Abstract
We propose a deterministic Fredkin gate which can accomplish controlledswap operation between threequbit states. The proposed Fredkin gate consists of a photonic system (single photon) and quantum dots (QDs) confined in singlesided cavities (two electron spin states). In our scheme, the control qubit is the polarization state of the single photon, and two electron spin states in QDs play the role of target qubits (swapped states by control qubit). The interaction between a photon and an electron of QD within the cavity (QDcavity system) significantly affects the performance of Fredkin gate. Thus, through the analysis of the QDcavity system under vacuum noise and sideband leakage, we demonstrate that reliable interaction and performance of the QDcavity system with photonic state (photon) can be acquired in our scheme. Consequently, the Fredkin gate proposed in this paper can be experimentally implemented with high feasibility and efficiency.
Introduction
Quantum controlled operations play critical roles in quantum information processing schemes, such as quantum computation^{1,2,3,4,5,6,7} and quantum communication^{8,9,10,11,12,13,14,15}, to accomplish a reliable performance with high efficiency. One of the universal quantum operations is controlledNOT (CNOT) gate, which can perform the spinflipping of a target qubit with regard to the state of a control qubit, and has been researched theoretically and experimentally^{16,17,18,19,20,21,22}. However, when constructing multiqubit controlled operations (gates) with CNOT gates and single qubit gates, the error probability of controlled operation, or the realization difficulty, increases in multiqubit controlled gates.
Many researchers have proposed to directly design or realize multiqubit controlled gates, such as Fredkin gate^{23,24,25,26}, Toffoli gate^{27,28}, and universal multiqubit gates^{29,30,31,32}. The Fredkin gate, which is a controlledswap gate, can be widely applicable in various quantum information processing schemes (quantum communication^{33,34,35,36} and quantum computation^{37,38,39,40}).
To obtain scheme for quantum information processing with the reliable performance, the most important task is to maintain the coherence of quantum state when to operate the procedure of multiqubit controlled operations. In quantum dot (QD)cavity system, which consists of an injected excess electron and a confined negatively charged exciton (X^{−}) in optical cavity^{2,4,11,12,13,15,41,42,43,44,45,46,47,48,49,50,51,52,53,54}, quantum information can be well stored for a long electronspin coherence time (T_{2}^{e}~μs)^{43,45,46,55,56,57} and a limited spin relaxation period (T_{1}^{e}~ms)^{42,50,58,59} against the influence (decoherence effect) of environment. Therefore, several diverse quantum information processing schemes have been designed between photons and electrons in the QDcavity system, as quantum controlled gates^{21,24,25,26,30,31,47,48,49,52,60}, quantum communications^{10,11,12,13,15,42}, and quantum entanglement^{2,4,44,53,54,61,62,63}.
In this paper, we represent an optical Fredkin gate, which can perform a controlledswap operation, using QDcavity system and linearly optical devices. Because our gate employs the interactions between a photon (control qubit) and two electrons (target qubits) inside the QDcavity systems (QD within a singlesided optical cavity), the long coherence time of target qubits (swapped electron spin states) can be achieved from the storage of quantum information in QD, for the reliable performance of the Fredkin gate. Subsequently, for the reliable Fredkin gate, we demonstrate the high efficiency and reliable performance of the interaction of photonelectron in QD under the vacuum noise in the QDdipole operation, and leaky modes (sideband leakage and absorption) in the cavity mode through our analysis^{51,54,64,65,66}. Consequently, the proposed Fredkin gate using QDcavity systems has the feasibility and the reliable performance to experimentally realize the controlledswap gate under the vacuum noise and leaky modes (sideband leakage and absorption).
