A heralded and error-rejecting three-photon hyper-parallel quantum gate through cavity-assisted interactions

Hyper-parallel quantum computation is a promising and fruitful area of research with its high capacity and low loss rate characters. In this paper, we propose a heralded, compact, scalable, and deterministic error-rejecting scheme for implementing three-photon hyper-parallel Toffoli gate simultaneously acting on polarization and spatial degrees of freedom. It is a practical and unity gate without strong coupling strength limitations, since the undesired performances caused by the side leakage and the limited coupling strength are detected by the single-photon detectors. The success of our proposal can be heralded by the detectors, and the efficiency can be further improved by repeating the operation processes when the detectors are clicked. The evaluation of gate performance with experimental parameters shows that it is feasible with current experimental technology.


Results
The optical property of a QD-microcavity platform. Now we consider a singly charged QD [e.g., a self-assembled in(Ga)As QD or GaAs QD] incorporated into the center of a double-sided optical resonator with tens to hundreds GHz (high speed) 29,52 . As shown in Fig. 1, the QD is confined in the center of the double-sided microcavity. When an excess electron is injected, a negative charged exciton (X − ) consisting of two electrons and a hole is created by optical excitation 53 . Here, both the ground states (the electron spin states) and the excited states (the spin states of the X − ) are twofold degenerate due to the Kramer's theorem 54,55 . There are two spin-dependent transition interacting with circularly polarized lights in QD-doubled-sided unite 56 due to the Pauli's exclusion principle and the conservation of the total spin angular momentum. The s z = +1 polarized photon (marked by |L ↓ 〉 and |R ↑ 〉) and s z = −1 polarized photon (marked by |L ↑ 〉 and |R ↓ 〉) only couple the transition |↑〉 → |↑↓ 〉 and |↓〉 → |↓↑ 〉, respectively, and then the incident photons are reflected by the cavity. Otherwise, the photons can not couple to the QD and feel the cold cavity (g = 0), and then it is transmitted through the cavity. Upon reflection, both the polarization and the propagation direction of the incident photon will be flipped. Whereas, the spin of the electron remains unchanged upon both the reflection and transmission. Here, |R〉 and |L〉 denote the right-and left-handed circularly polarized light, respectively. The superscripts |↑〉 and |↓〉 of |M ↑ 〉 and |M ↓ 〉 indicate the propagation direction of the M-polarized photon parallel and antiparallel to the z axis (the growth and optical axis). | 〉 and | 〉 represent the heavy-hole spin states with ± 3 2 , respectively. |↑〉 and |↓〉 represent the electron spin states with ± 1 2 , respectively. The reflection and the transmission coefficients of double-sided cavity can be obtained from the Heisenberg equations of motion for the cavity field operator â, the (X − ) dipole operator σ − and the input-output relations between the output fields â r , â t , and the input fields â in , ′ a in 57 , Figure 1. (a) A schematic diagram of the quantum dot-cavity coupled system. (b) Schematic description of the spin-dependent optical transition rules. |L ↑ 〉 (|L ↓ 〉) and |R ↑ 〉 (|R ↓ 〉) represent the left-and right-handed circularly polarized photons parallelled (antiparallelled) with the growth axis, respectively. |↑〉 and |↓〉 represent the electron spin states with ± where ω, ω c , and ω − X denote the frequencies of the external field (probe photon), the cavity mode, and the X − transition, respectively. g denotes the cavity coupling strength between X − and cavity mode. γ/2, κ, and κ s /2 denote the decay rates of the X − dipole, the cavity field, and the leaky mode (side leakage), respectively. Ĥ and Ĝ are the noise operators.
In the approximation of a weak excitation condition with (X − ) predominantly staying in the ground state, and taking 〈σ z 〉 ≈ −1 and σ = −â a z , one can find that the reflection and the transmission coefficients of the (X − ) cavity system can be expressed as 29,58 , For convenience to discussions, we consider the cavity mode is resonant with optical transition of the QD, i.e., ω ω ω = = − c X 0 , and then the interaction between the incident photon and the cavity can be summarized as However, in the practical working, the imperfect birefringence of the cavity, caused by the nonzero photon bandwidth, the mismatch and the finite coupling rate between the photon and the cavity mode, makes Eq.  Here the reflection coefficient r (r 0 ) and the transmission coefficient t (t 0 ) are described by Eq. (2) with g ≠ 0 (g = 0). In the following, we introduce our practical proposal for implementing emission-based three-photon hyper-parallel controlled-controlled-phase-flip gate, step by step. It is known that Toffoli gate is equivalent to the controlled-controlled-phase-flip gate upon to two Hadamard gates acting on the target qubit.
