Introduction

Quantum computer offers great advantages over the classical computer1, such as tremendous speedup, efficiently searching for unordered database, and factorizing large numbers. Quantum gates are the fundamental elements for the quantum computer. Quantum computation, quantum networks, and quantum algorithms can be decomposed into a set of universal gates and single-qubit gates2. Controlled-NOT (CNOT) gate or controlled-phase gate is the most popular nontrivial universal gate3, and considerable progresses have been made on them4,5,6. In recent years, constructions of the Toffoli gate7 or Fredkin gate8 has been addressed from various perspectives, and these two universal multiqubit-control gates are the key operations in quantum networks, quantum phase estimate, fault-error quantum computing, quantum-error correction, quantum Shor’s algorithm9,10, quantum Grover’s algorithm11, and optimal Long’s algorithm12. A Toffoli or Fredkin gate can be implemented by series of nontrivial two-qubit gates along with single-qubit gates. However, this strategy requires a number of gate steps, additional qubits, and increases the noise channels. The synthesis of a n-qubit Toffoli gate requires O(n2) two-qubit gates3. The minimal cost of a three-qubit Fredkin gate is five two-qubit gates13. Therefore, it is significant to directly and efficiently implemented Toffoli and Fredkin gates without resorting to two- and single-qubit gates.

Finding an efficient quantum computing with minimal quantum resources is the central work in quantum information process. Encoding qubits in multiple degrees of freedom (DoFs) increases the capacity of the quantum channel, reduces the quantum resources, and are robust against the decoherence caused by the environments. Single photon is nowadays recognized as an excellent candidate for quantum information process in multiple DoFs as it carries many qubit-like DoFs14 including momentum (linear, orbital, spatial), frequency, time-bin, polarization, and availability single-qubit manipulation. Utilizing cross-Kerr nonlinearity, Sheng et al.15 and Liu et al.16 proposed schemes for complete hyperentangled-Bell-state analysis in two and three DoFs, respectively. In 2012, Ren et al.17 theoretically implemented hyperentangled-Bell-state analysis in polarization-spatial DoF via quantum dot. In 2014, Wang et al.18 proposed a hyperdistillation scheme in two DoFs. In 2015, Wang et al.19 demonstrated quantum teleportation of a single photon with polarization-orbit DoF. Recently, Scholz et al.20 and Zhang et al.21 demonstrated two-qubit Deutsch-Jozsa algorithm with a single photon in polarization-spatial DoF, respectively. Abouraddy et al.22 constructed one-photon three-qubit universal gates. The weak interaction in single photon level is the limitation of photonic quantum computing. Fortunately, in 2001, Knill, Laflamme, and Milburn23 proposed an interesting scheme for implementing linear optical controlled-phase gate with maximal success probability of 3/4. The strategy adopted to overcome undeterministic is to employ cross-Kerr nonlinearity24 or photon-matter platforms25,26. Up to now, the giant cross-Kerr nonlinearity is still a challenge in experiment. In recent years, wide attention has been drawn to couple flying photonic qubit to stationary matter qubit as it provides a platform for scalable photonic or matter quantum computing. Ren et al.27,28 proposed a spatial-polarization hyper-controlled-not gate with coupled quantum dot. Ren et al.29, Wei et al.30, and Wang et al.31 presented schemes for multiphoton hyper-parallel quantum computing assisted by matter qubits coupled with resonators.

Isolated electron-spin in diamond nitrogen-vacancy (NV) center has been recognized as an unique and promising candidate for solid-state quantum computing. The appeal feature of the diamond NV center is its urtralong coherence time (~ms) even at room temperature32,33, and diamond NV center has been received strong interest in recent years. The initialization34, manipulation35, storage36, readout37, and the nontrivial interactions between two diamond NV spins38,39, which are the necessary operations for quantum computing, have been well experimentally demonstrated. There are also a number of hallmark other demonstrations, such as the quantum Deutsch-Jozsa algorithm40, controlled-ROT gate41, decoherence-protected controlled-rotation gates42, and holonomic CNOT gate43,44. Based on photon-NV entangled platform45,46, Hanson et al.39,47 experimentally demonstrated quantum teleportation and Bell inequality between distant diamond NV centers, and several proposals for implementing parallel and hyper-parallel quantum computing were also proposed31,48,49,50.

