Introduction

A quantum computer1 can run the famous Shor’s algorithm2 for integer factorization and implement Grover-Long algorithm3,4 for unsorted database search. In past decades, it has attracted much attention. Quantum logic gates are the key elements in quantum computers and play a critical role in quantum information processing (QIP). Two-qubit controlled-not (CNOT) gates together with single-qubit gates are sufficient for universal quantum computing1,5. In 2004, Shende proposed a “small-circuit” structure which is used to construct CNOT gates6. In the domain of three-qubit gates, Toffoli gate has attracted much attention and it is universal. Together with Hadamard gates, it can realize unitary manipulation for a multi-qubit system7,8. Moreover, it plays an important role in phase estimation1, complex quantum algorithms2,3,4, error correction9 and fault tolerant quantum circuits10. In 2009, the optimal synthesis for a Toffoli gate with six CNOT gates was proposed11. Up to now, for a general three-qubit logic gate, the optimal synthesis requires twenty CNOT gates12, which means that this method increases the difficulty and complexity of experiments and the possibility of errors largely. It is significant to seek a simpler scheme to directly implement multi-qubit gates.

By far, many physical systems have been used to implement quantum logic gates, such as photons in the polarization degree of freedom (DOF)13,14,15 and those in both the polarization and the spatial-mode DOFs (the hyper-parallel photonic quantum computing)16,17,18, nuclear magnetic resonance19,20,57,58,59, generation of entangled states60 and quantum controlled teleportation61. ** position of the atom. The reflection coefficient of a single-photon pulse with the frequency ωp injected into the optical cavity can be obtained by solving the Heisenberg- Langevin equations of motion for the internal cavity field and the atomic operators in the interaction picture62:

Here the one-dimensional field operator ain(t) is the cavity input operator which satisfies the commutation relation . bin(t), with the commutation relation , is the vacuum input field felt by the three-level atom. aout is the output operator. κ and γ are the cavity dam** rate and the atomic decay rate, respectively.

The atom is prepared in the ground states initially. By making κ sufficiently large, one can ensure that the excitation by a single-photon pulse is a weak one and obtain the input-output relation of the cavity field54

Here is the reflection coefficient for the atom-cavity system. When the atom is uncoupled to the cavity or an empty cavity, that is, g = 0, one can obtain62

If the atom is initially prepared in the ground state |0〉, the left-circularly polarized single-photon pulse |L〉 will drive the transition |0〉 ↔ |2〉. The output pulse related to the input one can be expressed as |Φout〉L = r(ωp)|L〉 ≈ eiϕ|L〉. The phase shift ϕ is determined by the parameter values in Eq. (3). However, if the atom is initially prepared in the ground state |1〉, the left-circularly polarized single-photon |L〉 will only sense a bare cavity. As a result, the corresponding output governed by Eq. (4) is , with a phase shift different from ϕ. Considering the parameters of the atom-cavity system satisfy the relationship ω0 = ωc = ωp, the reflection coefficient can be expressed as

Considering a bad cavity κ ≫ g2/κ ≫ γ in the atom-cavity intermediate coupling region, phase shifts ϕ = 0 and ϕ0 = π from Eq. (5) can be produced.

CNOT gate on a two-qubit hybrid system

Our CNOT gate on a two-qubit hybrid system is used to complete a bit-flip on the atom trapped in the cavity when the flying photon is in the left-circular polarization |L〉; otherwise, it does nothing. The schematic setup for our CNOT gate is shown in Fig. 2. We will describe its principle in detail as follows.

Figure 2
figure 2

The schematic setup for a deterministic CNOT gate with a flying polarized photon as the control qubit and an atom trapped in a single-sided optical cavity as the target qubit.

CPBSi (i = 1, 2) is a circularly polarizing beam splitter which transmits the photon in the right-circular polarization |R〉 and reflects the photon in the left-circular polarization |L〉, respectively. M is a mirror. DL is a time-delay device which makes the two wavepackets coming from the paths 2 and 3 interfere with each other.

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Suppose that the initial states of the flying photon p and the atom a trapped in the single-sided cavity are

First, the flying photon is led to the device shown in Fig. 2. The circularly polarizing beam splitter CPBS1 transmits the photon in the right-circular polarization |R〉 to path 1 and reflects the photon in the left-circular polarization |L〉 to path 2. The state of the hybrid system composed of the flying photon p and the atom a is changed from |Ψ〉0 ≡ |ψp〉 ⊗ |ψa〉 to |Ψ〉1. Here

where the subscripts 1 and 2 represent the paths that the flying photon passes through. The subscript a represents the atom trapped in the cavity.

