## Abstract

Quantum computing may provide potential superiority to solve some difficult problems. We propose a scheme for scalable remote quantum computation based on an interface between the photon and the spin of an electron confined in a quantum dot embedded in a microcavity. By successively interacting auxiliary photon pulses with spins charged in optical cavities, a prototypical quantum controlled–controlled flip gate (Toffoli gate) is achieved on a remote three-spin system using only one Einstein–Podolsky–Rosen entanglement, and local operations and classical communication. Our proposed model is shown to be robust to practical noise and experimental imperfections in current cavity–quantum electrodynamics techniques.

## 1. Introduction

Quantum computation studies theoretical computation systems (quantum computers) that make direct use of quantum-mechanical phenomena. Quantum computers are different from digital computers, which are based on transistors. Different from binary digits (bits) used for encoding in classical computers, quantum computation uses qubits (quantum bits) such as spin-*n*-bit number in *O*(*n*^{3}) time of a logic gate, whereas the same problem has no polynomial time using a classical computer. Shortly thereafter, Grover's quantum searching algorithm [4] provided a quadratic speed-up when searching over unsorted data. The optimal quantum search algorithm is given by Long [5], as has been extensively shown by Toyama *et al.* [6]. Several other examples of provable quantum speed-up for query problems have subsequently been discovered, such as finding collisions in two-to-one functions and evaluating NAND trees. Recently, Lloyd *et al.* [7] performed quantum principal component analysis of an unknown low-rank density matrix, revealing eigenvectors corresponding to large eigenvalues in time exponentially faster than any existing algorithm. Generally, quantum algorithms offering a more than polynomial speed-up over the best-known classical algorithm have been found for several problems, including the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of Jones polynomials, or solving Pell's equation. These typical applications of quantum computers have attracted great interest.

Classically, any Boolean function can be constructed using only AND, NOT and FANOUT gates, which are said to be universal. In order to build a quantum computer, in theory the universal logic gates should be constructed first. In fact, the set of the CNOT gate and qubit operations is universal in the sense that any unitary on *n*-qubits can be synthesized by a series of such gates [8–11]. This simulation commonly requires an exponential number of elementary gates, much like the classical case. One method being used to address this problem is to find new elementary quantum logic gates. A Toffoli gate is a candidate [12] when combined with a Hadamard gate, which is a three-qubit operation inverting the target qubit conditioned on two controlling qubits. Since the decomposition of the multiple qubits’ evolution always results in multi-controlling logic gates, new elementary gates become available [10,11]. This has a central role in quantum error correction [13,14] and fault-tolerant computation [15]. A Toffoli gate has been implemented in ion trap systems [16], a superconducting circuit [17] and linear optics [18]. Generally, implementation of a Toffoli gate requires six CNOT gates and 10 single-qubit operations [19]. In recent years, Ralph *et al.* [20], Lanyon *et al.* [21] and Luo *et al.* [22] have demonstrated a general technique that harnesses multi-level information photonic carriers to realize a three-qubit Toffoli gate with reduced CNOT gates.

In this paper, we consider the implementation of a Toffoli gate in distributive scenarios. Recent progress in quantum networking [23–25] may stimulate large-scale quantum applications with remote parties using both the quantum resources and classical resources [26,27]. As an example, Shor's algorithm may be jointly implemented with multiple parties if distributed logic operations are applied. In experiments, photons show great convenience for long-distance transmission, whereas solids are more stable for information storage [13]. Hence, their hybrid systems are used in our schemes to assist each other. Motivated by the schemes in [20–22], we present compact circuits of a Toffoli gate on remote three-qubit systems using an auxiliary extra-level logic state. Recent results have shown that quantum-dot spins in one-sided optical microcavities or double-sided optical microcavities [28–32] may provide a giant optical circular birefringence. With its superiorities, the cavity–quantum electrodynamics (QED) system has been widely used in quantum information processing [33–42], entanglement concentration [43–49] and scalable quantum computing [50–56]. For simplicity, by exploring the giant optical-controlled gate induced by quantum-dot spins in one-sided optical microcavities, a Toffoli gate may be deterministically implemented on remote three-spin systems using only one photonic Einstein–Podolsky–Rosen (EPR) pair or EPR pair of spin systems and local quantum operations and classical communication (LOCC). In comparison with a previous remote CNOT gate with one EPR pair [57–60], our schemes have greatly reduced the entanglement cost for distribution quantum computation. These schemes may be also adapted to implement Toffoli gates on a remote hybrid three-qubit system. Thus our schemes will provide the distributive implementation of universal logic gates for distribution quantum computation.

