. Deng, “Scalable photonic quantum computing assisted by quantum-dot spin in double-sided optical microcavity,” Opt. Express 21(15), 17671–17685 (2013).
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G. R. Feng, G. F. Xu, and G. L. Long, “Experimental realization of nonadiabatic holonomic quantum computation,” Phys. Rev. Lett. 110(19), 190501 (2013).
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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(4), 042323 (2013).
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K. Muller, T. Kaldewey, R. Ripszam, J. S. Wildmann, A. Bechtold, M. Bichler, G. Koblmuller, G. Abstreiter, and J. J. Finley, “All optical quantum control of a spin-quantum state and ultrafast transduction into an electric current,” Sci. Rep. 3(1), 1906 (2013).
[Crossref]

2012 (3)

C. Arnold, V. Loo, A. Lematre, I. Sagnes, O. Krebs, P. Voisin, P. Senellart, and L. Lanco, “Optical bistability in a quantum dots/micropillar device with a quality factor exceeding 200.000,” Appl. Phys. Lett. 100(11), 111111 (2012).
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Y. Dong, R. Wang, H. Li, J. Shao, Y. Chi, X. Lin, and G. Chen, “Polyamine-functionalized carbon quantum dots for chemical sensing,” Carbon 50(8), 2810–2815 (2012).
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C. Wang, “Efficient entanglement concentration for partially entangled electrons using a quantum-dot and microcavity coupled system,” Phys. Rev. A 86(1), 012323 (2012).
[Crossref]

2011 (1)

C. Y. Hu and J. G. Rarity, “Loss-resistant state teleportation and entanglement swapping using a quantum-dot spin in an optical microcavity,” Phys. Rev. B 83(11), 115303 (2011).
[Crossref]

2010 (2)

S. Reitzenstein and A. Forchel, “Quantum dot micropillars,” J. Phys. D: Appl. Phys. 43(3), 033001 (2010).
[Crossref]

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(16), 160503 (2010).
[Crossref]

2009 (5)

C. Y. Hu, W. J. Munro, J. L. O’Brien, and J. G. Rarity, “Proposed entanglement beam splitter using a quantum-dot spin in a double-sided optical microcavity,” Phys. Rev. B 80(20), 205326 (2009).
[Crossref]

V. V. Shende and I. L. Markov, “On the CNOT-cost of toffoli gates,” Quantum Inf. Comput. 9, 461–486 (2009).

P. Aliferis, F. Brito, D. P. DiVincenzo, J. Preskill, M. Steffen, and B. M. Terhal, “Fault-tolerant computing with biased-noise superconducting qubits: a case study,” New J. Phys. 11(1), 013061 (2009).
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A. Greilich, S. E. Economou, S. Spatzek, D. R. Yakovlev, D. Reuter, A. D. Wieck, T. L. Reinecke, and M. Bayer, “Ultrafast optical rotations of electron spins in quantum dots,” Nat. Phys. 5(4), 262–266 (2009).
[Crossref]

J. A. Richardson, L. A. Grant, and R. K. Henderson, “Low dark count single-photon avalanche diode structure compatible with standard nanometer scale CMOS technology,” IEEE Photonics Technol. Lett. 21(14), 1020–1022 (2009).
[Crossref]

2008 (4)

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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)

