Introduction

Quantum computing is an active area of research because of its ability to efficiently solve difficult problems without efficient classical algorithms1,2,3,4. The quantum computer, the elementary quantum element in quantum applications, is still difficult to realize with the methods of modern science. Based on the qubit system in two-dimensional Hilbert space, most quantum algorithms1,2,3,4 require a large number of qubits to encode information5,6,7. These quantum algorithms may be realized by special quantum circuits consisting of basic gates corresponding to unitary matrices. In other words, the design of quantum algorithms is equivalent to the decomposition of a unitary matrix into a product of matrices chosen from a basic set8,9. From classical matrix decomposition, such as cosine-sine decomposition9, multiple controlling logic gates have been fundamental to the multiple-qubit evolution. Finding efficient ways to synthesize these controlling logic gates may allow large-scale quantum computing tasks to be performed on a shorter time-scale.

Because classical computing is designed around irreversible gates, it is impossible to directly translate this expertise into the quantum world. The Gottesman-Knill Theorem says that Clifford gates (CNOT, Hadamard, S) can be classically simulated efficiently, so they are probably not sufficiently universal for quantum computation. These gates, together with other one-qubit gates, not generated by the gates in the Clifford group, form a universal set of gates for quantum computation10. Based on classical reversible logic11, the Toffoli gate8,9 has played a central role in this field; it is a controlled controlled-NOT acting on three bits. The Toffoli gate is also of interest in other quantum applications, for example, as a building block in phase estimation12, error correction13 and fault tolerant quantum circuits14. Much progress has been made and various physical architectures have been used, including NMR systems13, ion traps15,16, linear optics17, superconductors18 and atoms19,20. These experiments may create opportunities to investigate efficient quantum circuits for synthesizing quantum operations.

Qubit-based quantum applications require a two-level structure on atom, ion or photon systems that naturally have many accessible degrees of freedom (DOFs). These DOFs may be regarded as high-dimensional systems. In fact, high-dimensional systems may provide different quantum correlations and may be useful in quantum information processing21,22,23,24,25,26,27,28,29. High-dimensional systems are flexible in terms of improvements to the channel capacity21,22 and communication security24,25. Moreover, they also provide an alternate way of scaling quantum computation. By extending a proposal29, Lanyon et al.30 recently demonstrated a general technique that harnesses multi-level information carriers to significantly reduce the realization complexity of multiple-control logic gates. By making use of a multiple-level target system, they showed that the Toffoli gate and general two-qubit controlled-unitary gates may be realized with linear optics. Regrettably, their multiple-level target system is unscalable for large-scale applications such as Shor’s algorithm. This flaw is then addressed by using multiple-level auxiliary states31, which may result in a high-dimensional quantum Fourier transformation.

Motivated by their scheme23,29,30,31, in this paper, we propose modified proposals of the Toffoli gate by using auxiliary photons with two DOFs as an auxiliary four-dimensional quantum state. Previous results have shown that two DOFs of photons may be used to fuse hybrid quantum information32, reduce quantum resources33,34,35 and construct a universal ququart quantum computer36. Our application using two DOFs of photons is for the scalability of qubit-based quantum computations23,30 and to avoid high-dimensional quantum Fourier transformations31. Moreover, from the strong field provided by a Fabry-Perot-type cavity, cavity QED may have a very strong effect even at the single photon level. This effect is very useful for large-scale quantum computation. In fact, by exploring the giant optical circular birefringence induced by quantum-dot spins in one-sided optical microcavities32,33,37,38,39,40,41,42,43,44,45, a spin may be interacted with a linearly circularly polarized photon. Based on the cavity QED, the Toffoli gate can be deterministically implemented on all combinations of photons and spins using an auxiliary photon with the polarization DOF and the spatial mode DOF. Our schemes extend previous schemes13,14,15,16,17,19,20,34,35 with six CNOT gates, recent proposals29,30,31 with three CNOT gates and the multiple-level logic state. All of our input quantum systems are qubits. The multiple-dimensional system, i.e., one photon with two DOFs, is used as an auxiliary system to carry the control information30. With these constructions, the multiple DOFs will not cause confusion in quantum information processing due to different dimensions of encoded quantum systems31. The disentangling operations only involve single photon operations and detectors31. Furthermore, our Toffoli gate may be realized on all combinations of photons and quantum spins. Thus they may be very useful for hybrid quantum information processing from recent experiments44,45,46,47,48,49,50,51,52,53,54.