The rest of this paper is organized as follows. We present a theoretical model of an optical cavity system in §2. The implementation of a Toffoli gate on a remote three-spin system with a photonic EPR pair is shown in §3, whereas its implementation using an EPR pair of spin systems is shown in §4. The experimental discussions and conclusions are presented in the final section.

## 2. Cavity-QED system

Figure 1 shows the set-up for the basic building blocks of our scheme. A charged quantum dot is located inside a one-sided micropillar microcavity [28,30,31]. Distributed Bragg mirrors and index guiding provide three-dimensional confinement of light. In the following, we consider two kinds of transition channel for the cavity photon. The first one is the cavity decay due to transmission through the cavity mirror, whose rate is *κ*. All the other unwanted photon losses, such as cavity absorption and scattering, are characterized by an overall loss rate *κ*_{s}. The initial state of the input pulse can be expressed as |*ψ*〉=*c*_{0}|*R*〉+*c*_{1}|*L*〉. The polarization component |*R*〉(|*L*〉) has the form *f*(*t*) as a function of time *t* is the normalized pulse shape, also taken to be Gaussian with *T* is the pulse duration, *vac*〉 denotes the vacuum of all the optical modes. The evolution of the system can be described by the non-Hermitian conditional Hamiltonian in the framework of the quantum trajectory method [28]. Taking into account the coupling through the cavity decay channel and neglecting the spatial dependence, the master equation for the whole system may be expressed by a Lindblad form as follows:
**H**=**H**_{1}+**H**_{2}+**H**_{3}+**H**_{4}. *σ*_{−} and *σ*_{+} are the Pauli raising and lowering operators, respectively. *g* is the coupling strength between the cavity and *X*^{−}. **H**_{3}=(*ω*_{c}/2)*σ*_{z} is the Hamiltonian of the dipole. *ω*_{c} is the resonant frequency of the dipole, and *σ*_{z} is the Pauli operator for the population inversion. *κ* is the decay rate of the cavity field due to ohmic losses in the metal. *κ*_{s} is the decay rate of the cavity-side leakage mode due to scattering into free-space modes, and may be calculated classically from the Larmor formula. *γ* is the spontaneous emission rate of the dipole. *ρ* is an arbitrary system operator. Using this Hamiltonian one can derive Heisenberg equations of the motion for all of the operators [28]
_{c}=*ω*_{c}−*ω* and Δ_{x}=*ω*_{X−}−*ω* denote the cavity and dipole detunings, respectively. *ω* and *ω*_{X−} are the frequencies of the input probe light and *X*^{−} transition, respectively.

Suppose that the quantum electron spin is initially prepared in its steady state. When a photon is reflected from the cavity, its pulse shape would be changed due to the interaction with the spin-cavity system. When both the adiabatic condition (*g*≫*κ*,*γ*) are satisfied [61–70], the electron spin always stays in the steady state, i.e. 〈*σ*_{z}〉=−1 [28]. From *g*=0) [30,31], *r*_{h}(*ω*) is reduced to

The complex reflection coefficient indicates that the reflected light may feel a phase shift through an optical cavity [56,67]. The phase shift as a function of the relative frequency detuning Δ_{c}/*κ* and decay ratio of cavity *κ*_{s}/*κ* is presented in figure 2. From figure 2, it follows that |*r*_{h}(*ω*)|≈1 and |*r*_{0}(*ω*)|≈1 under the resonant condition with |Δ_{c}|≪*g* (|Δ_{c}|≪5*κ*). For *g*=0, from figure 2*c* the phase shift *θ*_{0}=*π* at Δ_{c}=0, while from figure 2*d* the phase shift *θ*_{h}=0 at Δ_{c}=0 for small *κ*_{s}/*κ*. In the strong-coupling regime with *g*≫*κ*,*γ* [62–70], the *X*^{−} state and cavity mode are mixed to form two dressed states, which leads to the vacuum-Rabi splitting [30,61,64]. If the electron spin is in the state |↑〉, after reflection the input pulse |*L*〉 gets a phase shift of *θ*_{h}=*arg*(*r*_{h}(*ω*)), whereas the input pulse |*R*〉 gets a phase shift of *θ*_{0}=*arg*(*r*_{0}(*ω*)). Conversely, if the excess electron spin is in the state |↓〉, after reflection the pulse |*R*〉 gets a phase shift of *θ*_{h} while the pulse |*L*〉 gets a phase shift of *θ*_{0}. Generally, for initial electron spin in the state *α*_{2}|↑〉+*β*_{2}|↓〉, the optical interaction leads to the following transformation [32,40,50,52]:
*α*_{1}|*R*〉+*β*_{1}|*L*〉. By adjusting the frequencies Δ_{c}→0, the phase difference *θ*_{0}−*θ*_{f}=*π* is followed for the strong coupling *g*≫*κ*,*γ*. Thus the dynamics of the interaction between photon and electron spin in a microcavity coupled system [52,40] is described as follows:

## 3. Toffoli gate on a remote three-spin system via a photonic EPR pair

Many important problems may be reduced to computing a unitary matrix *U* in a large space (the celebrated Shor's algorithm being a prime example). Previous results provide both a framework for formulating quantum algorithms and an architecture for constructing quantum physical computers [1,2]. However, if the involved quantum systems are remotely distributed, a traditional quantum circuit model cannot be directly implemented. To address this problem, the elementary quantum gates should be remotely implemented by costing the least resources. In the following sections, we will show that only one shared-EPR entanglement is sufficient for a Toffoli gate on a remote three-spin system to build universal remote quantum computation with LOCC.

Consider three electron spins *e*_{1},*e*_{2} and *e*_{3} in the states
*Φ*〉_{C1C2} defined by
*e*_{1} and *e*_{2} are controlling qubits while electron spin *e*_{3} is the target qubit.

### (a) Remote target qubit

In this subsection, assume the controlling electron spins belong to Alice while the target electron spin *e*_{3} belongs to Bob. The compact circuit is shown in figure 3. The realization is completed with an auxiliary photon *A* with 2 d.f. in the state *a*_{i} denote two spatial modes and *e* is an auxiliary spin in the state |+〉. The detailed evolution of the joint system is defined as follows.

First, Alice let the auxiliary photon *A* from the spatial mode *a*_{1} (*a*_{2}) pass through the polarization beamsplitter *cPS*_{1} (*cPS*_{2}). The reflected pulse as an input pulse interacts with the electron spin *e*_{1} in the cavity *Cy*_{1} from path °1 (°2). Its output pulse and the transmitted pulse of *cPS*_{1} arrive at the polarization beamsplitter *cPS*_{3} (*cPS*_{4}) simultaneously. It then performs a half wave-plate *H*_{1} (*H*_{2}) on the photon *A* from the spatial mode *a*_{1}. These operations may change the joint system of the photon *A* and the spin *e*_{1} from

Second, Alice let the auxiliary photon *A* from the spatial mode *a*_{2} pass through the polarization beamsplitter *cPS*_{5}. The reflected pulse from the upper path goes through the wave plate *X*_{1} and then, as an input pulse, it interacts with the spin *e*_{2} in the cavity *Cy*_{2} from path °3. Its output pulse goes through the wave plate *X*_{2}. Now, the transmitted photon of *cPS*_{5} as another input pulse is interacted with the electron spin *e*_{2} in the cavity *Cy*_{2} from path °4. The output pulses from the two paths arrive at *cPS*_{6} simultaneously. These operations will change |*Ψ*_{1}〉 in equation (3.4) into
*A* from two spatial modes *a*_{1} and *a*_{2} arrive at *cBS*_{1} simultaneously.

Third, Alice let the auxiliary photon *A* from the spatial mode *a*_{2} pass through the polarization beamsplitter *cPS*_{7}. The reflected pulse goes through the wave plate *X*_{3} and then as an input pulse interacts with the electron spin *e* in an auxiliary cavity *Cy* from path °5. Its output pulse goes through *X*_{4}. The output pulse from *X*_{4} and the transmitted pulse of *cPS*_{5} arrive at *cPS*_{8} simultaneously. These operations may transform the joint system of photon *A* and three electron spins *e*_{1},*e*_{2} and *e* from |*Ψ*_{3}〉|+〉_{e} in equation (3.6) into
*W*_{1} is performed on the electron spin *e*, and the state |*Ψ*_{4}〉 in equation (3.7) is changed into