Optical Fredkin Gate via Quantum dot within a Singlesided Optical Cavity
Interaction of photon and QDcavity system
The interaction of a QDcavity system consists of a singly charged QD confined in a singlesided optical cavity^{2,4,11,12,13,15,41,42,43,44,45,46,47,48,49,50,51,52,53,54}. In Fig. 1(a), a QDcavity system is schematically represented with two GaAs/Al(Ga)As distributed Bragg reflectors (DBRs). The bottom DBR is partially reflective and the top DBR is 100% reflective (singlesided cavity). \({\hat{b}}_{{\rm{i}}{\rm{n}}}\) and \({\hat{b}}_{{\rm{o}}{\rm{u}}{\rm{t}}}\) are the input and output field operators, respectively. QD is confined at the center of the singlesided cavity where κ_{s} (the sideleakage rate of the cavity mode) and γ (the decay rate of a negatively charged exciton, X^{−}, which consists of two electrons bound to one hole. Figure 1(b) shows the interaction of QDcavity system between an incident photon (described as \({\hat{b}}_{{\rm{i}}{\rm{n}}}\)) and an excess electron injected into the QD, following the Pauli exclusion principle. If the spin state of the excess electron in the QD is in the state ↑〉(↓〉), the polarization \(L\rangle \,(R\rangle )\) of a photon can drive the state ↑↓⇑〉 (↑↓⇓) of X^{−}. In the approximation of weak excitation, after the interaction of the incident photon and QDcavity system, we can obtain the reflection coefficient, R(ω), of the photon and the QDcavity system from the Heisenberg equation of motion^{67}, with the ground state in the QD (〈\({\hat{\sigma }}_{z}\)〉 = −1) for the steady state, due to the spin selection rule, as follows:
where \({\omega }_{{{\rm{X}}}^{}}\), ω_{c}, and ω are the frequencies of X^{−}, cavity mode, and external field, respectively. Additionally, κ and g are the decay rate of the cavity mode and the coupling strength between X^{−} and cavity mode, assuming the resonant interaction, ω_{c} = \({\omega }_{{{\rm{X}}}^{}}\). When the spin state of the excess electron is in the state \(\,\uparrow \,\rangle \,(\,\downarrow \,\rangle )\), the polarization of the photon \(L\rangle \,(R\rangle )\) drive the hot cavity, according to the coupled QD with the cavity, while the photon having the polarization, \(R\rangle \,(L\rangle )\), feels the cold cavity (the QD is uncoupled to the cavity). In this case, the reflection coefficients, R_{h} (hot cavity) and R_{0} (cold cavity), are given by:
where r_{h} (r_{0}) and \({\varphi }_{{\rm{rh}}}\equiv \text{arg}[{R}_{{\rm{h}}}]\,({\varphi }_{{\rm{r0}}}\equiv \text{arg}[{R}_{0}])\) are the reflectance and the phase shift of the hot (cold) cavity. Thus, the reflection operator \(\hat{{\rm{R}}}({\rm{\omega }})\) is expressed as:
In the QDcavity system, when we take the experimental parameters as a small sideleakage rate, κ_{s} (κ_{s} << κ), and the strong coupling strength g >> (κ, γ) with small γ (~ several μ eV)^{68,69,70,71}, the reflection operator can be given as:
where the reflectances and phase shifts are r_{h}(ω) = r_{0}(ω)≈1 and φ_{rh}(ω)≈0, φ_{r0}(ω)≈π for g/κ = 2.4, and γ/κ = 0.1 with κ_{s} → 0 and ω = ω_{c} (frequencies: external field = cavity mode)^{2,4,11,12,13,15,41,42,43,44,45,46,47,48,49,50,51,52,53,54}. According to the reflection operator, Eq. 4, the result of the interaction between the photon and the QDcavity is expressed as:
Theoretical circuit of Fredkin gate
First, we introduce the theoretical Fredkin (controlledswap operation) gate^{23,24,25,26}. In Fig. 2, the Fredkin (theoretical) gate can perform a controlledswap operation between three qubits of a control qubit, c, and two target qubits, 1 and 2 (which are swapped by the state of control qubit). The operation of Fredkin gate is given by:
where \(\alpha {0\rangle }_{{\rm{c}}}+\beta {1\rangle }_{{\rm{c}}}\) is the control qubit with α^{2} + β^{2} = 1. Additionally, \({\varphi \rangle }_{1}\) and \({\psi \rangle }_{2}\) are arbitrary quantum states (target qubits). This result means that if the state of control qubit is in the \({1\rangle }_{{\rm{c}}}\), two target states 1 and 2 are exchanged for each other (controlledswap). In Fig. 2, the Fredkin gate (theoretical circuit) can be implemented the multiqubit (two or three) controlled gates and the single qubit gates for the experimental realization. If we attempt to design Fredkin gate using CNOTs and Toffoli gates, we can reduce the number of controlled operations needed (the middle circuit in Fig. 2). However, it is challengeable to directly implement Toffoli gate (threequbit controlled gate) in practice. Thus, in this point of view, the methods of using only twoqubit controlled gates for the Fredkin gate have the experimental advantage, despite the number of controlled gates (or operations) increase (the end of right side in Fig. 2). For example, in Fig. 2, though the middle circuit has three controlled operations, the task to realize Toffoli gate is experimentally difficult. Meanwhile, in the circuit of right side, the number of controlled operations needed is eight interactions. And also this circuit can guarantee the feasibility for the implementation, due to consisting of the only twoqubit controlled gates. Therefore, we will propose the method of utilizing twoqubit (a photon and an electron in QD: QDcavity system) controlled gates to design an optical Fredkin gate with the experimental feasibility and reducing the number of controlled operations needed.