Second, a Hadamard operation is performed on the atom trapped in the cavity before the photon interacts with the atom-cavity system. The Hadamard operation on the atom is used to complete the transformations and . Thus, the state of the hybrid system is changed to be

Third, the photon interacts with the atom trapped in the single-sided cavity and the state of the system becomes

After the interaction between the flying photon and the atom trapped in the cavity, a Hadamard operation is performed on the atom again. At last, the two wavepacks split by CPBS1 reunion at CPBS2 from path 1 and path 3. The state of the system is transformed into

Here |Ψ〉4 is the objective state. One can see that the state of the atom (the target qubit) is flipped when the photon (the control qubit) is in the left-circular polarization |L〉; otherwise, nothing is done on the atom. That is, the schematic setup shown in Fig. 2 can be used to deterministically achieve a quantum CNOT gate on the photon-atom hybrid system by using the flying photon as the control qubit and the atom as the target qubit in principle.

Toffoli gate on a three-qubit hybrid system

Our Toffoli gate on a three-qubit hybrid system is used to complete a bit-flip operation on the atom trapped in cavity2 (the target qubit) when the polarization of the flying photon (the first control qubit) is in the left-circular polarization |L〉 and the atom trapped in cavity1 (the second control qubit) is in the state |1〉 at the same time; otherwise, it does nothing on the atom trapped in cavity2. The schematic setup of our Toffoli gate is shown in Fig. 3. Assume that the initial states of the flying photon qubit and the two atoms trapped in cavity 1 and cavity 2 are prepared in |ϕp〉, |ϕa1〉 and |ϕa2〉, respectively. Here,

Figure 3
figure 3

The schematic setup for constructing a deterministic Toffoli gate with the polarization of a flying photon and an atom trapped in a single-sided cavity (cavity1) as the two control qubits and anther atom trapped in another single-sided cavity (cavity2) as the target qubit.

Hpi(i = 1, 2, 3, 4) is a half-wave plate with the axis at 22.5° and it performs a Hadamard operation on the photon. Xi (i = 1, 2) represents a half-wave plate which performs a bit-flip operation on the photon. Mi (i = 1, 2, 3) is a mirror. cavityi (i = 1, 2) represents the atom-cavity system.

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The principle of our Toffoli gate can be described in detail as follows.

First, the photon is led into our device from the port in. CPBS1 reflects the photon in the left-circular polarization |L〉 to path 1 and transmits the photon in the right-circular polarization |R〉 to path 2. The photon passing through path 2 will not interact with the atoms trapped in cavities. After the photon passes through CPBS1, the state of the system is changed from |Φ〉0 ≡ |ϕp〉 ⊗ |ϕa1〉 ⊗ |ϕa2〉 to |Φ〉1. Here,

Second, a Hadamard operation is performed on the photon in path 1 and CPBS2 transmits the photon in |R〉 to M1 and reflects the photon in |L〉 to cavity 1. Here the Hadamard operation on the photon completes the transformations and . Subsequently, the flying photon interacts with the atom trapped in cavity 1. After the interaction, the two components of the photon reunion at CPBS2. Also, a Hadamard operation and a bit-flip operation σx = |L〉〈R| + |R〉〈L| are performed on the photon in path 3. The state of the whole system becomes

Third, one can perform a Hadamard operation on the atom trapped in cavity 2 and lead the photon in |L〉 reflected by CPBS3 to cavity 2 and the photon in |R〉 transmitted by CPBS3 to M2. The photon in |L〉 and the atom trapped in cavity 2 interact with each other. After the interaction, a Hadamard operation is performed on the atom trapped in cavity 2 again. The two components of the photon reunion at CPBS3. The state of the whole system is changed into

Finally, a bit-flip operation and a Hadamard operation are performed on the photon which emerges in path 4. CPBS4 transmits the photon in |R〉 to M3 and reflects the photon in |L〉 to cavity 1. The photon in |L〉 interacts with the atom trapped in cavity 1 again. After the interaction between the atom-cavity system and the photon, CPBS4 reflects the photon in |L〉 and transmits the photon in |R〉 to path 5. The former is reflected by cavity 1 and the latter is reflected by M3. A Hadamard operation is performed on the photon in path 5. At this time, the two components of the photon from paths 2 and 5 pass through CPBS5 simultaneously and then the photon is led out of our device. The final state of the whole system composed of the flying photon and the two atoms trapped in two cavities separately can be expressed as

From Eq. (15), one can see that the state of the atom trapped in cavity 2 (the target qubit) is flipped only when the photon (the first control qubit) is in the left-circular polarization |L〉 and the atom trapped in cavity 1 (the second control qubit) is in |1〉 at the same time. That is, the schematic setup shown in Fig. 3 can achieve a quantum Toffoli gate on a photon-atom-atom hybrid system by using the flying photon and the atom in cavity 1 as the two control qubits and the atom in cavity 2 as the target qubit in a deterministic way.

