Introduction

In various quantum algorithms and quantum computations, such as quantum phase estimation algorithm1,2,3, the factoring problem4,5,6,7, the discrete logarithm problem2,4,8,9, and the hidden subgroup problem10,11,12, a discrete quantum Fourier transform (DQFT)2,13,14,15,16,17,18,19 plays a critical role in accomplishing quantum information processing. Thus, for the experimental implementation of DQFT, a variety of physical resources have been used, including those based on linear optical systems20,21,22, nonlinear optical systems17,23,24,25, nuclear magnetic resonance or ion trap systems26,27,28,29, superconducting circuits30, and cavity-QED15,31,32,33.

In addition, for quantum information processing schemes, many researchers have theoretically proposed and experimentally designed quantum controlled gates34,35,36,37,38,39,40,41,42,43, which can apply an arbitrary operation to a target qubit according to a control qubit, via nonlinearly optical resources. In addition, in the DQFT scheme, the reliable quantum controlled operations are crucial components for transforming an input state of qubit into the linearly combined states of qubit. In particular, preserving the coherence against the decoherence effect and the extension of coherence time in quantum state are the most important challenges for the reliable performance of quantum controlled operation with the reliable performance.

From this point of view, for the coherence of quantum system, optical systems of micropillar cavities have been widely used to construct quantum controlled gates38,39,40,41,42. In particular, quantum information in the quantum dot (QD)-cavity system, which consists of an excess electron and a negatively charged exciton (X−) confined within an optical cavity40,42,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66, can be well isolated from the environment for a long electron-spin coherence time (\({{\rm{T}}}_{{\rm{2}}}^{{\rm{e}}}\)~μs)57,58,59,60,61,62 as well as a limited spin relaxation period (\({{\rm{T}}}_{{\rm{1}}}^{{\rm{e}}}\)~ms)63,64,65,66. Therefore, quantum controlled gates38,39,41,48,51,67,68,69 have been proposed via the QD-cavity system between photon-photon, electron-electron, and electron-photon.

In this paper, we design an optical controlled-rotation k (CRk) gate based on the interactions between two photons and two QD-cavity systems, as well as linearly optical devices. The proposed CRk gate can be directly applicable to comprise a scheme of DQFT for quantum computation and quantum algorithm. According to the expansion to arrange the CRk gates and the alternation of rotation operators (k), we can achieve scalability of the DQFT scheme, because our CRk gate using the QD-cavity systems serves as the basic module of the multi-qubit DQFT scheme. Subsequently, for the deterministic DQFT scheme, we analyze the efficiency and performance of the CRk gate using the interaction of photon-electron in QD under vacuum noise in the QD-dipole operation, and leaky modes (sideband leakage and absorption)40,42,53,54,55,56. Consequently, our DQFT scheme via the CRk gates shows scalability and experimental feasibility in practice.

Theoretical Circuit of Discrete Quantum Fourier Transform

The arbitrary quantum state, |ϕ〉, can be transformed by the operation of DQFT2,13,14,15,16,17,18,19, as follows:

$$\begin{array}{ccc}|\varphi \rangle =\mathop{\sum }\limits_{m=0}^{{2}^{t}-1}{\alpha }_{m}|m\rangle & \mathop{\to }\limits^{{\rm{DQFT}}} & \mathop{\sum }\limits_{f=0}^{{2}^{t}-1}{\beta }_{f}|f\rangle \end{array},$$
(1)

where \({\beta }_{f}=[{\sum }_{m=0}^{{2}^{t}-1}{\alpha }_{m}{e}^{2\pi i(mf)/{2}^{t}}]/\sqrt{{2}^{t}}\). Thus, DQFT on an orthonormal basis, |0〉, |1〉, …, |2t − 1〉 is defined as an operator, UDQFT, as follows:

$${{\rm{U}}}_{{\rm{DQFT}}}|m\rangle =\frac{1}{\sqrt{{2}^{t}}}\mathop{\sum }\limits_{f=0}^{{2}^{t}-1}{e}^{2\pi i(mf)/{2}^{t}}|f\rangle ,$$
(2)

where the |0〉, |1〉, …, |2t − 1〉 is the orthonormal basis consisting of the quantum state on t qubits. Using binary representation, state |m〉 can be written as |m〉 ↔ |j1〉 ⊗ … ⊗ |jt〉 where m ≡ j1 × 2t−1 + j2 × 2t−2 + \(\cdots \) + jt × 20 and jn ∈ {0, 1}. Thus, the product representation of DQFT on qubits can be given by

$$|m\rangle \leftrightarrow |{j}_{1}\rangle \otimes \cdots \otimes |{j}_{t}\rangle \mathop{\Rightarrow }\limits^{{\rm{D}}{\rm{Q}}{\rm{F}}{\rm{T}}}\Rightarrow \,\frac{1}{\sqrt{{2}^{t}}}(|0{\rangle }_{1}+{e}^{2\pi i(0.{j}_{t})}|1{\rangle }_{1})\otimes (|0{\rangle }_{2}+{e}^{2\pi i(0.{j}_{t-1}{j}_{t})}|1{\rangle }_{2})\otimes \cdots \otimes (|0{\rangle }_{t}+{e}^{2\pi i(0.{j}_{1}{j}_{2}\cdots \cdot {j}_{t})}|1{\rangle }_{t}),$$
(3)

where the notation, 0.j1j2 \(\cdots \) jt−1jt, is the binary fraction as 0.j1j2 \(\cdots \) jt ≡ j1 × 2−1 + j2 × 2−2 + \(\cdots \) + jt × 2−t.