Fourth, Alice let the photon *C*_{1} go through the wave plate *H*_{3} and the polarization beamsplitter *cPS*_{9}. The reflected pulse as another input pulse interacts with the electron spin *e* in the cavity *Cy* from path °6. Its output pulse and the transmitted pulse of the *cPS*_{9} arrive at the polarization beamsplitter *cPS*_{10} simultaneously. The output photon goes through the wave plate *H*_{4}. These operations may transform the joint system of three photons *A*,*C*_{1} and *C*_{2}, and three electron spins *e*_{1},*e*_{2} and *e* from |*Ψ*_{5}〉|*Φ*〉_{C1C2} shown in equations (3.2) and (3.8) into
*e* under the basis {|±〉}, and the Pauli phase flip *σ*_{Z} is performed on the photon *A* from the spatial mode *a*_{2} for the measurement outcome |−〉_{e}. And then she measures photon *C*_{1} under the basis {|*R*〉,|*L*〉} and sends out the measurement outcome to Bob through a classical channel, where the Pauli flip *σ*_{X} is performed on photon *C*_{2} by Bob when the photon is detected at *D*_{L}. This photonic measurement *M*_{C1} is completed with the polarization beamsplitter *cPS*_{11} and two single-photon detectors *D*_{R(L)}. After these measurements, the state |*Ψ*_{6}〉 in equation (3.9) collapses into

Fifth, Bob performs the Hadamard operation *W*_{2} on electron spin *e*_{3}, and allows photon *C*_{2} as an input pulse to be interacted with electron spin *e*_{3} in cavity *Cy*_{3} from path °8. After the photon arrives from *Cy*_{3}, Bob performs *W*_{3} on electron spin *e*_{3}. These operations may transform two photons *A* and *C*_{2}, and three electron spins *e*_{1},*e*_{2} and *e*_{3} from |*Ψ*_{7}〉|*ψ*_{3}〉 in equation (3.10) into
*C*_{2} under the basis *σ*_{Z} is performed on photon *A* from spatial mode *a*_{2} by Alice if photon *C*_{2} is detected at *D*_{L}. Here, the photonic measurement *M*_{C2} is completed with *H*_{5}, *cPS*_{12} and two single-photon detectors *D*_{R(L)}. After the measurement, |*Ψ*_{8}〉 in equation (3.11) collapses into

Finally, Bob needs to disentangle photon *A*. He measures photon *A* under the basis *M*_{A} is completed with *H*_{6} and *H*_{7}, *cBS*_{2}, *cPS*_{13} and *cPS*_{14}, and four single-photon detectors *D*_{Ra1}, *D*_{La1}, *D*_{Ra2} and *D*_{La2}. The recovery operations are shown in table 1. After the measurement, |*Ψ*_{9}〉 in equation (3.12) collapses into

### (b) Remote controlling qubits

In this subsection, assume that Alice has the electron spin *e*_{1} while Bob has the electron spins *e*_{2} and *e*_{3}. By using the photonic entanglement |*Φ*〉_{C1C2} in equation (3.2) shared by Alice and Bob, a compact circuit will be presented to implement Toffoli gate *T*_{e1e2,e3} in equation (3.3) on these remote electron spins.

The detailed circuit is shown in figure 4. First, Bob lets photon *C*_{2} go through the polarization beamsplitter *cPS*_{1}. The reflected pulse as an input pulse is interacted with the electron spin *e* in the auxiliary cavity *Cy* from path °1. Its output pulse and the transmitted pulse of *cPS*_{1} arrive at *cPS*_{2} simultaneously. These operations may transform the joint system of two photons *C*_{1} and *C*_{2}, and the electron spin *e* from |+〉_{e}|*Φ*〉_{C1C2} into
*A* in state *a*_{2} as another input pulse interact with electron spin *e* in cavity *Cy* from path °2. And then Bob measures electron spin *e* under the basis {|±〉} and sends measurement outcomes to Alice through a classical channel, where the Pauli flip *σ*_{X} is performed on photon *C*_{1} by Alice and *σ*_{X}*σ*_{Z} is performed on photon *C*_{2} by Bob for the measurement outcome |−〉. After these operations, the joint system consists of three photons *C*_{1},*C*_{2} and *A*, and the electron spin *e* is changed from

Second, by using a half waveplate, a circularly polarized beamsplitter and two single-photon detectors, Bob measures photon *C*_{2} under the basis {|+^{R}〉,|−^{L}〉} and sends measurement outcomes to Alice through a classical channel. Here, the Pauli phase flip *σ*_{Z} is performed on photon *C*_{1} by Alice if the photon is detected at *D*_{L}. After this measurement, state |*Ξ*_{2}〉 in equation (3.15) collapses into