Optical Fredkin gate using QDcavity systems
Figure 3 shows an optical scheme, the Fredkin gate, to perform the controlledswap operation as Eq. 6. The proposed Fredkin gate consists of a swap gate (SG), which has two QDcavity systems (QD 1 and 2), and linearly optical devices, as described in Fig. 3. Let us assume the initial (input) state, \({{{\boldsymbol{\Psi }}}_{{\rm{in}}}\rangle }_{{\rm{P}}12}\), of a Fredkin gate, as follows:
where α_{i}^{2} + β_{i}^{2} = 1. The control qubit is the flying photon P, and two target qubits are two electron spin states 1 and 2, which are stationary states confined within the QDcavity systems (QD 1 and 2), respectively. We also define the relations between the polarizations of photon and electron spin states, as:
where the linear polarization, \(\{H\rangle ,\,V\rangle \}\), and the circular polarization, \(\{R\rangle ,\,L\rangle \}\) of photon. As described in Fig. 3, after the state, \({{{\boldsymbol{\Psi }}}_{{\rm{in}}}\rangle }_{{\rm{P}}12}\), passes through PBS1, the state will be changed to:
where the path of the photon P is split into two paths, a and b. Subsequently, the photon state on the path b of the state \({{{\boldsymbol{\Psi }}}_{1}\rangle }_{{\rm{P}}12}\) is applied to SG, while the state on path a is delayed by DL, in Fig. 3. In SG, the state on path b will have three interactions (three times) with the QDcavity systems (QD 1 and 2).
[First cycle: time Table (1) {t = 0~t_{1}}]
Figure 4 represents the interaction between the photon state and two electron spin states (1 and 2) in the first cycle. DL1 and the switches (1, 2, 3, 4, and 5) are operated according to the time Table (1)), for the interval from t = 0 to t_{1}. After the interaction of two QDcavity systems (QD 1 and 2) with the photon, the state, \({{{\boldsymbol{\Psi }}}_{1}\rangle }_{{\rm{P}}12}\), is transformed to:
where the interaction between the photon and the QDcavity system is described from the reflection operator, \(\hat{{\rm{R}}}\), in Eqs. 4 and 5. Afterward, the pulses (microwave pulse or optical pulse in Fig. 3) are applied to the two electrons, 1 and 2, of QD1 and QD2 in the state \({{{\boldsymbol{\Psi }}}_{2}\rangle }_{{\rm{P}}12}\). Subsequently, we can obtain the output state, \({{{\boldsymbol{\Psi }}}_{3}\rangle }_{{\rm{P}}12}\) (t = t_{1}), of the first cycle in SG after the state \({{{\boldsymbol{\Psi }}}_{2}\rangle }_{{\rm{P}}12}\) passes through CPBS2 and QWP1, as follows:
[Second cycle: time table (2) {t = t_{1}~t_{2}}]
Figure 5 represents the interaction between the photon state on path d2 and two electron spin states (1 and 2) in the second cycle. The DLs and switches (1, 2, 3, 4, 5, 6, and 7) are operated according to the time table (2), for the interval from t = t_{1} to t_{2}. After the state, \({{{\boldsymbol{\Psi }}}_{3}\rangle }_{{\rm{P}}12}\), passes through CPBS3, the state will be changed to:
Afterward, the interactions, due to the reflection operator \(\hat{{\rm{R}}}\) (Eqs. 4 and 5), of two QDcavity systems (QD 1 and 2) are sequentially performed to the state, \({{{\boldsymbol{\Psi }}}_{4}\rangle }_{{\rm{P}}12}\), as follows:
where two electron spin states, which are linked with the photon state on path c, are exchanged (swapped), compared with the state \({{{\boldsymbol{\Psi }}}_{4}\rangle }_{{\rm{P}}12}\) in Eq. 12. Then, the pulses (microwave pulse or optical pulse in Fig. 3) are applied to the two electrons, 1 and 2, of QD1 and QD2 in the state \({{{\boldsymbol{\Psi }}}_{5}\rangle }_{{\rm{P}}12}\). After the state, \({{{\boldsymbol{\Psi }}}_{5}\rangle }_{{\rm{P}}12}\), passes through CPBS2 and QWP2, we can obtain the output state, \({{{\boldsymbol{\Psi }}}_{6}\rangle }_{{\rm{P}}12}\) (t = t_{2}), of the second cycle in SG, as follows:
[Third cycle: time table (3) {t = t_{2}~t_{3}}]
In Fig. 6, the interactions between the photon and two QDcavity systems (QD 1 and 2) are operated in the third (final) cycle for time (t_{2}~t_{3}). The procedure of DL and switches (1, 2, 4, 5, 6, and 7) is performed in accordance with the time table (3). The output state, \({{{\boldsymbol{\Psi }}}_{6}\rangle }_{{\rm{P}}12}\), from the second cycle is transformed after passing through CPBS3, as follows:
Then, the photon of the state \({{{\boldsymbol{\Psi }}}_{7}\rangle }_{{\rm{P}}12}\) interacts with two QDcavity systems (QD 1 and 2), according to Eqs. 4 and 5, as follows:
Finally, after the state \({{{\boldsymbol{\Psi }}}_{8}\rangle }_{{\rm{P}}12}\) passes through CPBS2, S6, and SF, we can obtain the output state, \({{{\boldsymbol{\Psi }}}_{9}\rangle }_{{\rm{P}}12}\) (t = t_{3}), of the third (final) cycle in SG, as follows:
Here, the state \({{{\boldsymbol{\Psi }}}_{9}\rangle }_{{\rm{P}}12}\) from SG (through the cycles, three times) shows that two electron spin states in QD 1 and QD 2 are swapped with each other, according to the polarization of photon P (\({V\rangle }_{{\rm{P}}}\) is to swap two electron spin states). Consequently, after the state, \({{{\boldsymbol{\Psi }}}_{9}\rangle }_{{\rm{P}}12}\), passes PBS2 in Fig. 3, we can show that our optical Fredkin gate in Fig. 3 can generate the operation of controlledswap through the final state, \({{{\boldsymbol{\Psi }}}_{{\rm{out}}}\rangle }_{{\rm{P}}12}\), as follows:
Compared with the input state, \({{{\boldsymbol{\Psi }}}_{{\rm{in}}}\rangle }_{{\rm{P}}12}\) in Eq. 7, this result, \({{{\boldsymbol{\Psi }}}_{{\rm{out}}}\rangle }_{{\rm{P}}12}\) (output state), means that the controlledswap operation, which performs the exchange of two states of target qubits (electron spin states 1 and 2) in the case of \({V\rangle }_{{\rm{P}}}\) (control qubit: photon’s polarization), is conducted by the proposed Fredkin gate by utilizing the QDcavity systems.