Fig. 1.
Fig. 1. (a) Structure of a singly charged QD coupled to a single-sided micropillar cavity. (b) The energy level of a QD-cavity system and the optical transition rules of negatively charged exciton $X^-$. $|\uparrow \rangle \rightarrow |\uparrow \downarrow \Uparrow \rangle$ and $|\downarrow \rangle \rightarrow |\downarrow \uparrow \Downarrow \rangle$ are driven by the left-circularly photon $|L\rangle$ and the right-circularly photon $|R\rangle$, respectively.
Fig. 2.
Fig. 2. The absolute amplitude $|r_-(\Delta )|$ of $r_-(\Delta )$ with $\gamma =0.01\kappa$. (a) is for different coupling rate with $\kappa _s=0$. (b) is for different cavity-decay rate with the coupling rate $g=\kappa$.
Fig. 3.
Fig. 3. The schematic diagram of hyper-CNOT gate type I. BLOCK$_1$ and BLOCK$_2$ represent two different units of interactions. In BLOCK$_1$, only the $|V\rangle$ photon interacts with the electron spin confined in the QD-cavity. In BLOCK$_2$, both the $|V\rangle$ and $|H\rangle$ photon interact with the electron spin of the cavity in sequence. PBS$_l$ ($l$=1, 2, 3, 4) is a polarization beam splitter, which transmits the photon in the horizontal polarization $|H\rangle$ and reflects the photon in the vertical polarization $|V\rangle$, respectively. H$_m$ ($m$=1, 2, 3, 4) represents a half-wave plate set at $22.5^{\circ }$ to perform a Hadamard operation on the polarization of a photon. BS$_n$ ($n$=1, 2) represents a 50:50 beam splitter to perform a Hadamard operation on the spatial mode DoF of a photon. X$_f$ ($f$=1, 2) is a half-wave plate set at $45^{\circ }$ to perform a polarization bit-flip operation $\sigma ^P_X=|H\rangle {\langle }V|+|V\rangle {\langle }H|$. R$_\theta =r_-(\Delta )|c\rangle$ complete a rotation that modifies the shape and intensity of the photons passing through it by $r_-(\Delta )$, ($c=H, V$). $P_\theta =|H\rangle {\langle }H|+e^{-\frac {\pi }{2}i}|V\rangle {\langle }V|$ denotes a quantum phase gate on the polarization of the photon. VBS$_v$ ($v$=1, 2) denotes an adjustable beam splitter with transmission coefficient $r_-(\Delta )$ and reflection coefficient $\sqrt {1-r_-^2(\Delta )}$. D$_w$ ($w$=1, 2, 3, 4) is a single-photon detector. DLs are delay lines, which make the photon in two spatial modes arrive simultaneously. u and d denote the spatial mode in BLOCK$_1$ and BLOCK$_2$.
Fig. 4.
Fig. 4. The schematic diagram of the hyper-CNOT gate type II. The polarization and the spatial mode of photon $a$ are the control qubits, while the spatial mode and the polarization of photon $b$ are the target qubits, respectively. The elements have the same function as the ones depicted in Fig. 3.
Fig. 5.
Fig. 5. The schematic diagram of hyper-CNOT gate type III. The spatial mode of photon $a$ $(b)$ is the control qubit, when the polarization mode of photon $b$ $(a)$ is the target qubit. The elements have the same function as the ones depicted in Fig. 3.
Fig. 6.
Fig. 6. The schematic diagram of hyper-CNOT gate type IV. The polarization mode of photon $a$ $(b)$ is the control qubit, when the spatial mode of photon $b$ $(a)$ is the target qubit. The elements have the same function as the ones depicted in Fig. 3.
Fig. 7.
Fig. 7. The schematic diagram of hyper-CNOT gate type V. The polarization of photon $a$ is the control qubit, when that of photon $b$ is the target qubit. The spatial mode of photon $b$ is the control qubit when that of photon $a$ is the target qubit. The elements have the same function as the ones depicted in Fig. 3.
Fig. 8.
Fig. 8. (a) and (b) are the relation between the coupling rate $g/\kappa$ and the probability amplitude $|r_-(\Delta )|$ at the frequency detuning $\Delta =0$ and $\Delta =g$, respectively. (c) and (d) are the relation between $g/\kappa$ and the efficiency $\eta$ at the frequency detuning $\Delta =0$ and $\Delta =g$, respectively. We take $\gamma =0.01\kappa$ here.

Equations (28)