Third, Bob performs the Hadamard operation *W*_{1} on electron spin *e*_{2}. He lets auxiliary photon *A* from the spatial mode *a*_{1} pass through *H*_{1}→*cPS*_{3}→*Cy*_{2}→*cPS*_{4}, while auxiliary photon *A* from spatial mode *a*_{2} passes through *H*_{2}→*cPS*_{5}→*Cy*_{2}→*cPS*_{6}. The joint system of two photons *C*_{1} and *A* and the electron spin *e*_{2} are transformed from |*ψ*_{2}〉|*Ξ*_{3}〉 into
*A* goes through *H*_{3} and *H*_{4}, and the Hadamard operation *W*_{2} is performed on electron spin *e*_{2}. |*Ξ*_{4}〉 in equation (3.17) is changed into

Fourth, Alice lets photon *C*_{1} pass through *cPS*_{7}→*Cy*_{1}→*cPS*_{8} from path °5. These operations may transform the joint system of two photons *C*_{1} and *A* and two electron spins *e*_{1} and *e*_{2} from |*ψ*_{1}〉|*Ξ*_{5}〉 in equation (3.18) into
*C*_{1} under the basis {|+^{R}〉,|−^{L}〉} and sends out measurement outcomes to Bob through a classical channel, where the phase flip −*I* is performed on photon *A* from spatial mode *a*_{2} by Bob for the measurement outcome |−^{L}〉. After these operations, the state |*Ξ*_{6}〉 in equation (5.1) collapses into

Fifth, Bob lets photon *A* from spatial modes *a*_{1} and *a*_{2} arrive at the 50/50 beamsplitter *cBS*_{1} simultaneously, and the state |*Ξ*_{7}〉 in equation (3.20) may be changed into
*W*_{3} on electron spin *e*_{3} to obtain |*ϕ*′_{3}〉_{e3}= *α*_{3}′|↑〉+|*β*_{3}′|↓〉, where *A* from spatial mode *a*_{2} goes through *cPS*_{9}→*Cy*_{3}→*cPS*_{10}. These operations transform the joint system of photon *A* and three electron spins *e*_{1},*e*_{2} and *e*_{3} from |*Ξ*_{8}〉|*ψ*_{3}〉 in equation (3.21) into

Finally, Bob performs the Hadamard operation *W*_{4} on electron spin *e*_{3}. And then he measures photon *A* under the basis *M*_{A} in figure 3, and sends out measurement outcomes to Alice through a classical channel. The recovery operations are shown in table 1. Here, the recovery operation on spin *e*_{1} is performed by Alice, whereas the recovery operation on spin *e*_{2} is performed by Bob. The state |*Ξ*_{9}〉 in equation (3.22) collapses into

## 4. Toffoli gate on a remote three-spin system via an EPR pair of spin systems

Consider three electron spins *e*_{1},*e*_{2} and *e*_{3} shown in equation (3.1). Different from the photonic entanglement channel |*Φ*〉 in equation (3.2) used in §3, in this section assume that Alice and Bob share an EPR pair |*Ψ*〉_{eAeB} defined by
*T*_{e1e2,e3} on three remote electron spins *e*_{i}, where *e*_{1} and *e*_{2} are controlling qubits and *e*_{3} is the target qubit.

### (a) Remote target qubit

Consider that *e*_{1} and *e*_{2} belong to Alice, while *e*_{3} belongs to Bob. A detailed circuit is shown in figure 5 using an auxiliary photon *A* in the state

First, since two controlling electron spins belong to Alice, she can implement the subcircuit from the polarization beamsplitter *cPS*_{1} to the 50/50 beamsplitter *cBS*_{1} (the same as that defined in figure 3), in order to get
*A* and two electron spins *e*_{1} and *e*_{2}.

Second, Alice performs *W*_{1} on electron spin *e*_{A} to change |*Ψ*〉_{eAeB} in equation (4.1) into *A* from spatial mode *a*_{2} go through *cPS*_{5}. The reflected pulse interacts with spin *e*_{A} in cavity *Cy*_{A} from path °4. Its output pulse and the transmitted pulse of *cPS*_{5} arrive at *cPS*_{6} simultaneously. These operations may transform the joint system of photon *A*, and four spins *e*_{1},*e*_{2},*e*_{A} and *e*_{B} into
*e*_{A} under the basis {|±〉} and sends out the measurement outcomes to Bob through a classical channel, where the Pauli flip *σ*_{X} is performed on electron spin *e*_{B} by Bob for the measurement outcome |−〉_{eA}. The state |*Ψ*_{4}〉 in equation (4.3) collapses into