So far, we have proposed an optical Fredkin gate, consisted of the QDcavity systems, to feasibly exploit the controlledswap gate for the high efficiency and the reliable performance. In our scheme, two electrons 1 and 2 confined within QDcavity systems (QD1 and QD2) play the roles of the target qubits. When a flying photon P (control qubit) interacts with the electrons respectively, the result of this interaction is identical with a controlledphase operation (twoqubit controlled gate), due to the reflection operator \(\hat{{\rm{R}}}(\omega )\) in Eq. 3. Thus, from each cycle of the three cycles in Figs. 4–6, we can calculate the total number of twoqubit controlled operations, as six interactions (a cycle has two interactions, as described in Figs. 4–6). Compared with the Fredkin gate having eight CNOT (twoqubit controlled) gates in Fig. 2, the proposed scheme in Fig. 3 can reduce the number of twoqubit controlled operations needed for the feasibility. Furthermore, in our Fredkin gate, swap gate (SG) having two QDcavity systems (two target qubits) plays the role of main part to store quantum information with coherence and to optically interact the photon with the electron in QD. Thus, for the practical usage of Fredkin gate, the quantification for the interaction of the QDcavity system (QD within a singlesided cavity) should be required by the analysis about the affections of the vacuum noise, sideband leakage, and absorption.
Analysis of the Interaction of QD within Singlesided Cavity under Noise
In an optical Fredkin gate performing controlledswap operation, the significant element is the QDcavity system, which can interact with a flying photon and an electron confined within QD, inducing a difference in reflectances (r_{h}, r_{0}) and phase shifts (φ_{rh}, φ_{r0}) from Eqs. 2–4, according to the hot (coupled to) cavity and cold (uncoupled to) cavity. Figure 7 shows the reflectances and phase shifts of a reflected photon, which feels hot (g ≠ 0) and cold (g = 0) cavity, for frequency detuning, 2(ω − ω_{c})/κ, and the different sideleakage rates (κ_{s}/κ = 0.0, 0.5, and 1.0) with fixed values of coupling strength, g/κ = 2.4, and decay rate, γ/κ = 0.1, of X^{−}, from Eq. 2. When κ_{s} is negligible and ω = ω_{c} (the frequency, ω, of the external field, photon, can be adjusted to the identical frequency, ω_{c}, of cavity mode), the values of reflectance and phase shift can be acquired, as r_{h}(ω) = r_{0}(ω) ≈ 1, φ_{rh}(ω) = 0, and φ_{r0}(ω) = π, as described in Fig. 7. The reflection operator, \(\hat{{\rm{R}}}\), which is utilized in our Fredkin gate, can be given as Eq. 4, referring to the interaction, Eq. 5, of QDcavity system.
In practice, for the high efficiency and reliable performance of the QDcavity system, which can induce the interaction an input photon and an confined electron inside cavity, the affections of the vacuum noise, N(ω), of the QDdipole and leaky modes, S(ω) (sideband leakage and absorption) should be quantified^{51,54,64,65,66}. For the reflection coefficient, \(\hat{{\rm{R}}}(\omega )\), with the noise, N(ω), and leakage, S(ω), coefficients, a solution of the Heisenberg equation of motion for a cavity field operator, â, a dipole operator, σ̂^{−}, of X^{−}, and the inputoutput relations^{67}, is given by:
where \({\hat{S}}_{in}\) (\({\hat{S}}_{out}\)) is an input (output) field operator from leaky modes, due to sideband leakage and absorption in the cavity mode, and \(\hat{N}\) is the vacuum noise operator for σ̂^{−}. The output field operator, \({\hat{b}}_{out}\), of reflected photon, will be given by \({\hat{b}}_{{\rm{o}}{\rm{u}}{\rm{t}}}\) = R(ω)\({\hat{b}}_{{\rm{i}}{\rm{n}}}\) + S(ω)\({\hat{S}}_{{\rm{i}}{\rm{n}}}\) + N(ω)\(\hat{S}\) from Eq. 19 in the weak excitation approximation^{50,53}, with ω_{c} = \({\omega }_{{{\rm{X}}}^{}}\) and 〈σ̂_{Z}〉≈−1 (ground state in QD). The reflection, R(ω), noise, N(ω), and leakage, S(ω), coefficients can be expressed, according to a hot and cold cavity, as
where the reflectances (hot: R_{h} and cold: R_{0}) of R(ω), are written in Eq. 2. n_{h} (n_{0}), and φ_{nh} (φ_{n0}) are noise rate and phase shift, corresponding vacuum noise the of QDdipole, in hot (cold) cavity. Additionally, s_{h} (s_{0}) and φ_{sh} (φ_{s0}) are leakage rate and phase shift, respectively, due to the sideband leakage and absorption for cavity mode, in a hot (cold) cavity. Thus, we can revise the reflection operator, \(\hat{{\rm{R}}}(\omega )\) in Eq. 4, to practical reflection operator, \(\hat{{\rm{R}}}\)_{P}(ω), including noise, N(ω), and leakage, S(ω), coefficients, as follows:
where R_{h}(ω) and R_{0}(ω) are in Eq. 2, and \({N}_{0}(\omega )\equiv {n}_{0}(\omega )\exp [i{\varphi }_{{\rm{n0}}}(\omega )]=0\) is from Eq. 20.