Figure 1 shows the quantum circuit of DQFT to transform the input state, \(|{j}_{1}\rangle \otimes |{j}_{2}\rangle \otimes \cdots \otimes |{j}_{t}\rangle \), into the product representation of DQFT on t qubits. As shown in Fig. 1, the significant important components in a circuit of DQFT are the CRk operations, UCRk, between two qubits to transform the input state. For example, let us assume an initial state of two qubits as \({|\varphi \rangle }_{{\rm{in}}}={x}_{1}{|0\rangle }_{{\rm{A}}}{|0\rangle }_{{\rm{B}}}+{x}_{2}{|0\rangle }_{{\rm{A}}}{|1\rangle }_{{\rm{B}}}+{x}_{3}{|1\rangle }_{{\rm{A}}}{|0\rangle }_{{\rm{B}}}+{x}_{4}{|1\rangle }_{{\rm{A}}}{|1\rangle }_{{\rm{B}}}\). As described in Fig. 1, the CRk operation (UCRk) can transform the initial state, |ϕ〉in, into a final state, |ϕ〉fin, as follows:

$$\begin{array}{ccc}{|\varphi \rangle }_{{\rm{in}}} & \mathop{\Rightarrow }\limits^{{\rm{CRk}}({{\rm{U}}}_{{\rm{CRk}}})} & {|\varphi \rangle }_{{\rm{fin}}}={x}_{1}{|0\rangle }_{{\rm{B}}}{|0\rangle }_{{\rm{A}}}+{x}_{2}{|1\rangle }_{{\rm{B}}}{|0\rangle }_{{\rm{A}}}+{x}_{3}{|0\rangle }_{{\rm{B}}}{|1\rangle }_{{\rm{A}}}+{x}_{4}{e}^{2\pi i/{2}^{k}}{|1\rangle }_{{\rm{B}}}{|1\rangle }_{{\rm{A}}},\end{array}$$
(4)

where the paths of the two qubits are swapped and the operation of rotation (Rk) is applied into the target qubit when a control qubit is in state |1〉 by CRk operation (two qubits controlled operation). Thus, if the CRk (k = 2, …, t) and Hadamard operations are arranged as shown in Fig. 1, we can obtain the theoretical circuit of multi-qubit DQFT with scalability. Consequently, for the high efficiency and reliable performance of DQFT based on CRk operations, the main issue is to design the experimentally implemented CRk operation in a feasibility manner.

Figure 1
figure 1

Plot presents the theoretical circuit to implement the operation of DQFT on t qubits. This circuit consists of controlled-rotation k [CRk (k = 2, …, t)] operations and Hadamard operations for DQFT on t qubits.

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Quantum Dot Inside Optical Cavity and Controlled-Rotation k Gate for Discrete Quantum Fourier Transform

Interaction of photon and QD-cavity system

We introduce a QD-cavity system42,44,45,46,47,48,49,50,51,52,63, which can induce the interaction of a photon and a singly charged QD (a negatively charged exciton: X−) confined in a single-sided optical cavity, and a reflection operator (\(\hat{{\rm{R}}}\)) from the interaction between a photon and the QD-cavity system. Figure 2 represents that the QD-cavity system is composed of two GaAs/Al(Ga)As distributed Bragg reflections (DBRs: bottom DBR – partially reflective, and top DBR – 100% reflective) as a single-sided cavity. And QD is confined at the center of the single-sided cavity (between two DBRs), where κs and γ are the side-leakage rate of the cavity mode and the decay rate of X− (two electrons bound to one hole), respectively. When a photon pulse is incident into the QD, an interaction occurs between the photon and an excess electron into the QD, according to the Pauli Exclusion Principle, where \({\hat{{b}}}_{{\rm{in}}}\) (\({\hat{{b}}}_{{\rm{out}}}\)) is the input (output) field operators of the photon pulse. As illustrated in Fig. 2, if the spin state of the excess electron in the QD is in the state |↑〉, then the polarization |L〉 of a photon can drive the state |↑↓⇑〉 of X−. Also, if the spin state |↓〉 of the excess electron in the QD and the polarization |R〉 of a photon, the interaction, the state |↓↑⇓〉, of X− can be created. Let us assume the approximation of weak excitation with the ground state, 〈σZ〉 = −1, in the QD for the steady state42,45,46,50,51,52. Then, due to the spin selection rule, we can calculate the reflection coefficient, R(ω), of the reflected photon and the QD from the Heisenberg equation of motion70, and also the reflectance, |rh| (|r0|), and phase shift ϕrh (ϕr0) of the hot (cold) cavity, as follows:

$$R(\omega )=\frac{{\hat{{b}}}_{{\rm{out}}}}{{\hat{{b}}}_{{\rm{in}}}}=\frac{[i({\omega }_{{{\rm{X}}}^{-}}-\omega )+\gamma /2][i({\omega }_{c}-\omega )-\kappa /2+{\kappa }_{s}/2]+{g}^{2}}{[i({\omega }_{{{\rm{X}}}^{-}}-\omega )+\gamma /2][i({\omega }_{c}-\omega )+\kappa /2+{\kappa }_{s}/2]+{g}^{2}},$$
(5)

where \({\omega }_{{{\rm{X}}}^{-}}\), ωc, and ω are the frequencies of X−, cavity mode, and external field, respectively. Also, g is the coupling strength between X− and cavity mode while κ is the decay rate of the cavity mode. In the interaction between a photon and an electron in the QD, when the spin state of the excess electron is in the state |↑〉(|↓〉), the polarization of the photon |L〉(|R〉) drives the hot cavity (the QD is coupled with the cavity: g ≠ 0), while the polarization of the photon |R〉(|L〉) feels the cold cavity (the QD is uncoupled with the cavity: g = 0). Thus, for the resonant interaction (\({\omega }_{{{\rm{X}}}^{-}}={\omega }_{c}\)), the reflection coefficients Rh(ω) and R0(ω) can be calculated as follows:

$$\begin{array}{ccc}{\rm{h}}{\rm{o}}{\rm{t}}\,{\rm{c}}{\rm{a}}{\rm{v}}{\rm{i}}{\rm{t}}{\rm{y}}:{R}_{{\rm{h}}}(\omega )\equiv |{r}_{{\rm{h}}}(\omega )|\exp [i{\phi }_{{\rm{r}}{\rm{h}}}(\omega )] & = & \frac{[i({\omega }_{c}-\omega )+\gamma /2][i({\omega }_{c}-\omega )-\kappa /2+{\kappa }_{s}/2]+{g}^{2}}{[i({\omega }_{c}-\omega )+\gamma /2][i({\omega }_{c}-\omega )+\kappa /2+{\kappa }_{s}/2]+{g}^{2}}\\ & = & R(\omega ),\\ {\rm{c}}{\rm{o}}{\rm{l}}{\rm{d}}\,{\rm{c}}{\rm{a}}{\rm{v}}{\rm{i}}{\rm{t}}{\rm{y}}:{R}_{0}(\omega )\equiv |{r}_{0}(\omega )|\exp [i{\phi }_{{\rm{r}}0}(\omega )] & = & \frac{i({\omega }_{c}-\omega )-\kappa /2+{\kappa }_{s}/2}{i({\omega }_{c}-\omega )+\kappa /2+{\kappa }_{s}/2},\end{array}$$
(6)

where \({\phi }_{{\rm{r}}{\rm{h}}}\equiv \arg [{R}_{{\rm{h}}}]\,({\phi }_{{\rm{r}}0}\equiv \arg [{R}_{0}])\) is the phase shift in the hot (cold) cavity. Using these reflectances (|rh|, |r0|) and phase shifts (ϕrh, ϕr0) in the reflection coefficients (Rh, R0), we can obtain the reflection operator, \(\hat{{\rm{R}}}(\omega )\), with the resonant interaction (\({\omega }_{{{\rm{X}}}^{-}}={\omega }_{c}\)), as follows:

$$\hat{{\rm{R}}}(\omega )=|{r}_{{\rm{h}}}(\omega )|{e}^{i{\phi }_{{\rm{rh}}}(\omega )}(|R\rangle \langle R|\otimes |\downarrow \rangle \langle \downarrow |+|L\rangle \langle L|\otimes |\uparrow \rangle \langle \uparrow |)+|{r}_{{\rm{0}}}(\omega )|{e}^{i{\phi }_{{\rm{r0}}}(\omega )}(|R\rangle \langle R|\otimes |\uparrow \rangle \langle \uparrow |+|L\rangle \langle L|\otimes |\downarrow \rangle \langle \downarrow |).$$
(7)
Figure 2
figure 2

Singly charged QD within a single-sided cavity with a photon \({\hat{{b}}}_{{\rm{in}}}\) and \({\hat{{b}}}_{{\rm{out}}}\). Due to the spin selection rule, the interaction between a polarization of incident photon and a spin state of excess electron in QD can create the transition that the photon |L〉 drives the state as |↑〉 → |↑↓⇑〉 (or the photon |R〉 drives the state as |↓〉 → |↓↑⇓〉), where |↑〉 ≡ |+1/2〉, |↓〉 ≡ |−1/2〉 are the spin states of the excess electron while |⇑〉 and |⇓〉(Jz = +3/2, −3/2) represent heavy-hole spin states.

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When the experimental parameters are κs ≪ κ (small side-leakage rate) and g ≫ (κ, γ) (large coupling strength) with small γ (~several μ eV)71,72,73,74, two kinds of reflection operators (\({\hat{{\rm{R}}}}^{1}\) and \({\hat{{\rm{R}}}}^{2}\)), obtained by adjusting the frequencies of the external field, ω, and cavity mode, ωc, can be given as

$$\begin{array}{lll}\omega -{\omega }_{c}=\kappa /2 & : & {\hat{{\rm{R}}}}^{1}(\omega )\approx |R\rangle \langle R|\otimes |\downarrow \rangle \langle \downarrow |+|L\rangle \langle L|\otimes |\uparrow \rangle \langle \uparrow |-i|R\rangle \langle R|\otimes |\uparrow \rangle \langle \uparrow |-i|L\rangle \langle L|\otimes |\downarrow \rangle \langle \downarrow |,\\ \omega -{\omega }_{c}=0 & : & {\hat{{\rm{R}}}}^{2}(\omega )\approx |R\rangle \langle R|\otimes |\downarrow \rangle \langle \downarrow |+|L\rangle \langle L|\otimes |\uparrow \rangle \langle \uparrow |-|R\rangle \langle R|\otimes |\uparrow \rangle \langle \uparrow |-|L\rangle \langle L|\otimes |\downarrow \rangle \langle \downarrow |,\end{array}$$
(8)

where the reflectances and phase shifts are |rh| = |r0| ≈ 1(≈1) and ϕrh ≈ 0 (≈0), ϕr0 ≈ −π/2 (≈π), respectively, in the detuning of frequency as ω − ωc = κ/2 (ω − ωc = 0) for g/κ = 2.4 and γ/κ = 0.1 with κs → 0 42,44,45,46,47,48,49,50,51,52,63. Subsequently, we will utilize the interaction of the QD-cavity system, the reflection operators (\({\hat{{\rm{R}}}}^{1}\) and \({\hat{{\rm{R}}}}^{2}\)) in Eq. 8, on the CRk gate using the QD-cavity systems for the DQFT scheme.