In this article, we investigate the implementations of the four-qubit Toffoli and Fredkin gates acting on two-photon systems via the diamond NV centers coupled to resonators. The Toffoli gate flips the polarization or spatial state of the target qubit conditional on the polarization and spatial states of the control qubits. The Fredkin gate swaps the polarization and spatial states of the target photon depending on the polarization and spatial states of the control photon. In our schemes, the two gates are heralded by the spins of the electrons in diamond NV centers. Additional photons, necessary for the cross-Kerr-based and parity-check-based optical quantum computing, are not required. Compared to the ones in one DoF, our schemes only require the half quantum resource, and are robust against the decoherence. The estimations of our devices show that our schemes are possible in experiment with current state of technology.

Results

The optical property of an NV-cavity platform

The Toffoli gates we considered are equivalent to the controlled-controlled-controlled-phase flip (CCCPF) gate up to two Hadamard (H) gates acting on the target qubits, i.e., UToffoli = H. CCCPF. H. Here, we employ the NV-cavity entangled platform to overcome the weak interactions between two single photons involved in the Toffoli gate. As shown in Fig. 1(a), |ms〉 = |±1〉 and |ms〉 = |0〉 (marked by |±〉, |0〉) are the ground triple states of the diamond NV center. In absence of a magnetic field, |±1〉 and |0〉 are split with 2.87 GHz due to the spin-spin interaction, and the corresponding transitions, |0〉 ↔ |±1〉, are in the microwave regime. The six electron excited states include51 |E1〉 = |E−〉|−1〉 − |E+〉|+1〉, |E2〉 = |E−〉|−1〉 + |E+〉|+1〉, |Ex〉 = |X〉|0〉, |Ey〉 = |Y〉|0〉, |A1〉 = |E−〉|+1〉 − |E+〉|−1〉, and |A2〉 = |E−〉|+1〉 + |E+〉|−1〉. |E±〉, and |X〉, |Y〉 are orbital states with angular momentum projections ±1 and 0, respectively. The energy gap arising from the spin-spin and spin-orbit interactions protects state |A2〉 against small strain and magnetic filed. The transitions, |±〉 → |A2〉, are driven by the resonator optical pluses with σ− and σ+-polarized at 637 nm, respectively (see Fig. 1(b)). Therefore, the device shown in Fig. 1 can be used as a quantum interface between optical and solid-state systems for scalable quantum information process.

Figure 1
figure 1

(a) Schematic diagram for a diamond NV center coupled to a double-sided resonator. (b) A ∧ -type electron energy structure of a trapped diamond NV center. Transitions |±1〉 ↔ |A2〉 are driven by the σ− and σ+ photons, respectively.

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Compared with the coupling of an unconfined diamond NV center, trapping a diamond NV center into an optical cavity enhances the coupling between the incident photon and the diamond NV center. Nowadays, various of optical microcavities, such as photonic crystal cavity52, fiber-based Fabry-Perot cavity53, microsphere54, toroidal resonators55, coupling to the diamond NV centers have been experimentally demonstrated. Suppose that the diamond NV center coupling with a whispering-gallery-mode microresonator with two tapered fibers (see Fig. 1). If the diamond NV in the state |+1〉, the incoming σ+-polarized photon via port a (port c) senses a cold cavity, and departs through port b (d) with phase shift , while the incoming σ−-polarized photon via port a (port c) senses a hot cavity, and leaves from the cavity via port d (b) with phase shift . In the same way, if the diamond NV center in the state |−〉, the input σ−-polarized photon senses a cold cavity and it is transmitted, while the σ+-polarized photon senses the hot cavity and it is reflected. The reflection and transmission coefficients of the NV-cavity system can be obtained by solving the Heisenberg equations for the cavity mode a with frequency ωc and the diamond NV lowering operator σ− with frequency ω0, that is56,

Here, κ and γ/2 are the decay rates of the cavity field and the diamond NV dipole, respectively. κs/2 is the cavity intrinsic loss rate (side leakage). g is the coupling rate between the diamond NV center and the resonator. σz is the pauli operator. h and f are the noise operators. The input field is connected with the output filed by and . Taking the weak excitation limitation, i.e., <σz> ≈ −1, through the operation, the transmission and reflection coefficients of the NV-cavity system can be respectively written as57,58

Under the resonator condition ωc = ω0 = ωp, if the Purcell factor g2/(κγ) ≫ 1 with κs ≈ 0 is ignored, the cavity mode of the hot cavity is reflected, i.e., r(ωp) → 1 and t(ωp) → 0. If κs/κ ≪ 1, the cavity mode of the bare cavity is transmission, i.e., r0(ωp) → 0 and t0(ωp) → −1. That is, the input-output relations between the single photon and NV center can be summarized as

Here, |R, a〉 and |L, d〉 with sz = −1, |L, a〉 and |R, d〉 with sz = +1, the polarization of the photon is defined with respect to the z axis (the diamond NV center axis).