Figure 8 shows a graph of noise rate, phase shift (blue, magenta, and brown), and leakage rate, phase shift (yellow, green, and gray), from Eq. 20, for frequency detuning, 2(ω−ω_{c})/κ, and the different sideleakage rates (κ_{s}/κ = 0.0, 0.5, and 1.0), when experimental parameters are strength coupling, g/κ = 2.4, and decay rate, γ/κ = 0.1, of X^{−} with ω_{c} = \({\omega }_{{{\rm{X}}}^{}}\). When κ_{s} is negligible (κ_{s} → 0) and 2(ω − ω_{c})/κ = 0, we can obtain the values of noise and leakage rates, phase shifts as n_{h} → 0, n_{0} = s_{h} = s_{0} = 0, and φ_{nh} → 0, φ_{n0} = φ_{sh} = φ_{s0} = 0 in QDcavity system, as described in Fig. 8. This result means that we can ignore the effects of the vacuum noise, N(ω), of QDdipole operation and leakage (sideband leakage and absorption), S(ω), for high efficiency and reliable performance of QDcavity system by choosing the parameters g/κ = 2.4 (strong coupling strength) and κ_{s}≈0 (small sideleakage rate) with γ/κ = 0.1.
Therefore, we can analyze the efficiency and performance of a QDcavity system to calculate the value of the average of fidelity (AoF) between two output states from the reflection operators \(\hat{{\rm{R}}}(\omega )\) (Eq. 4: ideal case) and \({\hat{{\rm{R}}}}_{{\rm{P}}}(\omega )\) (Eq. 21: practical case). For example, let us assume an arbitrary input state (photonelectron) as \((\cos \,\theta R\rangle +\,\sin \,\theta L\rangle )\otimes (\cos \,\vartheta \,\uparrow \,\rangle +\,\sin \,\vartheta \,\downarrow \,\rangle )\) where cos^{2}θ + sin^{2}θ = cos^{2}ϑ + sin^{2}ϑ = 1.
After the interaction of QDcavity system, we can show two kinds (\({\phi }_{{\rm{Id}}}\rangle \): ideal case, and \({\phi }_{{\rm{\Pr }}}\rangle \): practical case) of output states from the reflection operators (\(\hat{{\rm{R}}}\) and \({\hat{{\rm{R}}}}_{{\rm{p}}}\)) in Eqs. 4 and 21, as follows:
where N = R_{h} + N_{h} + S_{h}^{2}(cos ^{2}θ sin ^{2}ϑ + sin ^{2}θ cos ^{2}ϑ) + R_{0} + S_{0}^{2}(cos^{2}θ cos^{2}ϑ + sin^{2}θ sin^{2}ϑ). Then, the AoF between two output states (\({\phi }_{{\rm{Id}}}\rangle \) and \({\phi }_{{\rm{\Pr }}}\rangle \)) of QDcavity system is given by:
In Fig. 9, the values of the AoF are listed for the difference in κ_{s}/κ (g/κ), with fixed g/κ = 2.4 (κ_{s}/κ = 2.0) in γ/κ = 0.1 and ω = ω_{c}. The values of AoF approach 1 if the coupling strength is strong, g >> (κ, γ), and the sideleakage rate is small, κ_{s} << κ, as shown in Fig. 9. On the other hands, in the small range (0.1 ≤ g/κ ≤ 0.3 and 3.0 ≤ κ_{s}/κ ≤ 4.5), we can acquire the high AoF (i.e., g/κ = 0.1 and κ_{s}/κ = 4.5 → AoF ≈ 0.998). This means that the value of AoF can increase with small (weak) coupling strength and large sideleakage rate. However, when we experimentally implement the QDcavity system for our scheme, it’s quite challengeable to maintain the values of parameters into the small range for the operation of the QDcavity system. If the values of parameters deviate (less or over) from the small range, the value of AoF rapidly decreases, as described in Fig. 9, because of the tiny area of range. Rather, for the experimental realization, to improve the conditions (tendencies: strong coupling strength and small sideleakage) is more advantageous than to maintain the values of parameters fixed in the small