Design of CRk gate using two QD-cavity systems for DQFT scheme

In this section, we propose a feasible controlled-rotation k (CRk) gate using the interaction, in Sec. 3.1, of the QD-cavity system, in order to implement the CRk operation of Fig. 1 for direct applications in the DQFT scheme. First, we assume the definition of binary representation in the polarization of the photon as {|R〉, |L〉} ≡ {|0〉, |1〉}, and the relation between the circular- (|R〉 is right and |L〉 is left) and linear-polarization (|H〉 is horizontal and |V〉 is vertical), as \(|R\rangle \equiv (|H\rangle +|V\rangle )/\sqrt{2}\) and \(|L\rangle \equiv (|H\rangle -|V\rangle )/\sqrt{2}\).

Figure 3 shows a schematic of the CRK gate, which consists of two QD-cavity systems (QD1 and QD2 gates) and an Rk (rotation k) operator with linearly optical devices, for the DQFT scheme. In order to demonstrate the CRk gate, let us suppose an input state of two photons A and B with the defined polarization, {|R〉, |L〉} ≡ {|0〉, |1〉}, as follows: \({|\phi \rangle }_{{\rm{in}}}={x}_{1}{|R\rangle }_{{\rm{A}}}^{1}{|R\rangle }_{{\rm{B}}}^{2}+{x}_{2}{|R\rangle }_{{\rm{A}}}^{1}{|L\rangle }_{{\rm{B}}}^{2}+{x}_{3}{|L\rangle }_{{\rm{A}}}^{1}{|R\rangle }_{{\rm{B}}}^{2}+{x}_{4}{|L\rangle }_{{\rm{A}}}^{1}{|L\rangle }_{{\rm{B}}}^{2}\). After a BS and an SF (path 2) on photon B, as described in Fig. 3, the transformed state, |φ〉1, is given by

$$\begin{array}{ccc}{|\phi \rangle }_{{\rm{in}}} & \mathop{\Rightarrow }\limits^{\mathrm{BS},\mathrm{SF}} & {|\phi \rangle }_{{\rm{1}}}=\frac{1}{\sqrt{2}}({x}_{1}{|R\rangle }_{{\rm{A}}}^{1}{|R\rangle }_{{\rm{B}}}^{1}+{x}_{2}{|R\rangle }_{{\rm{A}}}^{1}{|L\rangle }_{{\rm{B}}}^{1}+{x}_{3}{|L\rangle }_{{\rm{A}}}^{1}{|R\rangle }_{{\rm{B}}}^{1}+{x}_{4}{|L\rangle }_{{\rm{A}}}^{1}{|L\rangle }_{{\rm{B}}}^{1}+{x}_{1}{|R\rangle }_{{\rm{A}}}^{1}{|L\rangle }_{{\rm{B}}}^{2}+{x}_{2}{|R\rangle }_{{\rm{A}}}^{1}{|R\rangle }_{{\rm{B}}}^{2}+{x}_{3}{|L\rangle }_{{\rm{A}}}^{1}{|L\rangle }_{{\rm{B}}}^{2}+{x}_{4}{|L\rangle }_{{\rm{A}}}^{1}{|R\rangle }_{{\rm{B}}}^{2}).\end{array}$$
(9)
Figure 3
figure 3

Controlled-rotation k (CRk) gate: For the implementation of the CRk operation illustrated in Fig. 1, this CRk gate employs two QD-cavity systems (QD1 and QD2 gates) and an Rk operator with linearly optical devices (BSs, SFs, and feed-forwards: PF-R, PF-L, path switches). In the QD1 gate, the interactions between photons (A and B) and QD1 are used as the reflection operator of \({\hat{{\rm{R}}}}^{1}\) (ω − ωc = κ/2), in Eq. 8. Further, the operations of the switches (S1~S6) on the paths of photon A and B comply with the time table. After measuring QD1 (electron spin 1), the feed-forward (PF-R and path switch 1) is performed by the outcome of QD1. In the QD2 gate, the reflection operator of \({\hat{{\rm{R}}}}^{2}\) (ω−ωc = 0), shown in Eq. 8, applies to the interaction of a photon B and QD2. Then, according to the outcome of measurement on QD2 (electron spin 2), PF-R, PF-L, and path switch 2 (feed-forward) are performed on photon A and B.

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[QD1 gate using reflection operator \({\hat{{\bf{R}}}}^{{\bf{1}}}\) (ω − ω c  = κ/2)]

In QD1, the preparation of electron spin 1 state is in \({|{+}_{{\rm{e}}}\rangle }_{{\rm{1}}}\) where \(|{\pm }_{{\rm{e}}}\rangle \equiv (|\uparrow \rangle \pm |\downarrow \rangle )/\sqrt{2}\). Then, the interaction, \({\hat{{\rm{R}}}}^{1}\), between QD1 and photons (A and B) is applied, and the procedure of switches (S1~S6) is performed in accordance with the time table shown in Fig. 3. After the state, |φ〉1, passes through SF on path 2 of photon B, the state is transformed into

$$\begin{array}{ccc}{|\phi \rangle }_{1}\mathop{\Rightarrow }\limits^{{\rm{Q}}{\rm{D}}1,\,{\rm{S}}{\rm{F}}}{|\phi \rangle }_{2} & = & \frac{1}{\sqrt{2}}(-{x}_{1}{|R\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}-{x}_{2}{|L\rangle }_{{\rm{B}}}^{2}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{3}{|R\rangle }_{{\rm{B}}}^{2}{|L\rangle }_{{\rm{A}}}^{1}+{x}_{4}{|L\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1})\otimes {|{-}_{{\rm{e}}}\rangle }_{1}\\ & - & \frac{i}{\sqrt{2}}({x}_{1}{|R\rangle }_{{\rm{B}}}^{2}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{2}{|L\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{3}{|R\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1}+{x}_{4}{|L\rangle }_{{\rm{B}}}^{2}{|L\rangle }_{{\rm{A}}}^{1})\otimes {|{+}_{{\rm{e}}}\rangle }_{1},\end{array}$$
(10)