Results

The Toffoli gate is an important three-qubit entangling gate in quantum logic gates11,12,13. It will flip the target qubit conditional on the two control qubits. Combined with the one-qubit Hadamard, the Toffoli gate offers a simple universal quantum gate set in comparison to the CNOT gate and one-qubit rotations10,55. Generally, a Toffoli requires at least five two-qubit gates or six CNOT gates11,54. If an additional logic state is permitted for the target, a reduced decomposition requires only three two-qubit gates29,30,31. The enhanced decomposition is achieved by harnessing a third level of the target information carrier, i.e., a qutrit with logical states and . Motivated by this idea29,30,31, two DOFs of one photon as a multiple-dimensional system will be used as the control information carrier but not the target information carrier. Four logic states are encoded with , respectively. and denote bases of the polarization DOF and spatial mode DOF of one photon respectively, where and denote right and left circularly polarizing photons, respectively and di denotes the spatial modes of one photon. In the following, we also denote with or for convenience. By exploring the interaction of quantum-dot spins and a circularly polarized photon32,33,37,38,39,40,41,42,43,44,45, a Toffoli gate may be realized on the spins and photons regardless of the type of control and target qubits, using three general CNOT gates. These hybrid CNOT gates are typical controlling flip operations on the different DOFs of one photon or different types of quantum systems. These schemes show hybrid implementations of the Toffoli gate with photons and quantum spins using a reduced number of controlling qubit gates.

QD-cavity system

Consider a singly charged GaAs/InAs quantum dot (QD) inside a micropillar cavity37,38,39, which consists of a λ-cavity between two GaAs/Al(Ga)As distributed Bragg reflectors. The QD is located in the center of the cavity to achieve maximal light-matter coupling. If the QD is neutral, optical excitation generates a neutral exciton. If the QD is singly charged, i.e., a single excess electron is injected, optical excitation can create a negatively-charged exciton (X−), which consists of two electrons bound to one hole37,38,39. Due to Pauli’s exclusion principle, for the spin state , X− in the state with the two electron spins antiparallel is created by resonantly absorbing a left circularly polarized photon , where the heavy-hole spin state ; for the spin state , X− in the state with the two electron spins antiparallel is created by resonantly absorbing a right circularly polarization photon , where heavy-hole spin state , as shown in Fig. 1. In the limit of a weak incoming field40,41,42, the spin cavity system behaves like a beam splitter. Based on the transmission and reflection rules of the cavity for an incident circular polarization photon conditioned on the QD-spin state, the dynamics of the interaction between the photon and spin in a QD-microcavity coupled system is described as below32,33,43,44,45

Figure 1
figure 1

Schematic energy level and optical selection rules due to Pauli’s exclusion principle.

and are the input and output field operators of the waveguide, respectively. and represent the left circularly and right circularly polarized photons, respectively.  and represent the spins of the excess electron.  and represent the negatively charged exciton X−1.

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under ideal conditions. In the following, this ideal spin-cavity unit is used to realize the Toffoli gate on photons and quantum-dot spins for efficient quantum information processing. Then, the experimental spin-cavity unit will be discussed in the last section.