range, in practice. Therefore, we can conclude that high efficiency and performance (according to the values of AoF) can be acquired by increase in the coupling strength, and also, with decrease in the sideleakage rate under the vacuum noise, N(ω), for operation of the QDdipole and leaky modes, S(ω) (sideband leakage and absorption)^{51,54,64,65,66}. Also, our optical Fredkin gate is utilizing the interactions of QDcavity systems (QD1 and QD2) with high efficiency and performance when to increase the coupling strength, g >> (κ, γ), and decrease the sideleakage rate, κ_{s} << κ. If the conditions of QDcavity systems for high values of AoF (approaching to 1) are maintained in our scheme, we can acquire the efficient controlledswap operation from the interactions between a photon (control) and QDs (QD1 and QD2: targets) through the three cycles, as described in Figs. 4–6.
Moreover, for the high efficiency and performance of QDcavity system in our scheme, the requirements of the reflection operator and interactions should be \(\hat{{\rm{R}}}(\omega )\) in Eq. 4, and as Eq. 5. As shown In Table 1, we can calculate the values of the reflection, R_{h}(R_{0}), noise, N_{h}(N_{0}), and leakage, S_{h}(S_{0}), coefficients in hot (cold) cavity from Eqs. 2 and 20, according to the experimental parameters (g/κ and κ_{s}/κ) with γ/κ = 0.1 and ω − ω_{c} = 0. Through the values in Table 1, we can compare with the reflection operators having high coupling strength and small leakage or not (i.e. g/κ = 2.4, κ_{s}/κ = 0.01 or g/κ = 0.01, κ_{s}/κ = 2.0), as follows:
where the practical reflection operator, \({\hat{{\rm{R}}}}_{{\rm{P}}(\omega )}\), including noise, N(ω), and leakage, S(ω), coefficients is given by Eq. 21. These obviously mean that the practical reflection operator, \({\hat{{\rm{R}}}}_{1}\), is closer to the ideal reflection operator, \(\hat{{\rm{R}}}\), in Eq. 4 when the coupling strength is strong, g >> (κ, γ), and the sideleakage rate is small, κ_{s} << κ, as shown in Fig. 9, Table 1, and Eq. 24, under the vacuum noise \(\hat{N}\) in dipole operation and sideband leakage, absorption \(\hat{S}\) in cavity.
Consequently, by our analysis, which is to quantify the efficiency and the reliable performance of the QDcavity system via the reflection operator \({\hat{{\rm{R}}}}_{{\rm{P}}(\omega )}\) in Eq. 21, we demonstrate that the optical Fredkin (controlledswap) gate using QDcavity systems can be experimentally realized with feasibility.
Conclusions
In this paper, we have proposed an optical scheme of Fredkin gate, which can realize controlledswap gate for the interaction with a control qubit (photon) and two target qubits (electrons), using the QDcavity systems. For the Fredkin gate, we also analyzed (quantified) the efficiency and performance (AoF: average of fidelity) of the QDcavity system by calculating fidelity under the vacuum noise, N(ω), in the QDdipole operation and leaky modes, S(ω) (sideband leakage and absorption)^{51,54,64,65,66}, as described in Sec. 3. According to our analysis, when utilizing large coupling strength g >> (κ, γ) and low sideleakage rate κ_{s} << κ, we obtained high fidelity, F, of the output state from the QDcavity system, due to the reduced (negligible) effect of the vacuum noise N(ω) and leaky modes S(ω).
