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Teleportation of a Toffoli gate among distant solid-state qubits with quantum dots embedded in optical microcavities

Teleportation of unitary operations can be viewed as a quantum remote control. The remote realization of robust multiqubit logic gates among distant long-lived qubit registers is a key challenge for quantum computation and quantum information processing. Here we propose a simple and deterministic scheme for teleportation of a Toffoli gate among three spatially separated electron spin qubits in optical microcavities by using local linear optical operations, an auxiliary electron spin, two circularly-polarized entangled photon pairs, photon measurements, and classical communication. We assess the feasibility of the scheme and show that the scheme can be achieved with high average fidelity under the current technology. The scheme opens promising perspectives for constructing long-distance quantum communication and quantum computation networks with solid-state qubits.

qubits have been proposed and made some interesting progress both in theory and in experiment [15][16][17][18][19][20] . From our point of view, therefore, it is very important and useful to investigate how to extend the teleportation of quantum gate operations to the cases of multiqubits.
Over the past decade, many physical systems have been studied in order to realize quantum computation, such as cavity quantum electrodynamics (QED) systems [21][22][23][24][25] , ion trap systems 26,27 , nuclear magnetic resonance (NMR) systems 28 , linear optics systems 29 , superconducting circuit systems 30,31 , and so on. Recently, semiconductor quantum dots (QDs) have been widely studied as promising candidates for solid-state-based quantum information processing and quantum computation. Bonato et al. 32 reported an important work that introduced an interface between the polarization degree of freedom of a photon and the spin of an electron confined in a quantum dot embedded in a microcavity operating in the weak-coupling regime. The interface is based on spin selective photon reflection from the cavity and can be used to construct a CNOT gate, a multiphoton entangler and a photonic Bell-state analyzer. Hu and Rarity 33 proposed efficient loss-resistant schemes for heralded state teleportation and entanglement swapping using a single quantum-dot spin in an optical microcavity based on giant circular birefringence. Based on such quantum-dot-microcavity coupled system, furthermore, some other attractive schemes have also been proposed to realize quantum computation and quantum information processing [34][35][36][37][38][39][40] .
In this paper, inspired by the above works, we propose an efficient scheme to teleport a Toffoli gate among three distant electron spins with quantum dots in optical microcavities, resorting to local linear optical operations, an auxiliary electron spin, and two circularly-polarized entangled photon pairs. The Toffoli gate can be successfully teleported from acting on local qubits to acting on remote qubits in a deterministic way by performing measurements on the photons and the auxiliary electron spin. The proposed scheme may open promising possibilities for distributed quantum computation, long-distance quantum communication, and the construction of remote quantum-information-processing networks.