Toffoli gate on a three-photon system

Consider three linearly circularly polarized photons A, B and C in the states

Our goal is to realize the Toffoli gate with the following form

where the photons A and B are the controlling qubits while the photon C is the target photon. The detailed circuit is shown in Fig. 2. This construction is completed with three auxiliary quantum electron spins ei in the state and an auxiliary photon D in the state . The Toffoli gate TAB,C is completed with the following three controlled gates.

Figure 2
figure 2

Toffoli gate on a three-photon system assisted by one photon with two DOFs.

di denote spatial modes of the auxiliary photon D. ei denote auxiliary electron spins in the state . Hi denote half waveplates to perform the Hadamard transformation and . Xi denote wave plates to perform the polarization flip transformation . Zi denote waveplates to perform the phase flip transformation . cPSi represent circularly polarizing beamsplitters that transmit and reflect . cBSi represent 50%50 circularly polarizing beamsplitters to perform the Hadamard operation and . Cyi denote the QD-cavity charged the electron spin ei. If there are two input lines of one cavity, the photon represented with red lines passes through the cavity firstly and then the photon represented with black lines passes through the cavity.

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First, from the subcircuit S1 shown in Fig. 2(a), the photon A as an input pulse passes through the cPS1, cavity Cy1, cPS2, sequentially. Then W1 is performed on the spin e1. Now, the pulse D from the spatial mode d1 passes through the H1, cPS3, cavity Cy1, cPS4, H2, sequentially. After these operations, the joint system consisting of the photons A and D and the spin e1 is changed from into ; the detailed computations are shown in SI. This joint state may collapse into

after the measurement of the electron spin e1 under the basis , where a Pauli phase flip is performed on the photon A for the measurement outcome . This circuit has realized the controlled-NOT gate on the input photon A and the polarization DOF of the auxiliary photon D, which is different from previous CNOT gate on the same type of input system.

Second, from the subcircuit S2 shown in Fig. 2(b), the photon B passes through the cPS5, cavity Cy2, cPS6, sequentially. Then W2 is performed on the spin e2. Now, the photon D passes through the BS1, cPS7, X1, cavity Cy2, X2, cPS8, BS2, sequentially. After these operations, the joint system consisting of the photons A, B and D and the spin e2 is changed from into ; the detailed computations are shown in the SI. This state may collapse into

after the measurement of the electron spin e2 under the basis , where a Pauli phase flip is performed on the photon B for the measurement outcome . This circuit has realized the controlled-NOT gate on the input photon B and the spatial mode DOF of the auxiliary photon D.

Third, from the subcircuit S3 shown in Fig. 2(c), the pulse D from the spatial mode d2 passes through the cPS9, cavity Cy3, cPS10, sequentially. Then W3 is performed on the spin e3 Now, the photon C passes through the H3, cPS11, cavity Cy3, cPS12, H2, sequentially. After these operations, the joint system consisting of the photons A, B, C and D and the spin e3 is changed from into

where a Pauli flip . This state may collapse into

after the measurement of the spin e3 under the basis , where a phase flip is performed on the photon D from the spatial mode a2 for the measurement outcome . This circuit may be viewed as the controlled-NOT gate CNOTD,C performed on the auxiliary photon D and the input photon C as follows

which is an essential three-qubit operation.

Finally, by performing the single qubit measurements on the photon D under the basis . In the experiment, this measurement may be completed with the 50%50 circularly polarizing beamsplitter cBS3, two circularly polarizing beamsplitters cPS13 and cPS14, two half waveplates H5 and H6 and four single photon detectors and . The recovery operations are shown in Table 1. The entanglement shown in equation (7) may collapse into

Table 1 The relations between the measurement outcomes of the auxiliary photon D and the feed-forward operations for implementing the Toffoli gate on three photons A, B and C.
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Thus, the Toffoli gate TAB,C shown in equation (3) has been deterministically realized with three general controlled gates and CNOTD,C.