Three-photon Toffoli gate acting on polarization DOF. Let us first introduce the performance of the key building block in our scheme. As shown in Fig. 2(a), the circularly polarizing beam splitter, PBS 1 , transmits the input R-polarized wave packet into spatial mode k 1 and reflects the L-polarized wave packet into spatial mode i 1 . Before and after the wave packets emitted from spatial mode k 1 and k 2 interact with the QD, Hadamard operations are performed on the spatial mode and the polarized mode respectively. The performance of the polarized-(spatial-) Hadamard operation can be written as   Subsequently, the bit-flip operation X, which can be implemented by using a half wave plate oriented at 45°, is performed on the wave packet emitted from the spatial mode k 2 . The wave packet emitted from the spatial mode . And then, the wave packets emitted from the spatial mode k 2 and j 2 arrive at PBS 2 , simultaneously. Therefore, X, VBS, and PBS 2 make Eq. (7) changed as  A schematic diagram for implementing a polarized-Toffoli gate assisted by double-sided microcavities. PBS j (j = 1, …, 13), a circularly polarizing beam splitter, which is used to transmit the R-polarized wave packets and reflect the L-polarized wave packets. BS j (j = 1, … 5), balanced nonpolarizing beam splitter, transforms the input modes as → represents a Hadamard operation on polarization, and it can be completed by using a half wave plate rotating at 22.5°. X j (j = 1, 2, …, 5) represents a bit-flip operation on polarization, and it can completed by using a half wave plate rotating at 45°.
From Eqs (5-9), when the detectors are not clicked, the transformation of the building block composed of PBS 1 , PBS 2 , BS, VBS, H p1 , H p2 , QD, X, D 1 , and D 2 , can be written as block 0 in the basis {|R ↑〉, |R ↓〉, |L ↑〉, |L ↓〉}. The efficiency of the key building block can be further improved by repeating the operation processes when the detectors are clicked. Now we introduce the performance of our polarized-Toffoli gate, step by step. As shown in Fig. 2(b), suppose that the initial states of gate photons a, b, and c are prepared as |ϕ a 〉, |ϕ b 〉, |ϕ c 〉, respectively. Electron mediums 1 and 2 are initially prepared in the states |↓ 1 〉 and |↓ 2 〉, respectively. Here, Where a 1 and a 2 (b 1 , b 2 or c 1 , c 2 ) represent the two spatial modes of the photon a (b or c).
When detectors D 1 and D 2 are not clicked, the above operations (H e1 → building block → H e2 ) transform the system composed of photon a, b, c, and QD 1 and QD 2 from the initial state |ϕ 0 〉 into |ϕ 1 〉. Here, Second, photon c in spatial modes c 1 and c 2 is injected and arrives at PBS 3 . PBS 3 transmits the R 3 -polarized wave packet to PBS 5 and reflects the L 3 -polarized wave packet into the building block composed of PBS 4 , VBS 2 , BS 2 , H p5 , H p6 , QD 1 , X 2 , PBS 6 , D 3 , and D 4 . Before and after the wave packets interact with the block, Hadamard operations, H p3 and H p4 , are performed on it, respectively. When D 1,…,4 are not clicked, operations (PBS 3 → PBS 5 → H p3 → building block → H p4 ) make |ϕ 1  Third, photon a in the spatial modes a 1 or a 2 is launched into the building block composed of PBS 7 , VBS 3 , BS 3 , H p7 , H p8 , QD 2 , X 3 , PBS 8 , D 5 , and D 6 . Before and after the wave packet interacts with the block, H e3 and H e4 are performed on QD 2 , respectively. When D 1,…,6 are not clicked, operations (H e3 → building block → H e4 ) make |ϕ 2 〉 changed as Fourth, photon c is leaded into the building block composed of PBS 9 , VBS 4 , BS 4 , H p9 , H p10 , QD 2 , X 4 , PBS 10 , D 7 , and D 8 . If D 1,…,8 are not clicked, the joint state is collapsed into Fifth, after photon c passes through VBS 6 and the building block composed of PBS 11 , VBS 5 , BS 5 , H p13 , H p14 , QD 1 , X 5 , PBS 12 , D 9 , and D 10 , the desired wave packets are mixed at PBS 13 . If D 1,…,11 are not clicked, the state of the system is then becoming Sixth, the QD 1 and QD 2 are measured in the basis |±〉 = | ↑〉±| ↓〉 { ( )/ 2 }. On detecting the QD 1 and QD 2 in the state |+ 1 〉 and |+ 2 〉, one disentangled |ϕ 5 〉 into the desired outcomes of the polarized-Toffoli gate, that is, As for the |+ 1 〉 and |− 2 〉 case, classical feed-forward single-qubit operations σ z = |R〉 〈R| − |L〉 〈L|, which can be implemented by using half wave plate oriented at 0°, are performed on the outing photon a to lead the outcomes to the desired state described by Eq. (19). As for the |− 1 〉 and |+ 2 〉 case, σ z s are performed on the outing photon b. As for the |− 1 〉 and |− 2 〉 case, σ z s are performed on the outing photon b and a, respectively.