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H = ω X σ ^ + σ ^ + ω c a ^ a ^ + i g ( σ ^ + a ^ a ^ σ ^ ) .
d a ^ d t = [ i ( ω c ω ) + κ 2 + κ s 2 ] a ^ g σ ^ κ a ^ i n + H ^ , d σ ^ d t = [ i ( ω X ω ) + γ 2 ] σ ^ g σ ^ z a ^ + G ^ .
r ( ω , g ) = [ i ( ω X ω ) + γ 2 ] [ i ( ω c ω ) κ 2 + κ s 2 ] + g 2 [ i ( ω X ω ) + γ 2 ] [ i ( ω c ω ) + κ 2 + κ s 2 ] + g 2 .
r ( Δ , g ) = ( i Δ + γ 2 ) ( i Δ κ 2 + κ s 2 ) + g 2 ( i Δ + γ 2 ) ( i Δ + κ 2 + κ s 2 ) + g 2 ,
r ( Δ , 0 ) = i Δ κ 2 + κ s 2 i Δ + κ 2 + κ s 2 .
| H | ± [ r + ( Δ ) | H | ± + i r ( Δ ) | V | ] / p 1 , | V | ± [ i r + ( Δ ) | V | ± + r ( Δ ) | H | ] / p 1 ,
| Ψ a = | ψ a p | ψ a s = ( α 1 | H + β 1 | V ) a ( γ 1 | a 1 + δ 1 | a 2 ) , | Ψ b = | ψ b p | ψ b s = ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) ,
| Ψ a b e 1 e 2 0 = ( α 1 | H + β 1 | V ) a ( γ 1 | a 1 + δ 1 | a 2 ) ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) | + 1 | + 2 , | Ψ a b e 1 e 2 1 = [ r ( Δ ) ( α 1 | H d | + 1 + β 1 | V u | 1 ) a + i r + ( Δ ) β 1 | H a u | + 1 ] ( γ 1 | a 1 + δ 1 | a 2 ) ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) | + 2 .
| Ψ a b e 1 e 2 2 = i r + ( Δ ) β 1 | H a u | + 1 ( γ 1 | a 1 + δ 1 | a 2 ) ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) | + 2 .
| Ψ a b e 1 e 2 3 = r ( Δ ) ( α 1 | H d | + 1 + β 1 | V u | 1 ) a ( γ 1 | a 1 + δ 1 | a 2 ) ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) | + 2 .
| Ψ a b e 1 e 2 4 = r 2 ( Δ ) ( α 1 | H a | + 1 + β 1 | V a | 1 ) ( γ 1 | a 1 | + 2 + δ 1 | a 2 | 2 ) ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) .
| Ψ a b e 1 e 2 5 = r 2 ( Δ ) ( α 1 | H a | + 1 + | 1 2 + β 1 | V a | + 1 | 1 2 ) ( γ 1 | a 1 | + 2 + | 2 2 + δ 1 | a 2 | + 2 | 2 2 ) ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) .
| Ψ a b e 1 e 2 6 = r 2 ( Δ ) ( α 1 | H a | + 1 + | 1 2 + β 1 | V a | + 1 | 1 2 ) ( γ 1 | a 1 | + 2 + | 2 2 + δ 1 | a 2 | + 2 | 2 2 ) ( α 2 | H + β 2 | V ) b ( γ 2 | b 1 + δ 2 | b 2 ) ,
| Ψ a b e 1 e 2 7 = r 3 ( Δ ) 2 { [ α 1 | H a ( α 2 | H b + β 2 | V b ) + β 1 | V a ( β 2 | H b + α 2 | V b ) ] | + 1 + [ α 1 | H a ( α 2 | H b + β 2 | V b ) β 1 | V a ( β 2 | H b + α 2 | V b ) ] | 1 } ( γ 1 | a 1 | + 2 + δ 1 | a 2 | 2 ) ( γ 2 | b 1 + δ 2 | b 2 ) ,
| Ψ a b e 1 e 2 8 = r 4 ( Δ ) 2 { [ α 1 | H a ( α 2 | H b + β 2 | V b ) + β 1 | V a ( β 2 | H b + α 2 | V b ) ] | + 1 + [ α 1 | H a ( α 2 | H b + β 2 | V b ) β 1 | V a ( β 2 | H b + α 2 | V b ) ] | 1 } { [ γ 1 | a 1 ( γ 2 | b 1 + δ 2 | b 2 ) + δ 1 | a 2 ( δ 2 | b 1 + γ 2 | b 2 ) ] | + 2 + [ γ 1 | a 1 ( γ 2 | b 1 + δ 2 | b 2 ) δ 1 | a 2 ( δ 2 | b 1 + γ 2 | b 2 ) ] | 2 } .
| Ψ a b 1 = r 4 ( Δ ) [ α 1 | H a ( α 2 | H b + β 2 | V b ) + β 1 | V a ( β 2 | H b + α 2 | V b ) ] [ γ 1 | a 1 ( γ 2 | b 1 + δ 2 | b 2 ) + δ 1 | a 2 ( δ 2 | b 1 + γ 2 | b 2 ) ] = r 4 ( Δ ) ( α 1 | H a + β 1 | V a σ X b P ) ( α 2 | H b + β 2 | V b ) ( γ 1 | a 1 + δ 1 | a 2 σ X b S ) ( γ 2 | b 1 + δ 2 | b 2 ) .