Toffoli gate on a three-spin system

Consider three electron spins ei in the states

This section is to realize the Toffoli gate

where the electron spins e1 and e2 are the controlling qubits, while the electron spin e3 is the target qubit. The detailed circuit is shown in Fig. 3 by using an auxiliary photon D in the state . This Toffoli gate is realized with the following three controlled gates on electron spins.

Figure 3
figure 3

Toffoli gate on a three-spin system assisted by one photon with two DOFs.

cPSi, cBSi, Xi, Hi and Wi are the same as those defined in Fig. 2. ei denote input electron spins. di denote spatial modes of an auxiliary photon D in the state .

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First, the auxiliary photon D from the spatial mode d1 passes through the half waveplate H1 to H2 sequentially. The joint system consisting of the photon D and the electron spin e1 changes from into

This subcircuit (denoted as S4) has realized the controlled-NOT gate on the spin e1 and the polarization DOF of the auxiliary photon D under the joint basis .

Moreover, by letting the photon D pass the cBS1 to cBS2 sequentially, the joint system may be changed into

This subcircuit (denoted as S5) has realized the controlled-NOT gate on the spin e2 and the spatial mode DOF of the photon D under the joint basis .

Furthermore, let the photon D pass the W1 to W2 sequentially. The joint system may be changed into

where the Pauli flip . This subcircuit (denoted as S6) has realized the controlled-NOT gate on the auxiliary photon D and the input spin e3 under the joint basis .

Finally, the joint system shown in the equation (14) may collapse into

by measuring the auxiliary photon D under the basis . Similarly, this measurement may be implemented in the experiment with the 50%50 circularly polarizing beamsplitter cBS3, two circularly polarizing beamsplitters cPS7 and cPS8, two half waveplates H3 and H4 and four single photon detectors and . The recovery operations are shown in Table 2. Thus, the three-spin Toffoli gate shown in the equation (13) has been deterministically realized with three control gates and .

Table 2 The relations between the measurement outcomes of the auxiliary photon D and the feed-forward operations for implementing the Toffoli gate on three electron spins e1, e2 and e3.
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Toffoli gate on hybrid three-qubit systems

The present Toffoli gate on a three-photon system shown in the Fig. 2 and a three-spin system shown in Fig. 3 may be combined to realize Toffoli gate on hybrid three-qubit systems. Thus, the three input qubits may be an arbitrary combination of photons and quantum spins. Because of the symmetry of two control qubits, four different cases are to be considered, as shown in Fig. 4.

Figure 4
figure 4

Toffoli gate on hybrid three-qubit systems assisted by one photon with two DOFs.

(a) Two photons A and B jointly control an electron spin e. (b) Two electron spins e1 and e2 jointly control a photon A. (c) One photon A and one electron spin e jointly control a photon B. (d) One photon A and one electron spin e1 jointly control an electron spin e2. S1, S2 and S3 denote the subcircuits shown in Fig. 2(a–c), respectively. S4, S5 and S6 are shown in Fig. 3. The bule lines denote the controlling qubits while the red lines denote the target qubits. The black lines denote an auxiliary photon D in the state . MD denotes the measurement of the photon D shown in Fig. 2(c).

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First, let two photons A and B jointly control an electron spin e; their initial states are , and , respectively. The detailed circuit is shown in Fig. 4(a). From the realized with the subcircuit S1, the joint system consisting of three qubits and an auxiliary photon D changes from into

Moreover, from the CNOT gate realized with the subcircuit S2, may change into

Furthermore, from the CNOT gate realized with the subciruit S6, the joint system shown in the equation (17) changes into

which may collapse into

by performing the single qubit measurement MD on the photon D under the basis . In the experiment, this measurement may be implemented in experiments with a 50%50 circularly polarizing beamsplitter, two circularly polarizing beamsplitters, two half waveplates and four single photon detectors, as shown in Fig. 2(c). The recovery operations are similar to these shown in Table 1. Thus, a Toffoli gate has been realized on the two photons and one spin using three CNOT gates.