where the reflection operator, \({\hat{{\rm{R}}}}^{1}\) (ω − ωc = κ/2), in Eq. 8 applies to the above interaction of QD1 and photons. According to the measurement result regarding QD1, if the electron spin state is in \({|{-}_{{\rm{e}}}\rangle }_{{\rm{1}}}\), feed-forward (S1 and PF-R) is performed on photons A and B; otherwise, \({|{+}_{{\rm{e}}}\rangle }_{{\rm{1}}}\), it is not, as described in Fig. 3. Thus, the output state, |φ〉O:1, of QD1 gate is expressed as

$${|\phi \rangle }_{2}\mathop{\Rightarrow }\limits^{{\rm{f}}{\rm{e}}{\rm{e}}{\rm{d}}-{\rm{f}}{\rm{o}}{\rm{r}}{\rm{w}}{\rm{a}}{\rm{r}}{\rm{d}}\,{\rm{o}}{\rm{r}}\,{\rm{n}}{\rm{o}}{\rm{t}}}{|\phi \rangle }_{{\rm{O}}:1}={x}_{1}{|R\rangle }_{{\rm{B}}}^{2}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{2}{|L\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{3}{|R\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1}+{x}_{4}{|L\rangle }_{{\rm{B}}}^{2}{|L\rangle }_{{\rm{A}}}^{1}.$$
(11)

[Rk operator and BS]

After the Rk operator on path 2 and BS is applied to photon B, as described in Fig. 3, the output state, |φ〉O:1, of QD1 gate is transformed into the state, |φ〉3, as follows:

$$\begin{array}{ccc}{|\phi \rangle }_{{\rm{O}}:1}\mathop{\Rightarrow }\limits^{{\rm{R}}{\rm{k}},\,{\rm{B}}{\rm{S}}}{|\phi \rangle }_{3} & = & \frac{1}{\sqrt{2}}({x}_{1}{|R\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{2}{|L\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{3}{|R\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1}+{x}_{4}{e}^{2\pi i/{2}^{k}}{|L\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1})\\ & + & \frac{1}{\sqrt{2}}(-{x}_{1}{|R\rangle }_{{\rm{B}}}^{2}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{2}{|L\rangle }_{{\rm{B}}}^{2}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{3}{|R\rangle }_{{\rm{B}}}^{2}{|L\rangle }_{{\rm{A}}}^{1}-{x}_{4}{e}^{2\pi i/{2}^{k}}{|L\rangle }_{{\rm{B}}}^{2}{|L\rangle }_{{\rm{A}}}^{1}).\end{array}$$
(12)

Subsequently, this state, |φ〉3, is injected into the QD2 gate in order to merge the split paths (1 and 2) of photon B.

[QD2 gate using reflection operator \({\hat{{\bf{R}}}}^{{\bf{2}}}\) (ω−ω c  = 0)]

After the operation of the interaction, \({\hat{{\rm{R}}}}^{2}\), between QD2 and photon B with the preparation of electron spin 2 state as \({|{+}_{{\rm{e}}}\rangle }_{{\rm{2}}}\), the state, |φ〉3, is transformed into

$$\begin{array}{ccc}{|\phi \rangle }_{3}\mathop{\Rightarrow }\limits^{{\rm{Q}}{\rm{D}}2}{|\phi \rangle }_{4} & = & \frac{1}{\sqrt{2}}(-{x}_{1}{|R\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{2}{|L\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}-{x}_{3}{|R\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1}+{x}_{4}{e}^{2\pi i/{2}^{k}}{|L\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1})\otimes {|{-}_{{\rm{e}}}\rangle }_{2}\\ & + & \frac{1}{\sqrt{2}}(-{x}_{1}{|R\rangle }_{{\rm{B}}}^{2}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{2}{|L\rangle }_{{\rm{B}}}^{2}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{3}{|R\rangle }_{{\rm{B}}}^{2}{|L\rangle }_{{\rm{A}}}^{1}-{x}_{4}{e}^{2\pi i/{2}^{k}}{|L\rangle }_{{\rm{B}}}^{2}{|L\rangle }_{{\rm{A}}}^{1})\otimes {|{+}_{{\rm{e}}}\rangle }_{2},\end{array}$$
(13)

where the reflection operator, \({\hat{{\rm{R}}}}^{2}\) (ω−ωc = 0), in Eq. 8 applies to the above interaction of QD2 and photon B. When measuring an electron spin 2 state in QD2, if the electron spin state is in \({|{-}_{{\rm{e}}}\rangle }_{{\rm{2}}}\), feed-forward (PF-R) is performed to photon A; otherwise, \({|{+}_{{\rm{e}}}\rangle }_{{\rm{1}}}\), feed-forward (S2, PF-R, and PF-L) is performed on photons A and B, as described in Fig. 3. Finally, after the input state, |φ〉in, passes through the CRk gate (QD1 and QD2 gates) in Fig. 3, the final state, |φ〉fin, can be given by

$${|\phi \rangle }_{{\rm{in}}}\mathop{\Rightarrow }\limits^{{\rm{CRk}}\,{\rm{gate}}}{|\phi \rangle }_{{\rm{fin}}}={x}_{1}{|R\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{2}{|L\rangle }_{{\rm{B}}}^{1}{|R\rangle }_{{\rm{A}}}^{1}+{x}_{3}{|R\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1}+{x}_{4}{e}^{2\pi i/{2}^{k}}{|L\rangle }_{{\rm{B}}}^{1}{|L\rangle }_{{\rm{A}}}^{1}.$$
(14)

Consequently, the final state, |φ〉fin, which is applied to the CRk gate using two QD-cavity systems, is identical to the state, |ϕ〉fin, in Eq. 4, which is applied to the CRk operation (UCRk: swapping paths and operation of rotation k).