Third, Bob lets auxiliary photon *B* in state |+^{R}〉 as an input pulse interact with electron spin *e*_{B} in cavity *Cy*_{B} from path °5. The output photon goes through *H*_{2}. The joint system of three electron spins *e*_{1},*e*_{2} and *e*_{B} and photon *B* is changed into
*W*_{2} on electron spin *e*_{3} to get |*ψ*_{3}〉=*α*_{3}′|↑〉+*β*_{3}′|↓〉, where *B* goes through *cPS*_{7}→*Cy*_{3}→*cPS*_{8} from path °6. After these operations, Bob performs *W*_{3} on electron spin *e*_{3}. These operations may transform the joint system of photon *B* and four spins *e*_{1},*e*_{2},*e*_{B} and *e*_{3} into

Fourth, Bob measures electron spin *e*_{B} under the basis {|±〉}, where the Pauli flip *σ*_{Z} is performed on photon *B* for the measurement outcome |−〉_{eB}. And then he measures photon *B* under the basis {|+^{R}〉,|−^{L}} and sends out measurement outcomes to Alice, where a Pauli phase flip *σ*_{Z} is performed on photon *A* by Alice from the spatial mode *a*_{2} for the measurement outcome |−^{L}〉_{B}. After these measurements, the state |*Ψ*_{7}〉 in equation (4.6) collapses into

Finally, Alice measures photon *A* under the bases *M*_{A} in figure 3. The recovery operations are shown in table 1. The state |*Ψ*_{8}〉 in equation (4.7) collapses into

### (b) Remote controlling qubits

In this subsection, assume that Alice has electron spin *e*_{1}, while Bob has electron spins *e*_{2} and *e*_{3}. By using spin channel |*Ψ*〉_{eAeB} in equation (4.1) shared by Alice and Bob, we can present the compact circuit of the Toffoli gate *T*_{e1e2,e3} on these remote spins.

The detailed circuit is shown in figure 6. The realization is completed with an auxiliary photon *A* in the state *W*_{1} on electron spin *e*_{B} to change |*Ψ*〉_{eAeB} in equation (4.1) into *A* from spatial mode *a*_{2} be interacted with electron spin *e*_{B} from path °1. These operations may transform the joint system of photon *A* and electron spins *e*_{A} and *e*_{B} from *e*_{B} under the basis {|±〉} and sends measurement outcomes to Alice through a classical channel, where a Pauli flip *σ*_{X} is performed on electron spin *e*_{A} by Alice for the measurement outcome |−〉. The state |*Ξ*_{1}〉 in equation (4.9) collapses into

Second, Bob performs *W*_{2} on electron spin *e*_{2} to get |*ψ*′_{2}〉=*α*_{2}′|↑〉+*β*_{2}′|↓〉 with *A* from spatial mode *a*_{1} passes through *H*_{1}→*cPS*_{1}→*Cy*_{2}→*cPS*_{2}, while photon *A* from spatial mode *a*_{2} passes through *H*_{2}→*cPS*_{3}→*Cy*_{2}→*cPS*_{4}. These operations may transform the joint system of photon *A* and electron spins *e*_{2} and *e*_{B} from |*ψ*_{2}〉|*Ξ*_{2}〉 in equation (4.10) into
*A* goes through *H*_{3} and *H*_{4}, and *W*_{3} is performed on spin *e*_{2}. After these operations, the state |*Ξ*_{3}〉 in equation (4.11) is changed into

Third, Alice lets auxiliary photon *B* in state |+^{R}〉 interact with electron spin *e*_{B} from path °4. The output pulse goes through *H*_{5}. These operations may transform the joint system of two photons *A* and *B* and two spins *e*_{A} and *e*_{2} from |+^{R}〉_{B}|*Ξ*_{4}〉 in equation (4.12) into
*e*_{A} under the basis {|±〉}, where a Pauli phase flip *σ*_{Z} is performed on photon *B* for the measurement outcome |−〉. The state |*Ξ*_{5}〉 in equation (4.13) collapses into

Fourth, Bob performs *W*_{4} on electron spin *e*_{1} in cavity *Cy*_{1} to get |*ψ*′_{1}〉=*α*_{1}′|↑〉+*β*_{1}′|↓〉 with *B* go through *cPS*_{5}→*Cy*_{1}→*cPS*_{6} from path °5. These operations may transform the joint system of photons *A* and *B* and two electron spins *e*_{1} and *e*_{2} from |*ψ*′_{1}〉|*Ξ*_{6}〉 in equation (4.14) into
*W*_{5} on electron spin *e*_{1} in cavity *Cy*_{1}.