Second, consider two electron spins e1 and e2 in the states that jointly control one photon A in the state . The detailed circuit is shown in Fig. 4(b). From the CNOT gates realized with the subcircuit S4 and S5 in Fig. 3, the joint system consisting of three input qubits and the auxiliary photon D changes from into

Moreover, from the CNOT realized with the subcircuit S3 in Fig. 2(c), the joint system changes into

which may collapse into

after performing the measurement MD of the photon D under the basis . The recovery operations are shown in Table 2. Thus, a Toffoli gate has been realized on two electron spins and one photon.

Third, consider one photon A in the state and one spin e in the state that jointly control one photon B in the state . The detailed circuit is shown in Fig. 4(c). Similar to the subcircuits shown in Fig. 4(a,b), from the CNOT gates realized with the subcircuits S1 in Fig. 2(a), S5 in Fig. 3 and S3 in Fig. 2(c), the joint system of the three input qubits and the auxiliary photon D changes from into

which may collapse into

after the measurement MD of the photon D under the basis . The recovery operations are shown in Table 2. The difference is that the Pauli phase flip is performed on the controlling spin e. Thus, a Toffoli gate has been realized on two electron spins and one photon.

Finally, consider one photon A in the state and one electron spin e1 in the state that jointly control the other electron spin e2 in the state . The detailed circuit is shown in Fig. 4(d). Similar to the subcircuit shown in Fig. 4(c), from the CNOT gates realized with the subcircuits S1 in Fig. 2(a), S5 in Fig. 3 and S6 in Fig. 3, the joint system consisting of three input qubits and the auxiliary photon D changes from into

which may collapse into

by performing the measurement MD of the photon D under the basis and for the polarization DOF and spatial mode, respectively. The recovery operations are the same as those in Fig. 4(c). Thus, the spin qubit may be jointly controlled by one photon and one spin.

Discussion

The optical selection rules of a QD-cavity system shown in equation (1) play core roles in the present Toffoli gates. In the resonance conditions Δωx = Δωc = 0, if one neglects the cavity side leakage κs ≈ 0, it easily follows that |r0| → 1 and |r| → 1 when the cooperativity parameter g2/(κγ) of cavity QED is large enough. Thus, our six Toffoli gates are deterministic and faithful. However, the side leakage from the cavity is unavoidable in the experiment44,45,47,48,49,50,51,52,53,54. In the following, consider two kinds of transition channels for the cavity photon. The first is the cavity decay due to transmission through the cavity mirror, whose rate is κ. Every other unwanted photon loss, such as cavity absorption and scattering, are characterized by the overall loss rate κs. Taking into account the coupling through the cavity decay channel and neglecting the spatial dependence, the relation of the input field operator and output operator may be approximated with an experimental reflection coefficient37,38,39

where Δωc and Δωx are the frequency detunings of the cavity mode and dipole transition, respectively, in relation to the input probe light (See Method). When the quantum dot is uncoupled from the cavity (g = 0), r(ω) is reduced to37,38,39

These complex coefficients indicate that the reflected light may experience a phase shift32,33,34,35,36,44,45,46. Under resonant conditions Δωc = Δωx = 0, the reflection coefficients |r| and |r0| are evaluated in Fig. 5 and the phase shifts θ and θ0 are evaluated in Fig. 6 inrelation to the decay ratios of cavity κs/κ and the cooperativity parameter C = g2/(κγ) of cavity QED56,57, which is a geometric parameter that characterizes the absorptive, emissive, or dispersive coupling of an atom to the cavity mode. Based on Fig. 5, the reflection coefficients will satisfy |r| ≈ 1 and |r0| ≈ 1 when C ≫ 10 and κs/κ → 0 and these additional conditions are not required for relative phase shifts θ0 = π and θ = π because r and r0 are real under the resonant conditions Δωc = Δωx = 0. Hence, the real reflection coefficients r and r0 will be considered under the resonant conditions.