Now, using a simple example in Fig. 4, we show the implementation of the three-qubit DQFT scheme by directly utilizing the proposed CRk gates via the QD-cavity systems. As shown in Fig. 4, we can realize the theoretical circuit of the three-qubit DQFT to three-photon DQFT scheme employing CRk gates (QD-cavity systems) and HWPs by arranging our CRk gates in the manner shown in Fig. 3. Thus, for the scalability of the multi-photon DQFT scheme (for the implementation of multi-qubit DQFT), the structure of the three-photon DQFT scheme can be generalized (or expanded) to realize the multi-photon DQFT scheme by the composition of the CRk (k = 2, …, t) gates and HWPs.

Figure 4
figure 4

Three-qubit DQFT (the theoretical circuit of DQFT shown in Fig. 1), based on three CRk (two CR2 and a CR3) operations and three Hadamard operations, can be implemented in the three-photon DQFT scheme, which consists of three CPk (two CR2 and a CR3) gates using the QD-cavity systems, and three HWPs. The HWP, which can rotate the polarization of a photon, performs the Hadamard operation.

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The structure of our DQFT scheme, which consists of CRk gates using QDs, is based on the theoretical circuit of DQFT in Fig. 1. When our scheme is expended to multi-photon DQFT scheme, the method of expansion is identical with the procedure of the theoretical circuit of multi-qubit DQFT algorithm, as described in Fig. 4. Thus, we can calculate the complexity2 about how many gates our DQFT scheme is utilizing on t photons, as follows: The total of t gates (a HWP and t − 1 CRk gates) is used on the first photon. And the total of t − 1 gates (a HWP and t − 2 CRk gates) is used on the second photon. Finally, the tth photon only utilizes one gate (a HWP). In our DQFT scheme, we can confirm that the number of gates for DQFT is as t(t + 1)/2 = t + (t − 1) + ⋅⋅⋅ + 2 + 1. Consequently, our DQFT scheme provides a Θ(t2) algorithm for the operation of DQFT on t photons. Furthermore, compared with the classical best algorithm as fast Fourier transform (FFT)2, which provides a Θ(t2t) algorithm for the discrete Fourier transform, our DQFT scheme on quantum computer is more efficient than FFT, which requires exponentially more operations, on classical computer.

So far, we have proposed the use of the CRk gate, using the QD-cavity systems, to feasibly implement the CRk (controlled-rotation k) operation for the optical DQFT scheme with high efficiency and reliable performance. In the optical DQFT, our CRk gate with two QD-cavity systems (QD1 and QD2 gates) is the essential element with the stable storage of quantum information and the actual interaction between the photon and QD. Thus, for the experimental realization of our CRk gate, we should analyze the interaction of the QD-cavity system (QD within a single-sided cavity) under vacuum noise, sideband leakage, and absorption, in practice.

Analysis of QD-Cavity System under Vacuum Noise and Sideband Leakage

In our CRk gate, in order to implement a controlled-rotation k (CRk) operation, the critical component is the interaction of the QD-cavity system inducing a difference in reflectance |rh| [|r0|] with phase shift ϕrh [ϕr0] from the reflection coefficient R h [R 0] shown in Eq. 6 with respect to the hot [cold] cavity. Thus, the interaction between a photon and an electron spin state in QD should be analyzed to quantify the efficiency and reliable performance of the QD-cavity system under vacuum noise, N(ω), for operation of the QD-dipole and leaky modes, S(ω) (sideband leakage and absorption)40,42,53,54,55,56. For this analysis, we introduce the Jaynes-Cummings Hamiltonian (HJC) in the rotating frame at the input field (\({\hat{{b}}}_{{\rm{in}}}\)) frequency.

$${H}_{{\rm{JC}}}=\hslash ({\omega }_{c}-\omega ){\hat{{a}}}^{+}\hat{{a}}+\hslash ({\omega }_{{{\rm{X}}}^{-}}-\omega ){\hat{\sigma }}_{+}{\hat{\sigma }}_{-}+i\hslash g({\hat{\sigma }}_{+}\hat{{a}}-{\hat{{a}}}^{+}{\hat{\sigma }}_{-})+\hslash \int d\omega ^{\prime} \omega ^{\prime} {\hat{{b}}}^{+}\hat{{b}}+\frac{\hslash }{\sqrt{2\pi }}\int d\omega ^{\prime} \sqrt{\kappa }(\hat{{a}}{\hat{{b}}}^{+}+{\hat{{a}}}^{+}\hat{{b}}).$$
(15)

Using the solutions of the Heisenberg equations of motion (Bloch equation with damping, γ), we can calculate the quantum Langevin equations of a cavity field operator, \(\hat{{a}}\), a dipole operator, \({\hat{\sigma }}_{-}\), of X−, and the input-output relations with vacuum noise, N(ω), and leaky modes, S(ω), as follows:

$$\begin{array}{ccc}\frac{d\hat{a}}{dt} & = & -\frac{i}{\hslash }[\hat{a},\,{H}_{{\rm{J}}{\rm{C}}}]=-[i({\omega }_{c}-\omega )+\frac{\kappa }{2}+\frac{{\kappa }_{s}}{2}]\hat{a}-g{\hat{\sigma }}_{-}-\sqrt{\kappa }{\hat{b}}_{{\rm{i}}{\rm{n}}}-\sqrt{{\kappa }_{s}}{\hat{S}}_{{\rm{i}}{\rm{n}}},\\ \frac{d{\hat{\sigma }}_{-}}{dt} & = & -\frac{i}{\hslash }[{\hat{\sigma }}_{-},\,{H}_{{\rm{J}}{\rm{C}}}]=-[i({\omega }_{{{\rm{X}}}^{-}}-\omega )+\frac{\gamma }{2}]{\hat{\sigma }}_{-}-g{\hat{\sigma }}_{Z}\hat{a}+\sqrt{\gamma }{\hat{\sigma }}_{Z}\hat{N},\\ \frac{d{\hat{\sigma }}_{Z}}{dt} & = & \frac{d({\rho }_{22}-{\rho }_{11})}{dt}=0,\\ {\hat{b}}_{{\rm{o}}{\rm{u}}{\rm{t}}} & = & {\hat{b}}_{{\rm{i}}{\rm{n}}}+\sqrt{\kappa }\hat{a},\,{\hat{S}}_{{\rm{o}}{\rm{u}}{\rm{t}}}={\hat{S}}_{{\rm{i}}{\rm{n}}}+\sqrt{\kappa }\hat{a},\end{array}$$
(16)

where \({\hat{S}}_{{\rm{in}}}({\hat{S}}_{{\rm{out}}})\) is the 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 \({\hat{\sigma }}_{-}\). In Sec. 3.1, we assumed the approximation of weak excitation with the ground state, \(\langle {\hat{\sigma }}_{Z}\rangle =-1\) (no saturation), in the QD for the steady state42,45,46,50,51,52. Based on this condition and Eq. 16, the output field operator, \({\hat{{b}}}_{{\rm{out}}}\), can be given by

$${\hat{{b}}}_{{\rm{out}}}=R(\omega ){\hat{{b}}}_{{\rm{in}}}+N(\omega )\hat{N}+S(\omega ){\hat{S}}_{{\rm{in}}}.$$
(17)

Here, we consider the vacuum noise, N(ω), and leaky mode, S(ω) (sideband leakage and absorption) in the interaction of the QD-cavity, and contrast it with the case presented in Eq. 5 (the ideal case: \({\hat{{b}}}_{{\rm{out}}}=R(\omega ){\hat{{b}}}_{{\rm{in}}}\)). Therefore, the coefficients of noise, N(ω), and leakage, S(ω), can be calculated with \({\omega }_{{{\rm{X}}}^{-}}={\omega }_{c}\), as follows:

$$\begin{array}{rcl}N(\omega ) & = & \frac{\sqrt{\gamma \kappa }g}{[i({\omega }_{c}-\omega )+\gamma /2][i({\omega }_{c}-\omega )+\kappa /2+{\kappa }_{s}/2]+{g}^{2}},\\ S(\omega ) & = & \frac{-\sqrt{{\kappa }_{s}\kappa }[i({\omega }_{c}-\omega )+\gamma /2]}{[i({\omega }_{c}-\omega )+\gamma /2][i({\omega }_{c}-\omega )+\kappa /2+{\kappa }_{s}/2]+{g}^{2}},\end{array}$$
(18)

where the reflection coefficient, R(ω), is obtained from Eq. 5. Then, we can acquire the noise rate-phase shift (|nh| - ϕnh) [|n0| - ϕn0], as well as the leakage rate-phase shift (|sh| - ϕsh) [|s0| - ϕs0] from the coefficients of noise, \({N}_{{\rm{h}}}\equiv |{n}_{{\rm{h}}}|{e}^{i{\phi }_{{\rm{nh}}}}\) \([{N}_{{\rm{0}}}\equiv |{n}_{{\rm{0}}}|{e}^{i{\phi }_{{\rm{n0}}}}]\), and leakage, \({S}_{{\rm{h}}}\equiv |{s}_{{\rm{h}}}|{e}^{i{\phi }_{{\rm{sh}}}}\) \([{S}_{{\rm{0}}}\equiv |{s}_{{\rm{0}}}|{e}^{i{\phi }_{{\rm{s0}}}}]\), corresponding to the (hot: g ≠ 0) [cold: g = 0] cavity.

The graphs shown in Fig. 5 represent the values of the reflectances (|rh|, |r0|), noise rates (|nh|, |n0|), leakage rates (|sh|, |s0|), and phase shifts (ϕrh, ϕr0, ϕnh, ϕn0, ϕsh, ϕs0) for the detuning frequency as 2(ω − ωc)/κ in terms of the differences in the side-leakage rates (κs/κ = 0.01 and 1.00) with g/κ = 2.4, γ/κ = 0.1, and \({\omega }_{{{\rm{X}}}^{-}}={\omega }_{c}\). As illustrated in Fig. 5, if we take the small side-leakage, κs/κ = 0.01, with g/κ = 2.4 and γ/κ = 0.1, the values of the reflectances, noise rates, leakage rates, and phase shifts can be respectively acquired as |rh| ≈ 1 (≈1), |r0| ≈ 1 (≈1), |nh| ≈ 0.1 (≈0.1), |n0| = 0 (=0), |sh| ≈ 0 (≈0), |s0| ≈ 0.1 (≈0.2), and ϕrh ≈ 0 (≈0), ϕr0 ≈ −π/2 (=π), ϕnh ≈ 0 (≈0), ϕn0 = 0 ( = 0), ϕsh ≈ 11π/20 (=π), ϕs0 ≈ −3π/4 (=π) in the adjusted frequencies as ω − ωc = κ/2 (ω − ωc = 0). Based on these values, we can confirm that the affections of the vacuum noise, N(ω), and leaky modes, S(ω) (sideband leakage and absorption) will be small (ignored), due to the extremely small values of noise rates (|nh|, |n0|) and leakage rates (|sh|, |s0|) for g/κ = 2.4 and γ/κ = 0.1, respectively, with κs/κ = 0.01 (small side-leakage rate)40,42,53,54,55,56.