Discussion

Reiserer et al.47 exploited the atom-cavity system to complete a robust photon detection scheme experimentally. In their experiment, a single 87Rb atom is trapped at the center of a Fabry-Perot resonator63. Their experiment was completed in the experimental parameters [g, κ, γ]/2π = [6.7, 2.5, 3.0]MHz. In the same experimental parameters, they49 implemented a quantum CNOT gate that a flip of the photon is controlled by an atom trapped in a Fabry-Perot cavity. Tiecke et al.48 realized a scheme in which a single atom switches the phase of a photon and a single photon modifies the atom’s phase. Their experiment was implemented in the parameters [2g, κ, γ]/2π = [(1.09 ± 0.03) GHz, 25 GHz, 6 MHz]. Compared with intermediate coupling strength of the atom-cavity system, it is still challenging to realize the strong coupling strength in experiment. For obtaining shorter operation time, it is significant to realize the atom-cavity photon scattering with a bad cavity in experiment. Turchette et al.51 made a measurement on the conditional phase shifts for quantum logic in the experimental parameters [g, κ, γ]/2π = [20, 75, 2.5]MHz. These parameters satisfy the limit of a bad cavity κ ≫ g2/κ ≫ γ and an intermediate coupling region (g = 0.27κ). Based on these experimental parameters, the average fidelities of our CNOT gate and Toffoli gate are and , respectively. The average efficiencies of our CNOT gate and Toffoli gate are and , respectively. Dayan et al.52 demonstrated an intermediate atom-cavity coupling in experiment. In their experiment, a Cs atom is trapped in a microtoroidal resonator. They gave a set of parameters [g, κ, γ]/2π = [70,(165 ± 15), 2.6]MHz. The probe laser can be swept continuously over a range Δ = ωp − ωc = ±400 MHz and the atom-cavity detuning ω0 − ωc = 0 can be obtained. The parameters in their experiment satisfy the requirements of a bad cavity and an intermediate coupling regime (g = 0.38κ). Based on these experimental parameters, the average fidelities of our CNOT gate and Toffoli gate are and , respectively. The average efficiencies of our CNOT gate and Toffoli gate are and , respectively. The analyses above show that the average fidelities and the averages efficiencies of our two gates can remain high values in the intermediate coupling region with a bad cavity.

In contrary to the CNOT scheme presented by Bonant et al.35, in which a confined electron spin in a QD trapped in a cavity acts as a control qubit and the spin of the photon acts as a target qubit, we use a flying photon as a control qubit and use an atom trapped in an cavity as a target qubit. Our scheme is different from the CNOT scheme proposed by Reiserer et al.49, in which an atom trapped in a cavity acts as a control qubit and the polarization state of the photon acts as a target qubit. In our scheme, the two different polarizations of the photon are split by the CPBS before the photon interacts with the atom-cavity system, which will reduce the difficulty of the experiment. Our scheme is also different from the work by Su et al.59 in which an atom trapped in an cavity acts as the control qubit and an atom trapped in another cavity acts as the target qubit with an auxiliary atom qubit and measurements on the auxiliary qubit and the photon.

In summary, we have proposed two schemes for constructing a deterministic CNOT gate and a deterministic Toffoli gate on photon-atom hybrid systems, respectively, by utilizing the nonlinear interaction between the flying photon and the atom-cavity system and some linear optical elements. For our CNOT gate, the control qubit is encoded on the flying photon and the target qubit is encoded on the atom trapped in the cavity. For our Toffoli gate, the control qubits are encoded on the flying photon and an atom trapped in one cavity and the target qubit is encoded on an atom trapped in another cavity. The quantum circuits of our two gates are very simple. They do not need any auxiliary qubit and measurements to complete the CNOT and Toffoli gates on photon-atom hybrid systems. Our two schemes can work in the atom-cavity intermediate coupling region with bad cavities. The atom-cavity system working in the intermediate coupling region is achieved in experiment51,52. The ratio of coupling strength to dissipation factors affects the fidelities and efficiencies of our gates a little. Our calculations show that even in a worst condition or a reasonable experimental condition, the average of fidelities and the average efficiencies of our two gates can remain high values. What’s more, there exist experimental parameters that satisfy the requirements in this work.