Figure 5
figure 5

Reflection coefficients versus the cavity leakage ratio κs/κ and the cooperativity C under resonant conditions.

(a) Reflectance |r| and (b) reflectance |r0| under resonant conditions.

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Figure 6
figure 6

Phase shifts versus the cavity leakage ratio κs/κ and the cooperativity C under resonant conditions.

Here, the scale of the phase shift is π.

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In fact, the ideal optical selection rules shown in equation (1) are changed into

in the experiment. Based on these general optical selection rules, one can also complete the Toffoli gate from our schemes.

For our first Toffoli gate on the three photons shown in Fig. 2, three auxiliary electron spins e1, e2 and e3 in the state are used and four photons A, B, C and D are involved; the success of this protocol is heralded by the instance in which the detector or click. The efficiency of our Toffoli gate is defined by , where Pj is a successful reflection probability of the j-th photon from a micropillar cavity37,50,54,57 and denotes the index set of photons involved in each scheme. Its efficiency is evaluated in Fig. 7(a). To detail the influence of the practical input-output process on the fidelity of the final joint system after this Toffoli gate, we take the case in which the detector clicks as an example and obtain the average fidelity Fpp,p, as evaluated in Fig. 8(a). Here, , where the integral is evaluated over all possible input states, and are the ideal final state and the experimental final state with side leakages, respectively. For our second Toffoli gate on three electron spins shown in Fig. 3, three electron spins e1, e2 and e3 are involved and one photon D is used; its success is determined by the photon D, which is detected at the detector or click. The practical efficiency Ess,s is evaluated in Fig. 7(b) whereas the average fidelity Fpp,p is evaluated in Fig. 8(b) for the photon D detected at the detector as an example. For the other four cases, one can obtain similar results.

Figure 7
figure 7

Efficiencies of Toffoli gate versus the cavity leakage ratio κs/κ and the cooperativity C.

(a) Efficiency Epp,p of Toffoli gate on a three-photon system. (b) Efficiency Ess,s of Toffoli gate on a three-spin system.

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Figure 8
figure 8

Average fidelities of Toffoli gate versus the cavity leakage ratio κs/κ and the cooperativity C.

(a) The average fidelity Fpp,p of Toffoli gate on a three-photon system. (b) The average fidelity Fss,s of Toffoli gate on a three-spin system.

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Typically, the cavity side leakage may greatly affect the efficiency and fidelity of the Toffoli gate. As shown in the Figs 7 and 8, high efficiency and fidelity may be achieved even in the weakly coupling regime when . Otherwise, the strong coupling defined by g ≫ (κ, γ) is necessary39,40,41,48,49,50,51,52,53,54. The classical strong-coupling condition corresponds to the single-photon Rabi frequency 2g being larger than the geometric mean of the atomic and cavity line widths. In general, the system can be parameterized in terms of two dimensionless parameters, namely, the ratios g/κ and g/γ in the cavity QED description or, in the classical description, the cooperativity parameter C and the line width ratio κ/γ. The cavity QED strong-coupling condition 2g > (κ, γ) corresponds to a normal-mode splitting that is much larger than the line widths of the normal modes. The cooperativity parameter of cavity QED is shown to play a central role and is given a geometrical interpretation. The cooperativity has been realized up to 2758. Under this cooperativity, the efficiencies EPP,P and ESS,S are greater than 91.24% for κs/κ ≈ 0.248,54; the average fidelities FPP,P and FSS,S are greater than 93.47% for κs/κ ≈ 0.248,54. If one hopes to achieve a fault tolerance threshold of 7.5 × 10−3 on a two-dimensional lattice of qubits59, the cavity leakage ratio should be κs/κ < 0.04 and the cooperativity should be C > 28 for a photonic Toffoli gate, whereas the cavity leakage ratio should be κs/κ < 0.03 and the cooperativity should be C > 34 for a Toffoli gate on a three-spin system. When the fault tolerance threshold is reduced to 1 × 10−3 using controlled phase gates based on dipole-induced transparency60, the cavity leakage ratio should be reduced to 0.02 and the cooperativity should be improved to C > 38 for a photonic Toffoli gate, whereas the cavity leakage ratio should be reduced to 0.015 and the relative coupling strength should be improved to 4.2 for a Toffoli gate on a three-spin system. κs/κ = 0.05 has been reported, which could be achieved by taking a pillar microcavity with the quality factor of Q = 165000 demonstrated in ref. 54 and decreasing the reflection of the top mirror to reduce the quality factor to Q = 9000, which is still in the strong-coupling regime48.