Fifth, by using a half waveplate, circularly polarized beamsplitter and two single-photon detectors, Alice measures photon *B* under the basis {|+^{R}〉,|−^{L}〉} and sends out measurement outcomes to Bob through a classical channel, where one phase flip −*I* is performed on photon *A* from spatial mode *a*_{2} by Bob if the photon is detected at *D*_{L}. The state |*Ξ*_{7}〉 in equation (4.15) collapses into

Sixth, Bob performs *W*_{5} on electron spin *e*_{3} in cavity *Cy*_{3} to get |*ψ*′_{3}〉=*α*_{3}′|↑〉+ *β*_{3}′|↓〉 with *A* from spatial mode *a*_{2} goes through *cPS*_{7}→*Cy*_{3}→*cPS*_{8} from path °6. These operations may transform the joint system consisting of photon *A* and electron spins *e*_{1},*e*_{2} and *e*_{3} from |*ψ*′_{3}〉|*Ξ*_{8}〉 in equation (4.16) into
*W*_{7} is performed on electron spin *e*_{3} in cavity *Cy*_{3}.

Finally, as for measurement *M*_{A} in figure 3, Bob measures auxiliary photon *A* under the basis *e*_{1} is performed by Alice, while the recovery operation on spin *e*_{2} is performed by Bob. The state |*Ξ*_{9}〉 in equation (4.17) collapses into

## 5. Discussion

The optical selection rules of the quantum-dot cavity system shown in equation (2.8) play core roles in remote Toffoli gates. Under ideal conditions, one may neglect the cavity side leakage, i.e. |*r*_{0}|≈1 and |*r*_{h}|≈1 from figure 2. Thus our four Toffoli gates are deterministic and faithful. However, the side leakage from the cavity is unavoidable in the experiment [32–47]. In this case, the reflectance coefficients *r*_{0} and *r*_{h} are shown in figure 7. This figure shows that ideal reflectance coefficients *r*_{0} and *r*_{h} may be achieved with a small side leakage ratio *κ*_{s}/*κ* and weak coupling defined by the cooperativity *C*=*g*^{2}/(*κγ*)<100. Moreover, we will consider these non-ideal conditions for our schemes. Under resonant conditions, the ideal optical selection rules in equation (2.8) are changed into
*F* of the four Toffoli gates may be calculated as illustrated in figure 8 by making use of the cooperativity *C*=*g*^{2}/(*κγ*) [57–69], *r*_{h}=1−2/(1+*κ*_{s}/*κ*+4*C*) and *r*_{0}=1−2/(1+*κ*_{s}/*κ*). Here, *Ψ*_{i}〉 and |*Ψ*_{f}〉 are the ideal final state and the experimental final state with side leakages, respectively. The efficiencies *P* of the four Toffoli gates are calculated as illustrated in figure 9, where the efficiency *P* is defined as the ratio of the number of input photons to output photons [71]. The reflection probability for a photon from a cavity with the cooperativity *C*=*g*^{2}/(*κγ*) under the resonances Δ_{c}=Δ_{x}=0 is given by *P*≈1−(1+4*C*+(*κ*_{s}/*κ*)^{2})/(1+4*C*+4*C*^{2}+(*κ*_{s}/*κ*)^{2}) [71]. Generally, the cavity side leakage ratio has a great effect on the implementation fidelities and efficiencies. As shown in figures 8 and 9, high fidelity and efficiency may be achieved even in the weakly coupling regime *C*<100 when the cavity side leakage ratio *κ*_{s}/*κ*→0. Otherwise, strong coupling is necessary [32,61,66,67]. Fortunately, strong coupling has been realized up to 2.4 [65] in 1.5 μ*m* micropillar microcavities. If the experimental parameters, i.e. the cavity damping rate is (*κ*,*γ*,*g*)=2*π*×(165±15,2.6,70) *MHz* [72], are considered, the fidelities are greater than 85.6% and the efficiencies are greater than 83.2% for *κ*_{s}/*κ*≈0.2. Moreover, for experimental parameters (*κ*,*γ*,*g*)=2*π*×(25,0.006,1.09) *GHz* [38], the fidelities are greater than 88.6% and the efficiencies are greater than 84.6% for *κ*_{s}/*κ*≈0.2. Under this strong coupling, the fidelities are greater than 91.25% and the efficiencies are greater than 86.5% for *κ*_{s}/*κ*≈0.2.