| Ψ a b e 1 e 2 9 = r 4 ( Δ ) 2 { [ α 1 | H a ( γ 2 | b 1 + δ 2 | b 2 ) + β 1 | V a ( δ 2 | b 1 + γ 2 | b 2 ) ] | + 1 + [ α 1 | H a ( γ 2 | b 1 + δ 2 | b 2 ) β 1 | V a ( δ 2 | b 1 + γ 2 | b 2 ) ] | 1 } { [ γ 1 | a 1 ( α 2 | H b + β 2 | V b ) + δ 1 | a 2 ( β 2 | H b + α 2 | V b ) ] | + 2 + [ γ 1 | a 1 ( α 2 | H b + β 2 | V b ) δ 1 | a 2 ( β 2 | H b + α 2 | V b ) ] | 2 } .
| Ψ a b 2 = r 4 ( Δ ) ( α 1 | H a + β 1 | V a σ X b S ) ( γ 2 | b 1 + δ 2 | b 2 ) ( γ 1 | a 1 + δ 1 | a 2 σ X b P ) ( α 2 | H b + β 2 | V b ) .
| Ψ a b e 1 e 2 10 = r 4 ( Δ ) 2 { [ γ 1 | a 1 ( α 2 | H b + β 2 | V b ) + δ 1 | a 2 ( β 2 | H b + α 2 | V a ) ] | + 1 + [ γ 1 | a 1 ( α 2 | H b + β 2 | V b ) δ 1 | a 2 ( β 2 | H b + α 2 | V b ) ] | 1 } { [ γ 2 | b 1 ( α 1 | H a + β 1 | V a ) + δ 2 | b 2 ( β 1 | H a + α 1 | V a ) ] | + 2 + [ γ 2 | b 1 ( α 1 | H a + β 1 | V a ) δ 2 | b 2 ( β 1 | H a + α 1 | V a ) ] | 2 } .
| Ψ a b 3 = r 4 ( Δ ) ( γ 1 | a 1 + δ 1 | a 2 σ X b P ) ( α 2 | H b + β 2 | V b ) ( γ 2 | b 1 + δ 2 | b 2 σ X a P ) ( α 1 | H a + β 1 | V a ) .
| Ψ a b e 1 e 2 11 = r 4 ( Δ ) 2 { [ α 1 | H a ( γ 2 | b 1 + δ 2 | b 2 ) + β 1 | V a ( δ 2 | b 1 + γ 2 | b 2 ) ] | + 1 + [ α 1 | H a ( γ 2 | b 1 + δ 2 | b 2 ) β 1 | V a ( δ 2 | b 1 + γ 2 | b 2 ) ] | 1 } { [ α 2 | H b ( [ γ 1 | a 1 + δ 1 | a 2 ) + β 2 | V b ( δ 1 | a 1 + γ 1 | a 2 ) ] | + 2 + [ α 2 | H b ( γ 1 | a 1 + δ 1 | a 2 ) β 2 | V b ( δ 1 | a 1 + γ 1 | a 2 ) ] | 1 } .
| Ψ a b 4 = r 4 ( Δ ) ( α 1 | H a + β 1 | V a σ X b S ) ( γ 2 | b 1 + δ 2 | b 2 ) ( α 2 | H b + β 2 | V b σ X a S ) ( γ 1 | a 1 + δ 1 | a 2 ) .
| Ψ a b e 1 e 2 12 = r 4 ( Δ ) 2 { [ α 1 | H a ( α 2 | H b + β 2 | V b ) + β 1 | V a ( β 2 | H b + α 2 | V 2 ) ] | + 1 + [ α 1 | H a ( α 2 | H b + β 2 | V b ) β 1 | V a ( β 2 | H b + α 2 | V 2 ) ] | 1 } { [ γ 2 | b 1 ( γ 1 | a 1 + δ 1 | a 2 ) + δ 2 | b 2 ( δ 1 | a 1 + γ 1 | a 2 ) ] | + 2 + [ γ 2 | b 1 ( γ 1 | a 1 + δ 1 | a 2 ) δ 2 | b 2 ( δ 1 | a 1 + γ 1 | a 2 ) ] | 1 } .
| Ψ a b 5 = r 4 ( Δ ) ( α 1 | H a + β 1 | V a σ X b P ) ( α 2 | H b + β 2 | V b ) ( γ 2 | b 1 + δ 2 | b 2 σ X a S ) ( δ 1 | a 1 + γ 1 | a 2 ) .
| Ψ i = ( α 1 | H a + β 1 | V a σ X b P ) ( α 2 | H b + β 2 | V b ) ( γ 1 | a 1 + δ 1 | a 2 σ X b S ) ( γ 2 | b 1 + δ 2 | b 2 ) .
| Ψ r = r 4 ( Δ ) ( α 1 | H a + β 1 | V a σ X b P ) ( α 2 | H b + β 2 | V b ) ( γ 1 | a 1 + δ 1 | a 2 σ X b S ) ( γ 2 | b 1 + δ 2 | b 2 ) .
F ¯ = | Ψ r | Ψ i | 2 ¯ = 1 4 π 4 0 2 π d θ p 0 2 π d ϕ p 0 2 π d θ s 0 2 π d ϕ s | Ψ r | Ψ i | 2 Ψ r | Ψ r = 1.
η = | r ( Δ ) | 8 = | r ( Δ , 0 ) r ( Δ , g ) 2 | 8 = | 4 g 2 / κ 2 ( 2 i Δ / κ + 1 + κ s / κ ) [ ( 2 i Δ / κ + γ / κ ) ( 2 i Δ / κ + 1 + κ s / κ ) + 4 g 2 / κ 2 ] | 8 .

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