Putting all the pieces together, one can see that Fig. 2(b) can deterministically implement an error-rejecting three-photon polarized-Toffoli gate in a heralded way, without any influence on their spatial mode quantum states.
Three-photon Toffoli gate acting on spatial DOF. Up to now, we have discussed the implementation of the error-rejecting polarized-Toffoli gate without any influence on their spatial mode quantum states. In order to implement a hyper-parallel Toffoli gate performing controlled-controlled-NOT operations on the polarization and spatial DOFs, independently, a scheme for implementing error-rejecting three-photon spatial-Toffoli gate without any negative influence on their polarization states will be designed in this subsection.
The scheme we designed for implementing the spatial-Toffoli gate without influence on their polarization states is depicted by Fig. 3, and it can be completed by six steps. The three photons a, b, and c are initially prepared in the arbitrary product states with polarization and spatial DOFs (see Eq. (11)). QD 1 and QD 2 are prepared in the states |↑ 1 〉 and |↓ 2 〉, respectively.
First, the photon b is injected. The wave packets emitted from the spatial mode b 1 arrive at VBS 1 with transmission coefficient t − t 0 and reflection coefficient − − t t 1 ( ) 0 2 . While the ones emitted from the spatial mode b 2 arrive at PBS 1 , and PBS 1 transmits the R-polarized wave packet to the right round composed of BS 2 , H p3 , H p4 , QD 1 and reflects the L-polarized wave packet to the left round composed of X 1 , BS 1 , H p1 , H p2 , QD 1 , X 2 . Here the two rounds complete the transformations  Second, photon c emitted from the spatial mode c 1 passes through VBS 2 with transmit (t − t 0 ) 3 and reflect − − t t 1 ( ) 0 6 . Photon c emitted from the spatial mode c 2D passes through the round composed of PBS 1 , X 1 , H p1 , H p2 , BS 1 , X 2 , QD 1 , PBS 2 or the round composed of PBS 1 , H p3 , H p4 , BS 2 , QD 1 , PBS 2 . Before and after the photon passes through the two rounds, Hadamard operation H s s, which complete the transformation, are performed on the spatial mode of photon c by BS 1 and BS 2 . It is worthy to note that after BS 3 , only the wave packets emitted from the spatial mode c 2D interact with QD 1  Third, photon a emitted from the spatial mode a 1 passes through VBS 3 with transmit (t − t 0 ), and the one emitted from the spatial mode a 2 passes through the round composed of PBS 3 , X 3 , H p7 , H p8 , BS 5 , X 4 , QD 2 , PBS 4 or the round composed of PBS 3 , H p5 , H p6 , BS 6 , QD 2 , PBS 4 . Here, before and after the photon passes through the two rounds, H e s are performed on QD 2 , respectively. When D 5 and D 6 are not clicked, |ψ 2 〉 is collapsed into the desired state Fifth, after wave packets emitted from the spatial c 2U and c 2D are mixed at BS 7 , the wave packet emitted from c 2U passes through the block composed of PBS 1 , PBS 2 , X 1 , X 2 , BS 1 , BS 2 , H p1 , H p2 , H p3 , and H p4 , and then they are mixed with the wave packet emitted from c 2D passing through the VBS 6 and BS 8 . When D 9 , D 10 and D 11 are not clicked, one can see that |ψ 4 〉 is collapsed into the desired state

Discussion
QDs mimic the behavior of single atomic dipole-like transitions. However, unlike atoms, QDs can be easily incorporated into solid-state devices such as cavities or waveguide that enhance the light-matter interaction for scalable QIP. Moreover, the manipulation can be achieved with high-speed (up to THz). The coupling efficiency of the pillar microcavity is higher than waveguide as the pillar cavity mode is Guassian type and matches perfectly with the external laser beam. Compared to the devices via QD-double-sided emitters, the ones via QD-single-sided emitters are fragile due to the balanced reflectance, for the coupled and uncoupled cavities are necessary to get high fidelity 29 . The length and deep of the gates via QD-double-sided emitters usually are surpass the ones via QD-single-sided emitters as the maximum Faraday rotation π can be achieved in particular QD-double-sided emitter 29 , whereas π/2 is for the QD-single-sided oemitter 60 .