Two-photon four-qubit Toffoli gate acting on the polarization and spatial DoFs

The quantum circuit we designed, as shown in Fig. 2, performs the following operation on a two-photon four-qubit state

Figure 2
figure 2

Two-photon four-qubit Toffoli gate acting on the polarization and spatial DoFs.

Polarizing beam splitters in the circularly basis, CPBSi (i = 1, 2, 3, 4), transmits the R-polarized and reflects the L-polarized components, respectively. Single-qubit operations can be achieved by employing two half-wave plates oriented at 45° and 22.5°, respectively. The local single-qubit operations, heralded by the outcomes of the diamond NV center, via a feed-forward process are not shown in the sketch.

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where α1, α2, β1, β2, γ1, γ2, δ1, and δ2 are arbitrary complex parameters satisfying |α1|2 + |α2|2 = 1, |β1|2 + |β2|2 = 1, |γ1|2 + |γ2|2 = 1, and |δ1|2 + |δ2|2 =1. The indices 1 and 2 of R (or L) denote the R- (or L-) polarized wave-packets of the photon 1 and photon 2, respectively. a1 (b1) and a2 (b2) represent photon 1 (photon 2) emitted from the spatial modes 1 and 2, respectively.

In Fig. 2, we first use the circularly polarized beam splitter, CPBS1, to depart the wave-packets, R and L, of photon 1 emitted from the spatial mode a2., i.e., the R1-component is transmitted and the L1-component is reflected. The R1-component does not interact with the diamond NV center and arrives at CPBS2 directly. Before and after the L1-component interacts with the block composed of CPBS3, NV, and CPBS4, the two single-qubit rotations Ry(−π/2) and Ry(π/2) are respectively applied on photon 1 simultaneously with two Hadamard operations Hes applied on the diamond NV center. Subsequently, the R1-component and L1-component mix at CPBS2. The operations (CPBS1 → Ry(−π/2), He → CPBS3 → NV → CPBS4 → Ry(π/2), He → CPBS2) transform the system composed of the photon 1, photon 2, and diamond NV center from the initial state |ψ0〉 into |ψ1〉. Here,

The Hadamard transformation He can be achieved by employing π/2 pulse and it completes the following transformations

The single-qubit rotations Ry(±π/2) can be achieved by employing two half-wave plates oriented at 45° and 22.5°, respectively. In the standard basis {|R〉, |L〉}, Ry(±π/2) can be written as

Next, photon 2 in the spatial mode b2 is injected into the block. Before and after photon 2 interacts with the block, are applied on it, respectively. The above operations (Ry(−π/2) → CPBS3 → NV → CPBS4 → Ry(π/2)) transform |ψ1〉 into

Third, we measure the electron spin state of the diamond NV center in the basis . If the diamond NV center in the state , we perform a phase shift π on the photon emitted from the spatial mode b2, i.e., |R2〉|b2〉 → −|R2〉|b2〉 and |L2〉|b2〉 → −|L2〉|b2〉. If the diamond NV in the state , classical feed-forward operations, phase shift π and σz = |R〉〈R| − |L〉〈L|, are performed on the photons emitted from the spatial modes b2 and a2, respectively. After the above operations, the state of the system composed the two photons becomes

It is obvious that the quantum circuit shown in Fig. 2 determinately implementing a CCCPF gate which flips the phase of the input state if and only if the system initially is in the state |L1〉|a2〉|L2〉|b2〉, and has no change otherwise. Therefore, a two-photon four-qubit Toffoli gate which performs a NOT operation on the spatial (or polarization) state of the photon 2 depending on the states of other qubits can be achieved by the scheme depicted by Fig. 2 up to two Hadamard gates.