If the experimental electron spin decoherence and trion dephasing41,42 are considered, the real efficiency and fidelity are slightly decreased when the hole spin coherence time is longer than three orders of the cavity photon lifetime44,50,51. Moreover, by using the spin echo technique57,61 and the nanosecond spin resonance microwave pulse47 to protect the electron spin coherence, faithful Hadamard transformations may be implemented on the electron spin for our six Toffoli gates. The heavy-light hole mixing may be reduced by engineering the shape, size and type of the charged exciton61. The optical selection rule has been experimentally realized with the spin state of a single trapped atom and the polarization state44,45. To achieve weak excitation, some adiabatic conditions are used to ensure that the X− stays in the ground state for the most time. With a first-order approximation, we can adiabatically eliminate from the third subequation of equation (31) by substituting the steady-state solution to the first two subequations of equation (31). Under the adiabatic condition , the system may be unchanged between the ground state and excite state under the first-order approximation. Here, ΔE10 = E1 − E0. If the dephasing is considered for the atomic system, it may be modeled by introducing phenomenological decay terms or noise operators into three subequations of equation (31). Because the output modes are initially in a vacuum, the . By substituting the steady-state solution to the third subequation of equation (31), the only difference is one noise operator and the modified spontaneous emission rate of the dipole 62. Of course, the present Toffoli schemes are also conditional on the perfect overlap of the cavity mode with the two spatially separated optical beams, the phase stability of the interferometer composed of the cBS and the perfect time overlap of two beams passing through several interferometers.

In conclusion, we have investigated the possibility of hybrid quantum computation assisted by the quantum spins and photons with two DOFs. Six deterministic Toffoli gates are realized on the joint system of all combinations of the photon or the quantum spin systems. Compared with previous Toffoli gates13,14,15,16,17,19,20, our Toffoli gates may be realized with three general control-NOT gates, which are similar to the schemes in ref. 29, 30, 31. Unlike the multiple dimensional quantum target state of the photonic Toffoli gate18,30, all the input systems are qubit systems, whereas the additional multiple-dimension logic state is used as the auxiliary system. With the modification, one does not need to consider the different dimensional quantum systems to encode information in quantum applications. This method is similar to that in ref. 31. However, their disentangling operations are necessary and essential controlled operations or high-dimensional operations on the auxiliary system. If our photon with two DOFs is considered, their Fourier disentangling operations require two controlled operations. However, with our schemes, even if the photon with two DOFs is used as an auxiliary system, we do not need to implement controlled operations or high-dimensional operations on the auxiliary system. Our disentangling operations are only single-qubit operations. Moreover, the Toffoli gate may be realized on different quantum systems, which may be very useful depending on the specific requirements. Different from the Toffoli gate34 on the three-atom system, our Toffoli gate may be implemented on a hybrid photon and spin system. Our optical cavity system is easier than the Toffoli gate35 using the double-side cavity system. Compared with their six controlled qubit operations34,35,63, our circuits are also compact by as a result of the auxiliary high-dimensional system and cost only three controlled qubit operations. Our theoretical results show that photons and quantum spins may be used alternatively in quantum information processing. Of course, the optical selection rules may be affected by the cavity leakage and spin coherence in quantum dots or the exciton coherence in the experiment. With the recent experiments regarding QD-cavity system47,48,49,50,51,52,53,54 and the quantum gate between a flying optical photon and a single trapped atom32, our results are expected to be applicable for large-scale quantum computation.