Results
Cavity-induced photon-polarization electron-spin interface. We consider a singly charged GaAs/InAs QD embedded in a double-sided optical microcavity with two partially reflective mirrors in the top and bottom. The four relevant energy levels and optical selection rules is shown in Fig. 1. The optical excitation of the system will produce an exciton (X − ) with negative charges and the charged exciton consists of two electrons bound in one hole. There are two kinds of optical dipole transitions between the electron and the exciton X − , under Pauli's exclusion principle. For a photon with s z = − 1 ( ↓ R or ↑ L ), if the electron is in the state ↓ it feels a coupled (hot) cavity and will be reflected with both the polarization and propagation direction of the photon being flipped. While if the electron is in state ↑ , the photon feels an uncoupled (cold) cavity and will be transmitted by the cavity, acquiring a π phase shift relative to the reflected photon. Likewise, for the photon with s z = + 1 ( ↑ R or ↓ L ), if the electron is in the state ↓ , it will transmit the cavity. If the electron is in the state ↑ the photon is reflected by the optical cavity. That is, the electron-spin-cavity system behaves like a beam splitter. According to the above discussion the dynamics of the interaction between photon and electron in QD-microcavity coupled system is described as below 32 where L and R represent the states of the left-and right-circularly-polarized photons, respectively. The superscript arrow in the photon state indicates the propagation direction along the z axis, ↑ and ↓ denote the direction of the electron spins.
Teleportation of a Toffoli gate among distant solid-state qubits. We now show how to teleport a Toffoli gate among three spatially separated electron spin qubits using quantum-dot-microcavity coupled system based on the photon-spin interaction rules discussed above. The schematic is shown in Fig. 2. The Toffoli gate is among electron spins A, B and C in which electron spins A and B are control qubits and electron spin C is target qubit. In Fig. 2 and electron spins A, B, and C are initialized to the following product state where α i and β i satisfy |α i | 2 + |β i | 2 = 1. Here photon 1 and electron spin A are in Alice's site, photon 2 and electron spin B are in Bob's site, and photons 3 and 4 and electron spin C are in Charlie's site, respectively. To realize the teleportation process, Charlie introduces an ancillary electron spin a, initialized in the state ( ) . Firstly, let photons 1 and 2 pass through the PBS 1 and PBS 2 , respectively. Here PBS is the polarizing beam splitter in the circular basis, which transmits the right circularly polarized photon R and reflects the left circularly polarized photon L . Then photons 1 and 2 pass through a phase shifter PS, respectively. The action of PS is to complete the transformation: The state of the photons 1 and 3 (2 and 4) becomes Then photons 1 and 2 enter into the optical microcavities interacting with electron spins A and B, respectively. After that, photons 1 and 2 exit from the optical microcavities and pass through the PBS 1 and PBS 2 again. Then the state of the system is given by  a  a   1  1  1  3  1  1  3  1  1  3  1  1  3   2  2  4  2  2  4  2  2  4  2  2  4   3  3 Secondly, Alice and Bob perform a measurement on photons 1 and 2, respectively. If Alice measures photon 1 in the state L 1 , the photon 3 in Charlie's site is rotated by a half-wave plate HWP 1 , whose action is given by the transformation: ↔ L R; if the measurement result is R 1 , nothing is done on photon 3. While for the measurement results of Bob's, if photon 2 is measured in state L 2 , the photon 4 in Charlie's site is rotated by HWP 1 ; if the measurement result is |R〉 2 , nothing is done on photon 4. After photons 1 and 2 are detected, we obtain Thirdly, let photon 4 pass through HWP 2 , PBS 5 , and PS and then enter into the optical microcavities interacting with electron spin a. The action of HWP 2 is to complete the transformation After the interaction between photon 4 and spin a, Charlie performs a Hadamard gate operation, which can be achieved by a π/2 microwave pulse 41 , on electron spin a to accomplish the transformation Then the state of the system becomes Fourthly, let photon 3 pass through PBS 6 . When the state of photon 3 is |L〉 3 , it will be reflected by PBS 6 and pass through optical switch S 1 and PBS 7 successively, and then it interacts with electron spin a. When the state of photon 3 is |R〉 3 , it will be transmitted by PBS 6 and dose not interact with electron spins C and a during the following process. That is, PBS 6 splits photon 3 into two wave-packets. After the component |L〉 3 of the photon 3 interacts with spin a, the state of the system is changed to Fifthly, let the photon pass through S 2 , HWP 2 , and PBS 8 and enter into the optical microcavity interacting with electron spin C. Before and after the interaction between the photon and electron spin C, a Hadamard gate operation is performed on electron spin C, respectively. When the photon leaves the optical microcavity, it passes through PBS 8 , HWP 2 , and PS, the state of the system becomes Sixthly, the photon passes through S 1 and PBS 7 successively and then it interacts with electron spin a again. After that, the photon passes through S 2 and arrives at PBS 9 , and in the meantime another wave-packet of photon 3 arrives PBS 9 too. Namely, the two wave-packets of photon 3 pass through PBS 9 simultaneously. The state of the system is given by Seventhly, let photon 3 pass through HWP 2 and a Hadamard gate operation is performed on electron spin a. Finally, Charlie performs the measurements on photon 3, photon 4, and electron spin a. The measurement of electron spin is discussed in detail in the methods section. After the measurement and appropriate single-qubit gate rotation on electron spins A and B (see Table 1), the final state of electron spins A, B, and C is written as 7  1 2  3  3  1 2  3  3   1 2  3  3  1 2  3  3 which achieves the deterministic teleportation of a Toffoli gate among three remote electron spins successfully.

Discussion
We now begin to discuss the feasibility of the present scheme. The basic module in the present scheme is QD-cavity system, which determines the performance of the present scheme. In equation (1) we give the optical transition rules without regarding to the side leakage and cavity loss. In a realistic cavity, however, the factors are not negligible. In the weak excitation approximation the reflection and transmission coefficients of the cavity are described by 42 where g is the coupling strength between X − and the cavity field, κ, κ s , and γ are the cavity field decay rate, leaky rate, and X − dipole decay rate, respectively. ω, ω c , and ω − X are the frequencies of the input photon, cavity mode, and the spin-dependent optical transition, respectively. The coefficients can be calculated from the Heisenberg equations of motion for the cavity field operator â and X − dipole operator σ − in the interaction picture(see methods section). In our work we consider ω , the reflection and transmission coefficients of the coupled cavity and uncoupled cavity (g = 0) are given by