Two-photon four-qubit Fredkin gate acting on the polarization and spatial DoFs

The Fredkin gate swaps the states of the two target qubits conditional on the states of the control qubits. The quantum circuit depicted by Fig. 3 implements a Fredkin gate which completes the transformation

Figure 3
figure 3

Two-photon four-qubit Fredkin gate acting on the polarization and spatial DoFs.

Hadamard transformation HP can be achieved by employing a half-wave plate oriented at 22.5°. Phase shifter Pπ completes the operations |R〉 → −|R〉 and |L〉 → −|L〉. The local classical feed-forward single-qubit operations are not shown in the sketch.

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As illustrated in Fig. 3, such Fredkin gate can be implemented by three steps. We first inject the photon 1 (encoded as the control photon) emitted from the spatial mode a2 into the block composed of CPBS3, phase shifters and , NV, and CPBS4. Before and after the photon interacts with such block, Hadamard operations, and , are respectively applied on it with the half-wave plates oriented at 22.5° simultaneously with two Hadamard operations Hes applied on the diamond NV centers. The above operations () transform the system composed of the two single photons and one diamond NV center from the initial state |ϕ0〉 into |ϕ1〉. Here,

In the standard basis {|R〉, |L〉}, Hadamard operation Hp can be expressed as

Phase shifter, or , introduces a phase shift π to the passing photon, i.e., |R〉 → −|R〉 and |L〉 → −|L〉.

Subsequently, photon 2 emitted form the spatial mode b1 or b2 is injected and then arrives at CPBS5. CPBS5 transforms |φ1〉 into

The wave-packets emitted from the spatial mode b1 pass through the block composed of CPBS3, phase shifters and , NV, and CPBS4, and such block transforms the state of the whole system into

After the wave-packets of photon 2 mix at CPBS6, the state of the whole system becomes

We lastly measure the output state of the diamond NV center in the basis . If the diamond NV center in the state , a single-qubit operation σz is performed on the photon emitted from the spatial mode a2. If the diamond NV center in the state , no feed-forward operation is applied. After the above operations, the state of the composite system is projected onto

Therefore, the quantum circuit shown in Fig. 3 completes a two-photon four-qubit Fredkin gate which swaps the polarization and spatial states of the target photon when the control photon is in the state |L1〉|a2〉, and remain in their initial states otherwise.

Discussion

The most popular universal libraries for multiqubit system are {Toffoli or Fredkin gate and Hadamard gates}. Implementations of the Toffoli and Fredkin gates have been attracted strong interesting in recent years, and most of them focus on the three-qubit system with one DoF. The traditionally three-qubit Toffoli and Fredkin gates have been proposed in various systems, such as hybrid photon-matter31,59, linear optics60, single-photon48,61, nuclear magnetic resonance62, ion trap63, atom64, quantum dot65, diamond NV center49 and superconducting66,67. In this work, we have proposed two schemes for implementing two-photon four-qubit Toffoli and Fredkin gates via NV-cavity interactions. The individual interactions between computational qubits are achieved by employing linear optical elements and diamond NV centers. The diamond NV center can be created with excellent optical coherence property, i.e., the zero-phonon line (ZPL) exhibiting the spectral diffusion close to the optical line width (~MHZ)68. The enhancement of the ZPL emission for the diamond NV center is particularly important as the diamond NV center can be served as a photon-emitter for quantum information process, and it has been enhanced from 3–4% to 70% of the total emission69.

In summary, we have designed two compact quantum circuits for determinately implementing two-photon four-qubit Toffoli and Fredkin gates through cavity-assisted interactions. Compared to the traditional Toffoli and Fredkin gates with one DoF, the gates presented are acting on both the polarization DoF and the spatial DoF, are robust against the decoherence caused by their environments, and reduce the photons from four to two. The traditional two-body interactions in one DoF do not appear in naturally physical systems and can not be implemented by solely using linear optical elements. The two-body interactions involved in one single photon, which necessary for our Toffoli and Fredkin gates, can be achieved by employing linear optics. Additional photons, necessary for the cross-Kerr-based, cluster-based, or parity-check based quantum computing, are not required in our schemes, and the diamond NV center only provides a platform for photon-photon interactions. Compared to the ones synthesized by CNOT and single-qubit gates, our schemes are not only robust against the environment noises but also reduce the complexity in experiment. The length of the synthesis of a four-qubit Toffoli gate is 16 nontrivial two-qubit gates3. The average fidelities and efficiencies show that our schemes may be possible with current technology.