Methods Fidelities and efficiencies of the gates

The nonlinear interaction between the single photon and the atom-cavity system produces a phase shift between the output photon and the input photon. Utilizing this shift and some linear optical elements we construct a CNOT gate and a Toffoli gate on photon-atom and photon-atom-atom hybrid quantum systems, respectively. In the process of constructing these two universal quantum hybrid gates, we set and ϕ = 0. In this ideal case, the hybrid quantum gates are deterministic and the fidelity and the efficiency are 100% for each gate. However, the phase shift is an exact value when g = 0, while the phase shift ϕ = 0 is an approximate value when κ ≫ g2/κ ≫ γ. It is a function of , which is decided by the experimental condition. Considering the realistic condition, we will calculate the fidelities of our quantum gates to show their performance. The fidelity is defined as F = |〈Ψr|Ψi〉|2. Here |Ψr〉 and |Ψi〉 are final states of the hybrid quantum system in our schemes for quantum gates in the realistic condition and the ideal condition, respectively.

The fidelity of our CNOT gate is expressed as

The coefficients of the system can be expressed as αp = cos φ, βp = sin φ, α = cos θ and β = sin θ. The average fidelity of the CNOT gate is

The relationship between the average fidelity of our CNOT gate and on a logarithmic scale is shown in Fig. 4(a) with the solid line. For our Toffoli gate on a three-qubit hybrid system, its fidelity is

Figure 4
figure 4

(a) Average fidelity of our CNOT gate on a two-qubit hybrid system (solid line) and that of our Toffoli gate on a three-qubit hybrid system (dashed line) vs on a logarithmic scale. (b) Average efficiency of our CNOT gate on a two-qubit hybrid system (solid line) and that of our Toffoli gate on a three-qubit hybrid system (dashed line) vs on a logarithmic scale.

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Here ξ1 = |αp|2, , , , and . The coefficients of the system can be expressed as αp = cos φ, βp = sin φ, α1 = cos θ, β1 = sin θ, α2 = cos η and β2 = sin η. The average fidelity of our Toffoli gate is

The dashed line in Fig. 4(a) shows the relationship between the average fidelity of our Toffoli gate and on a logarithmic scale.

The efficiency of a quantum gate is defined as , where nout is the number of the photons coming out of the device and nin is the number of the photons led into the device. The efficiency of our CNOT gate is

The average efficiency of our CNOT gate is

The relationship between the average efficiency of our CNOT gate and on a logarithmic scale is shown in Fig. 4(b) with the solid line.

The efficiency of our Toffoli gate is

We can also calculate the average efficiency of our Toffoli gate

The dashed line in Fig. 4(b) shows the relationship between the average efficiency of our Toffoli gate and on logarithmic scale. From Fig. 4(a,b), one can see that the average fidelities and average efficiencies of these two universal quantum gates are affected by the cooperativity C () of the atom-cavity system. The average fidelities are relatively sensitive to the cooperativity when (0 on a logarithmic scale) and they are faintly affected by cooperativity when (0.17 on a logarithmic scale). If , which is not a difficult experimental requirement, the average fidelities of our CNOT and Toffoli gates can be higher than 0.9949 and 0.9896, respectively. The average efficiencies are relatively sensitive to the cooperativity when (0.3 on a logarithmic scale) and they are faintly affected by the cooperativity when (0.48 on a logarithmic scale). If , which is not a difficult experimental requirement, the average efficiencies of our CNOT and Toffoli gates can be higher than 0.9737 and 0.9609, respectively.

Except for the cooperativity C, some other realistic losses and imperfections, that would affect the fidelities and the efficiencies of our schemes, should be taken into account. The mismatching of spatial mode between cavity and the input photon, the quality of atomic state preparation and rotation will affect both of the fidelities and the efficiencies of our schemes49. The fidelities of our schemes will be also affected by the small probability of more than one photon in the input laser pulses49. The efficiencies can be also affected by the stability of difference between the cavity resonance and the frequency of the input photon and the imperfect absorption losses of the mirror of the cavity49. In our scheme, as only one polarization of a photon is injected to the atom-cavity system and the two polarizations are split by the CPBS, the precise timing of the arrival times from different photon paths is required in the realistic experiment.

Additional Information

How to cite this article: Wang, G.-Y. et al. Universal quantum gates for photon-atom hybrid systems assisted by bad cavities. Sci. Rep. 6, 24183; doi: 10.1038/srep24183 (2016).