Measurement results
Operations Fidelities  to the measurement results of photon 3, photon 4, and electron spin  a, the corresponding single-qubit operations on electron spins A and B, and the In order to quantitatively test the quality of the present scheme, we introduce the gate fidelity defined as 43 where the overline indicates average over all possible input states |Ψ 0 〉 , U is the ideal three-qubit Toffoli gate, and ρ t = |Ψ t 〉 〈 Ψ t |, with |Ψ t 〉 being the final state of the realistic Toffoli gate operation in the present scheme. Assume that the three electron spins are initially in the general state In Fig. 3 we plot the fidelities with the side leakage κ s /κ and the coupling strength g/κ corresponding to different measurement results (see Table 1), showing that in the case of κ s ≪ κ we can get a high average gate fidelity. Experimentally, the weak coupling with g < (κ + κ s )/4 can be easily achieved, while for the strong coupling with g > (κ + κ s )/4 has also been demonstrated in various microcavity-and nanocavity-QD systems recently [44][45][46][47] and g/(κ + κ s )  0.5 and g/(κ + κ s )  2.4 have been reported 44,47 . In The fidelity corresponding to that the measurement result of photon 3 is |L〉 3 (see Table 1). Here we have set γ = 0.1κ.
Scientific RepoRts | 5:11321 | DOi: 10.1038/srep11321 the present scheme, if setting κ s = 0.01κ, g = 2.4κ, we can obtain = . % F 98 66 ; even when setting κ s = 0.1κ, g = 0.27κ, we also can obtain = . % F 70 42 . Therefore, the present scheme can work in both the weak coupling and the strong coupling regimes. On the other hand, the preparation of electron spin superpositions and its fast initialization and manipulation have been demonstrated in Refs. [41,[48][49][50][51][52][53][54][55]. The entangled pairs of photons can be produced by the well-known spontaneous parametric down-conversion 56 and the photon detectors in our scheme are non-photon-number-resolving detectors, which greatly decreases the high-quality requirements of photon detectors in practical realization. Besides, some other small factors caused by spin decoherence and the trion dephasing also can reduce the gate fidelity, which has been discussed in detail in Ref. [33] and we thus dot not repeat here. Therefore, the present scheme is feasible with current technology.
In conclusion, we put forward an effective scheme to implement quantum remote control with quantum dots embedded in optical microcavities by using local linear optical operations, an auxiliary electron spin, two circularly-polarized entangled photon pairs, photon measurement, and classical communication. The scheme is achieved in a deterministic way by the sequential detection of photons and auxiliary electron spin and the single-qubit rotations of photon and electron spin in principle. The calculated results show that our scheme can work in both the weak coupling and the strong coupling regimes and when κ s ≪ κ we can get a fidelity near unity in strong coupling regime. The scheme is feasible with current technology and would open promising perspectives for long-distance quantum communication, distributed quantum computation and remote quantum-information-processing.

Methods
Manipulation and measurement of the electron spin in QD. The QD-spin superposition state can be prepared by performing single spin-qubit rotations with picosecond optical pulses 57,58 . Ultrafast optical coherent manipulation of a QD-spin qubit has been demonstrated in a picosecond or femtosecond time scale 58,59 , and an ultrafast π/2 spin rotation can be used to complete a Hadamard operation on a spin qubit. The projection measurement of the spin under the basis {|↑ 〉 ,|↓ 〉 } can be achieved with the help of an auxiliary circularly polarized photon. If a right-circularly polarized photon |R〉 is initially sent into the cavity along the z axis, after interacting with QD-cavity system, the joint state of the photon and electron becomes Obviously, the projection measurement of the electron spin can be completed by detecting the reflection and transmission of the photon. If the photon is detected on the reflection port, the electron spin is projected into the state |↑ 〉 ; if the photon is detected on the transmission port, the electron spin is projected into the state |↓ 〉 .
Input-output relation of QD double-sided cavity system. The reflection and transmission coefficients of the QD-cavity system can be calculated from the Heisenberg equations of motion for the cavity field operator â and X − dipole operator σ − in the interaction picture 60 where â in , ′ a in and â r , â t are the two input and the two output fields operators of the double-side cavity, respectively. And other parameters are the same as equation (14). The reflection and transmission coefficients in equation (14) can be obtained in the approximation of weak excitation where the charged QD is predominantly in the ground state with σ = − 1 z .