The unconstructed low theoretical lower bound for a generic traditional n-qubit gates is    − −    n (4 3 1) n 1 4 CNOT gates 61 . Up to now, the optimal cost of a traditional Toffoli gate acting on single DOF is 6 CNOT gates 12,62 . The scheme we designed for physical implementing a hyper-parallel Toffoli gate acting on two DOFs, and the essential cost of our scheme is 6 CNOT gates. Compared to the traditional one, our hyper-parallel Toffoli gate reduces the quantum resource, the operation time, and the influence of the noise by half. The previous works mainly were investigated via the optical transition rules in Eq. (3), that is, the side-leakage and the imperfect birefringence of the cavity are not taken into account 28,29,37 and t 0 → −1, r → 1. Our hyper-parallel Toffoli gate are constructed via Eq. (4), and the undesired performances caused by the side leakage and the imperfect birefringence are detected by the detectors. Moreover, the strong coupling limitation can be avoided in our scheme because the fidelity of our scheme is unity in principle.
In summary, we have designed a quantum circuit for implementing a heralded error-rejecting hyper-parallel Toffoli gate assisted by QD-double-sided cavities. It is a practical proposal, and the side leakage and the imperfect birefringence of the incident photons are detected. The fidelity of the present gate is always unity and the efficiency can be further improved by repeating the construction processes. In addition, our scheme overcomes the exciting strong coupling limitations, and can work in both the strong-coupling and the weak-coupling regimes. These interesting features make us believe that the present work may be useful for some practical hyper-parallel quantum information processing, including hyper-entanglement concentration 63 and purification 48 , hyper-entanglement state analysis, hyper-parallel quantum repeater and so on.
Evaluation of the performance. An efficient QD-photon has been recognized as a potential building block for QIP due to their inherent scalability and mature semiconductor technology 28,29 . QDs incorporated into photonic-crystal (PC) waveguides 64,65 , photonic nanowires 66 , pillar microcavities 37,67,68 , and photonic crystal cavities cite 69 have been achieved in experiment. In the present, the double-sided symmetric pillar cavity supports circularly polarized light, and it incorporates a negative charged QD. This type of cavity has been fabricated in experiment 68 . Some specific symmetry photonic crystal nanocavities 70 are suitable for proposal as well. Compared to weak coupling cavity-QED system, strong coupling cavity-QED system is a challenge in experiment with current technology. Fortunately, the strong coupling regime g/(κ + κ s ) = 2.4 (g = 80 μeV, κ + κ s = 33 μeV), has been  29,37 showed that the π phase shift, which is necessary for the emitter-based quantum gates, can be achieved in this particular double-sided-QD-cavity combination.
The performance of the gate can be characterized by the fidelity and efficiency. According to the arguments in result, one can find that the unity fidelity of our practical hyper-parallel Toffoli gate can be achieved in principle if the overall phase is ignored. The efficiency of the present gate depends on cavity QED parameters (g, κ, κ s , γ), and it can be further improved by repeating the operation processes when the undesired performances are detected. Here, the efficiency is defined as the ratio between the number of the output photons to the input photons. The shape functions η is plotted in Fig. 4 for the average efficiency of the present gate averaged over [0, 2π]. γ ~ μeV caused by the the spontaneous emission and the pure dephasing is usually smaller than κ + κ s in high-quality QD-cavity samples. Here γ = 0.1κ, which is experimentally achieved, is taken. Figure 4 suggests that we could make high efficiency in the strong coupling regime when the side leakage κ s /κ is small. The strong coupling QD-cavity system has been reported 67, 71 . The side leakage, κ s , can be reduced by engineering the fabrication and various cavity details such as materials, structures, size, etc.
Photon loss has no contribution to the fidelity of the gate. In realistic case, the fidelity of the emitter can be decreased by a factor 29  , due to spin decoherence. Therefore, the lower is the Δt T / e 2 , the higher is the F. Here T e 2 and Δt are the electron spin coherence time and the time interval between input photons, respectively. μ ∼ T s e 2 and Δt ~ ns have been achieved in experiment 72 . The imperfect spin-selection rules, induced by the heavy-light hole mixing, reduce emitter fidelity by a few percent 73,74 . The heavy-light hole mixing depends on the kind, shape and size of the QD. The emitter fidelity can be reduced by amount of F = [1 − exp(−τ/T 2 )] caused by the QD spin dephasing, and such influence can be neglected because T 2 ~ μs and τ ~ ps have been experimentally demonstrated in the coupling regime with 10 4 -10 5 75,76 . Here, τ and T 2 are the cavity photon lifetime and QD coherence time, respectively. Moreover, the technical imperfection such as photon detection events, the unbalanced PBSs, and the spatial mismatch between the cavity mode and the photon influence the success probability and the fidelity of the gate.