Also, the methods have been proposed to implement Fredkin gate^{23,24,25,26}. Milburn^{23} designed the optical Fredkin gate utilizing linearly optical devices under particular operating conditions (nondissipative and errorfree). But this scheme cannot be operated when to realize Fredkin gate in practice (dissipative and error). Afterward, many researchers have employed the cavity system (cavity QED) to compose Fredkin gate. In 2017, for the controlledswap operation between a photon and atoms (hybrid system), Song et al.^{24} exploited (giant) Faraday rotation, which can be occurred from the difference in phase shift, according to the polarization of input state in hot or cold cavity, for the Fredkin gate. However, In practice, when to implement quantum information processing schemes using cavity system, various conditions, which can affect to diminish the performance, should be considered. Thus, they, in ref. ^{24}, overlooked the affections of sideleakage rate (κ_{s}), vacuum noise in QDdipole operation, and leaky mode (sideband leakage and absorption) in the cavity mode. In ref. ^{25} a control qubit and target qubits correspond to a single superconducting flux and two resonator or nitrogenvacancy (NV) centers in a hybrid Fredkin gate with quantum memories. Here, for the feasibility, our scheme utilized a single flying photon, as control qubit, because the photons can be used to best carriers for fast and reliable processing. Also, in ref. ^{26}, the roles of NV centers are restricted to only the ancillary systems for the controlledswap operation between three photons. Whereas, in our scheme, two electrons in QDcavity systems (QD1 and QD2: target qubits) for the acquisition^{43,45,46,55,56,57} of the coherence of quantum state, when to perform the procedure of controlled operations. Thus, our optical scheme of Fredkin gate has the advantages (feasibility and efficiency), as the above mentions, compared with the previous works (the implementations of Fredkin gate)^{23,24,25,26}.
In the point of the experimental implementation, to obtain such experimental condition for high fidelity, we reviewed previous related studies. In a micropillar cavity having diameter of 1.5 μm and quality factor as Q = 8800^{68}, the coupling strength was achieved as g/(κ + κ_{s}) ≈ 0.5, and also, could be increased to g/(κ + κ_{s})≈ 2.4 for Q = 40000^{72}. Moreover, in ref. ^{73}, by the technics (etching process or improving the sample growth), the authors could improve the experimental quality of optical cavity (In_{0.6}Ga_{0.4}As) as g/(κ + κ_{s}) ≈ 2.4 and Q = 40000 by the decreasing rate of sideleakage. For small sideleakage rate, the method has been proposed to improve quality factor Q = 40000 (κ ≈ 6.2 μeV)^{74}. Thus, QDcavity systems in our scheme (optical Fredkin gate) can acquire the high efficiency and reliable performance (high fidelity), according to our analysis, in Sec. 3. Furthermore, the QDcavity systems have provided the initialization, manipulation and measurement of the spin state for the interaction between an input photon (photonic spin) and a confined electron (electron spin) in QD, in^{70}. Thus, for the preparation of electron spinsuperposition state (arbitrary quantum state) in Fredkin gate, we can employ the methods as optical pumping and optical cooling^{75}. And, the single spin of QD (the preparation of the excess electron spin state and operation on the spin) can be manipulated by using pulsed magnetic resonance techniques, nanosecond microwave pulses, or picosecond/femtosecond optical pulses^{76,77,78,79}.
Consequently, based on our analysis, the QDcavity system having the high efficiency and reliable performance can be operated in practice (under vacuum noise and sideband leakage). We have also demonstrated that controlledswap operation of our Fredkin gate can be experimentally feasible by using the QDcavity systems.
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Acknowledgements
This work was supported by the NRF programs (2019R1A2C2006381, 2019M3E4A107866011, 2019M3E4A1079777), and the KIST research program (2E30620), and Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (No. 2019R1I1A1A01042699).
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M.S. Kang, and J. Heo conceived the main idea. M.S. Kang, and J. Heo wrote the manuscript. M.S. Kang, and J. Heo calculated the main calculations. J. Heo, S.G. Choi, S. Moon, and S.W. Han analyzed the results. J. Heo supervised the entire project. All authors reviewed the manuscript.
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Kang, MS., Heo, J., Choi, SG. et al. Optical Fredkin gate assisted by quantum dot within optical cavity under vacuum noise and sideband leakage. Sci Rep 10, 5123 (2020). https://doi.org/10.1038/s41598020619388
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DOI: https://doi.org/10.1038/s41598020619388
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