Method

Optical selection rules

A singly charged GaAs/InAs QD32,33,37,38,39,40,41,42,43,44,45 has four relevant electronic levels , and . An exciton consisting of two electrons bound to one hole with negative charges can be created by the optical excitation of a photon and an electron spin. In theory, consider the interaction between a single cavity mode and a single two-level spin interacting with a single cavity mode at optical frequencies. By neglecting the spatial dependence37,44,45, taking into account the coupling through the cavity decay channel and neglecting the spatial dependence, the master equation of the whole system can be expressed by the Lindblad form

where H = H1 + H2 + H3. ρ is an arbitrary system operator. is the Hamiltonian of the input photon pulse. is the standard Jaynes-Cummings Hamiltonian for a two-level system interacting with a single electromagnetic mode by applying the rotating wave approximation and dropping the energy nonconserving terms. are cavity input operators with the standard commutation relations . σ− and σ+ are the Pauli raising and lowering operators respectively. is the system Hamiltonian for the dipole, ωc is the resonant frequency of the dipole and σz is the Pauli operator for the population inversion. κ is the decay rate of the cavity field due to ohmic losses in the metal. accounts for the damping of the input photon pulse. κs is the decay rate of the cavity side leakage mode due to scattering into free-space modes. The scattering rate κs may be calculated classically from the Larmor formula. accounts for spontaneous emission of the dipole. Using this Hamiltonian, the Heisenberg equations for first order field/spin moments easily follow

where Δωc = ωc − ω and , is the frequency dipole transition. The classical boundary condition is defined as 37,38,39 with the input and output field operators and , respectively. In the approximation of weak excitation (X− stays in the ground state for the most time39,40,41,42,43), i.e., , and are approximately related with the reflection coefficient

where r(ω) is defined in equation (27). If the quantum dot is uncoupled from the cavity (g = 0), r(ω) is reduced to r0(ω) as shown in equation (28). For the strong coupling regime g ≫ (κ, γ), one can get |r| ≈ 1 and |r0| ≈ 1 under resonant conditions by adjusting ω, ωx and ωc. Thus, if the excess electron spin lies in the spin state , the input light acquires a phase shift of θ = arg[r(ω)](θ0 = arg[r0(ω)]) by passing through the cavity. Conversely, if the excess electron spin lies in the spin state , the input light acquires a phase shift of θ = arg[r(ω)](θ0 = arg[r0(ω)]) by passing through the cavity. Thus, two phase shifts may be obtained as37,38,39

When the side leakage and cavity loss are ignored, the optical selection rules shown in equation (1) are followed by adjusting frequencies to achieve the phase shifts θ0 = π and θ = 032,33,44,45.

Measurement of the entangled excess electron spin in a QD-cavity

To complete our Toffoli gates, the entangled excess electron spins have to be measured under the basis . Generally, an auxiliary photon is used32,33,34,35,44,45. Consider a generally entangled system , where are orthogonal states of other systems except the electron spin e. The joint state is first represented by one Hadamard transformation W, i.e., . Then, let the auxiliary photon pass through one circularly polarizing beamsplitter, the QD-cavity and the other circularly polarizing beamsplitter. This joint system becomes . Thus, by measuring the photon under the orthogonal basis with one half waveplate, one circularly polarizing beamsplitter and two single photon detectors, the electron spin e can be faithfully disentangled. The experimental performances depend on the experimental optical selection rules shown in equation (1).

Additional Information

How to cite this article: Luo, M.-X. et al. Hybrid Toffoli gate on photons and quantum spins. Sci. Rep. 5, 16716; doi: 10.1038/srep16716 (2015).