Hybrid Toffoli gate on photons and quantum spins

Abstract

Quantum computation offers potential advantages in solving a number of interesting and difficult problems. Several controlled logic gates, the elemental building blocks of quantum computer, have been realized with various physical systems. A general technique was recently proposed that significantly reduces the realization complexity of multiple-control logic gates by harnessing multi-level information carriers. We present implementations of a key quantum circuit: the three-qubit Toffoli gate. By exploring the optical selection rules of one-sided optical microcavities, a Toffoli gate may be realized on all combinations of photon and quantum spins in the QD-cavity. The three general controlled-NOT gates are involved using an auxiliary photon with two degrees of freedom. Our results show that photons and quantum spins may be used alternatively in quantum information processing.

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 d_i 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. 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.

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 e_i in the state and an auxiliary photon D in the state . The Toffoli gate T_AB,C is completed with the following three controlled gates.

Figure 2. Toffoli gate on a three-photon system assisted by one photon with two DOFs.

d_i denote spatial modes of the auxiliary photon D. e_i denote auxiliary electron spins in the state . H_i denote half waveplates to perform the Hadamard transformation and . X_i denote wave plates to perform the polarization flip transformation . Z_i denote waveplates to perform the phase flip transformation . cPS_i represent circularly polarizing beamsplitters that transmit and reflect . cBS_i represent 50%50 circularly polarizing beamsplitters to perform the Hadamard operation and . Cy_i denote the QD-cavity charged the electron spin e_i. 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.

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 CNOT_D,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.

Qubit	Feed-forward
	Photon A	Photon B
	Ip	Ip
		Ip
	Ip

and .

Thus, the Toffoli gate T_AB,C shown in equation (3) has been deterministically realized with three general controlled gates and CNOT_D,C.

Toffoli gate on a three-spin system

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

cPS_i, cBS_i, X_i, H_i and W_i are the same as those defined in Fig. 2. e_i denote input electron spins. d_i denote spatial modes of an auxiliary photon D in the state .

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.

Qubit	Feed-forward
	Spin e1	Spin e2
	Ie	Ie
		Ie
	Ie

and .

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. 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 . M_D denotes the measurement of the photon D shown in Fig. 2(c).

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 M_D 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 M_D 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 M_D 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 M_D 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 Δω_x = Δω_c = 0, if one neglects the cavity side leakage κ_s ≈ 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 κ_s. 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 Δω_c and Δω_x 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 Δω_c = Δω_x = 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 κ_s/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 κ_s/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 Δω_c = Δω_x = 0. Hence, the real reflection coefficients r and r₀ will be considered under the resonant conditions.

Figure 5. Reflection coefficients versus the cavity leakage ratio κ_s/κ and the cooperativity C under resonant conditions.

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

Figure 6. Phase shifts versus the cavity leakage ratio κ_s/κ and the cooperativity C under resonant conditions.

Here, the scale of the phase shift is π.

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 P_j 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 F_pp,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 E_ss,s is evaluated in Fig. 7(b) whereas the average fidelity F_pp,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. Efficiencies of Toffoli gate versus the cavity leakage ratio κ_s/κ and the cooperativity C.

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

Figure 8. Average fidelities of Toffoli gate versus the cavity leakage ratio κ_s/κ and the cooperativity C.

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

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 E_PP,P and E_SS,S are greater than 91.24% for κ_s/κ ≈ 0.2⁴⁸,⁵⁴; the average fidelities F_PP,P and F_SS,S are greater than 93.47% for κ_s/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 κ_s/κ < 0.04 and the cooperativity should be C > 28 for a photonic Toffoli gate, whereas the cavity leakage ratio should be κ_s/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. κ_s/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, 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. accounts for the damping of the input photon pulse. κ_s is the decay rate of the cavity side leakage mode due to scattering into free-space modes. The scattering rate κ_s 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 Δω_c = ω_c − 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, ω_x and ω_c. 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).