A single electron spin is necessary to implement Toffoli gates on remote spins in our case. This superposition state may be prepared by optical pumping or optical cooling using an independent resonant laser [67,73]. When the hole spin coherence time is longer than three orders of the cavity photon lifetime [74,75], the hole spin dephasing [32] has little effect on experimental fidelities and efficiencies. For a micropillar cavity with oxide apertures [76,77], the optical losses due to the radiation and aperture scattering are much smaller than the photon escape loss through the mirror. In general, the dipole leak can be increased by reducing the cavity loss and increasing the Purcell factor [28,61] and the dipole lifetime. The Hadamard transformation of the spin can be realized by using a spin echo technique [32,78,79] and nanosecond spin resonance microwave pulse [62] to protect the spin coherence. Moreover, the heavy–light hole mixing is also avoidable by engineering the shape, size and type of the charged exciton [32]. When the optical coherence time of an exciton *T*≫1/*κ*, the pulse dephasing has a negligible effect on remote Toffoli gates [61,70]. Efficient quantum error correcting codes may be incorporated into this computation scheme to achieve fault tolerance [13] of photon loss during interactions, which may be modelled as a leakage error. This leakage error induces a small inefficiency for each gate because it only affects the probability of registering a photon from each pulse. In current experiments [80], typically (*κ*,*κ*_{s})≈(8,5.2)⋅2*π* MHz, and *g*≈25⋅2*π* MHz, which yields a very small leakage error probability [70]. Recent experiments on the controlled phase flip and CNOT on the joint system of the atom state and the polarization state [76] have provided possible candidates to build our models. Of course, in experiments, the remote computations are also conditional on several flawless elements and set-ups in the experiment and shared entanglement.

In conclusion, we have investigated the possibility that a cavity with a single-trapped spin can be exploited to realize scalable remote quantum computation. In contrast to previous remote CNOT gates [57–59], we have constructed four deterministic Toffoli gates operating on a bipartite three-spin system using LOCC and only a photonic EPR entanglement or spin EPR entanglement. If using previous remote CNOT gates [57–59] and classical decomposition of Toffoli gates [13], four EPR entanglements will be required. Hence, our remote quantum computation also cost only a quarter of the entanglement resources of quantum computation using previous schemes [57–59]. Moreover, different from previous linear optical models [57,58], our computation model is based on cavity quantum electrodynamics and is robust to various experimental sources of noise. These atomic gates have no limited fidelity. It means that near deterministic gates may be realized if certain experimental conditions are satisfied [62–74]. Furthermore, the optical transmission superiority may be taken to construct quantum entanglement resources in our schemes. Recently, Wang *et al*. [55] presented two deterministic schemes for constructing a CNOT gate and a Toffoli gate on photon–atom and photon–atom–atom hybrid quantum systems assisted by bad cavities, respectively. They are achieved by cavity-assisted photon scattering and work in the intermediate coupling region with bad cavities. Their Toffoli gate is simple and makes use of a quarter wave packet which is used to interact the photon with each of the atoms every time. Our motivations in this paper are to present remote Toffoli gates on the bipartite system with reduced quantum entanglements. To implement a Toffoli gate, an auxiliary photon with 2 d.f. is used as a ququart system [20–22] to reduce the CNOT gate. With this assistance, the entanglement resources are greatly reduced in contrast to previous remote CNOT gates [57–60]. Of course, with their results [55], similar remote Toffoli gate may be performed assisted by bad cavities. With little effort, it is easy to present compact circuits for Toffoli gates on a hybrid three-qubit system. Thus our remote quantum computation model is flexible and scalable for large-scale fault-tolerant quantum computation from different superiorities of the quantum-dot spin and photon.

## Data accessibility

Fidelities and efficiencies may be accessed via http://sist.swjtu.edu.cn/XYGK/teachers/infoshow.asp?id=1134.

## Authors' contributions

M.-X.L. proposed the theoretical method. M.-X.L. and X.W. wrote the manuscript. M.-X.L. reviewed the manuscript.

## Competing interests

We declare we have no competing interests.

## Funding

This work is supported by the National Natural Science Foundation of China (no. 61303039), Fundamental Research Funds for the Central Universities (no. 2682014CX095) and Science Foundation Ireland (SFI) under International Strategic Cooperation Award grant no. SFI/13/ISCA/2845.

- Received April 25, 2015.
- Accepted November 20, 2015.

- © 2015 The Author(s)