The average fidelities and efficiencies of the Toffoli and Fredkin gates

The key element of our schemes is the NV-cavity emitter. In the idealistic case, the optical transition rules of the emitter is perfect, however, the phase and amplitude of the incident photon are not perfect in experiment, and they are the functions of the κ, κs, and g. If the side leakage is taken into account, there may be the reflection (transmission) part with coefficient r0 (t) of incident photon in the cold (hot) cavity. Therefore, in the realistic case, the optical transition rules of the incident photon described by Eq. (3) should be changed with |R, a〉|+1〉 → r|L, d〉|+1〉 + t|R, b〉|+1〉, |R, a〉|−1〉 → t0|R, b〉|−1〉 + r0|L, d〉|−1〉, |L, c〉|+1〉 → r|R, b〉|+1〉 + t|L, d〉|+1〉, |L, c〉|−1〉 → t0|L, d〉|−1〉 + r0|R, b〉|−1〉, |R, c〉|+1〉 → t0|R, d〉|+1〉 + r0|L, b〉|+1〉, |R, c〉|−1〉 → r|L, b〉|−1〉 + t|R, d〉|−1〉, |L, a〉|+1〉 → t0|L, b〉|+1〉 + r0|R, d〉|+1〉, and |L, a〉|−1〉 → r|R, d〉|−1〉 + t|L, d〉|−1〉. It is note that the above undesirable parts will degrade the performance of the emitter. Hence, in order to quality the performances of our schemes, we should investigate the average fidelity, which is defined as in experiment. Here the overline indicates average over all possible input (output) states. |ψr〉 and |ψi〉 are the normalized output states of the system in the realistic and idealistic cases, respectively. Therefore, the average fidelity of our Toffoli or Fredkin gate, or , can be calculated by

where cos α = α1, sin α = α2, cos β = β1, sin β = β2, cos γ = γ1, sin γ = γ2, and cos δ = δ1, sin δ = δ2 are taken for Eqs (4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18). The average fidelity of our Toffoli and Fredkin gates as functions of g2/κγ and κs/κ are plotted in Fig. 4, respectively.

Figure 4
figure 4

The fidelities averaged over all the input (output) states for (a), Toffoli gate and (b), Fredkin gate .

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Efficiency is another powerful and practical tool for qualifying the performance of the setup. In experiment, the efficiency is defined as noutput/ninput, i.e., the yield of the incident photons. noutput and nintput are the numbers of gate’s output and input photons, respectively. The average efficiencies of our Toffoli and Fredkin gates, and , are calculated by

where cos α = α1, sin α = α2, cos β = β1, sin β = β2, cos γ = γ1, sin γ = γ2, and cos δ = δ1, sin δ = δ2. By calculating, one can find that

The date shows () against g2/κγ and κs/κ are presented in Fig. 5.

Figure 5
figure 5

The efficiencies averaged over all the input (output) states for Toffoli gate and Fredkin gate .

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From Figs 4, 5, one can see that the normalized side leakage and the normalized cavity coupling strength impact the gate’s fidelity and efficiency. The higher g2/κγ and the lower κs/κ, the higher fidelity and efficiency. The high fidelities and efficiencies of our Toffoli and Fredkin gates can be achieved. For example, in the case κs/κ = 0.1 and g2/(κγ) = 2.4, ,  0.979516 with , and κs/κ = 1 and g2/(κγ) = 2.4, , 0.868208 with .

Our schemes are deterministic if the unavoidable effect of photon loss and imperfections are neglected. Our schemes in practice working with photon loss, non-ideal single-photon sources, unbalanced CPBSs and BSs, timing jitter, spin flips during the optical excitation, the mix between state |A2〉 and other excited states, signal to noise in the ZPL channel, imperfect spin initialization and detection. These imperfections degrade the fidelities and efficiencies of our two gates, and the imperfections which are due to the technical imperfection will be improved with further technical advances. Given the current technology, our schemes are more efficient than the traditional ones with one DoF, the synthesis-based ones which are rely on CNOT gates and single rotations, and the parity-check-based ones, and they may be possible in experiment.

Additional Information

How to cite this article: Wei, H.-R. and Zhu, P.-J. Implementations of two-photon four-qubit Toffoli and Fredkin gates assisted by nitrogen-vacancy centers. Sci. Rep. 6, 35529; doi: 10.1038/srep35529 (2016).