Figure 5
figure 5

Reflectances (|rh|, |r0|), noise rates (|nh|, |n0|), leakage rates (|sh|, |s0|), and phase shifts (ϕrh, ϕr0, ϕnh, ϕn0, ϕsh, ϕs0) for the detuning frequency, 2(ω − ωc)/κ in terms of the differences in the side-leakage rates (κs/κ = 0.01 and κs/κ = 1.00). In these plots, the experimental parameters (the coupling strength and decay rate of X−) are fixed at g/κ = 2.4 and γ/κ = 0.1 [g ≫ (γ, κ)] with \({\omega }_{{{\rm{X}}}^{-}}={\omega }_{c}\).

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In addition, for the quantifying efficiency and performance of the QD-cavity system, we should modify the reflection operator, \(\hat{{\rm{R}}}(\omega )\) in Eq. 7, into the practical reflection operator, \({\hat{{\rm{R}}}}_{{\rm{p}}}(\omega )\), including to the affections of the vacuum noise, N(ω), and leaky modes, S(ω) (sideband leakage and absorption), as follows:

$$\begin{array}{rcl}{\hat{{\rm{R}}}}_{{\rm{p}}}(\omega ) & = & [|{r}_{{\rm{h}}}(\omega )|{e}^{i{\phi }_{{\rm{rh}}}(\omega )}+|{n}_{{\rm{h}}}(\omega )|{e}^{i{\phi }_{{\rm{nh}}}(\omega )}+|{s}_{{\rm{h}}}(\omega )|{e}^{i{\phi }_{{\rm{sh}}}(\omega )}]\\ & & \times \,(|R\rangle \langle R|\otimes |\downarrow \rangle \langle \downarrow |+|L\rangle \langle L|\otimes |\uparrow \rangle \langle \uparrow |)\\ & & +\,[|{r}_{0}(\omega )|{e}^{i{\phi }_{{\rm{r0}}}(\omega )}+|{s}_{{\rm{0}}}(\omega )|{e}^{i{\phi }_{{\rm{s0}}}(\omega )}](|R\rangle \langle R|\otimes |\uparrow \rangle \langle \uparrow |+|L\rangle \langle L|\otimes |\downarrow \rangle \langle \downarrow |),\end{array}$$
(19)

where \({N}_{{\rm{0}}}(\omega )\equiv |{n}_{{\rm{0}}}(\omega )|\exp [i{\phi }_{{\rm{n0}}}(\omega )]=0\) (cold cavity: g = 0), as shown in Eq. 18. Therefore, we can analyze the efficiencies and performances of the QD-cavity systems (QD1 and QD2 gates) to quantify the fidelity values of the output states from the reflection operators, \({\hat{{\rm{R}}}}^{1}\) (ω − ωc = κ/2) and \({\hat{{\rm{R}}}}^{2}\) (ω − ωc = 0) in Eq. 8 (ideal case), and the reflection operator, \({\hat{{\rm{R}}}}_{{\rm{p}}}\) in Eq. 19 (practical case). When the input state is assumed to be \((|R\rangle +|L\rangle )/\sqrt{2}\otimes (|\uparrow \rangle +|\downarrow \rangle )/\sqrt{2}\), the ideal output states, \(|{\psi }_{{\rm{ID}}}^{1}\rangle (\mathrm{QD1}:\,\omega -{\omega }_{c}=\kappa /2)\) and \(|{\psi }_{{\rm{ID}}}^{2}\rangle (\mathrm{QD2}:\,\omega -{\omega }_{c}=0)\), from Eq. 8, and a practical output state, |ψPR〉, from Eq. 19, can be given by

$$\begin{array}{rcl}|{\psi }_{{\rm{ID}}}^{1}\rangle & = & \frac{1}{\sqrt{2}}[\frac{1}{\sqrt{2}}(|R\rangle |\downarrow \rangle +|L\rangle |\uparrow \rangle )-\frac{i}{\sqrt{2}}(|R\rangle |\uparrow \rangle +|L\rangle |\downarrow \rangle )],\\ |{\psi }_{{\rm{ID}}}^{2}\rangle & = & \frac{1}{\sqrt{2}}[\frac{1}{\sqrt{2}}(|R\rangle |\downarrow \rangle +|L\rangle |\uparrow \rangle )-\frac{1}{\sqrt{2}}(|R\rangle |\uparrow \rangle +|L\rangle |\downarrow \rangle )],\\ |{\psi }_{{\rm{PR}}}\rangle & = & \frac{1}{\sqrt{{\rm{N}}}}[\frac{({R}_{{\rm{h}}}+{N}_{{\rm{h}}}+{S}_{{\rm{h}}})}{\sqrt{2}}(|R\rangle |\downarrow \rangle +|L\rangle |\uparrow \rangle )+\frac{({R}_{{\rm{0}}}+{S}_{{\rm{0}}})}{\sqrt{2}}(|R\rangle |\uparrow \rangle +|L\rangle |\downarrow \rangle )],\end{array}$$
(20)

where N = |Rh + Nh + Sh|2 + |R0 + S0|2. Then, we can calculate the fidelities F1 (QD1 gate) between \(|{\psi }_{{\rm{ID}}}^{1}\rangle \) and |ψPR〉, and F2 (QD2 gate) between \(|{\psi }_{{\rm{ID}}}^{2}\rangle \) and |ψPR〉 as follows:

$$\begin{array}{c}{{\rm{F}}}_{{\rm{1}}}\equiv |\sqrt{\langle {\psi }_{{\rm{PR}}}|{\psi }_{{\rm{ID}}}^{1}\rangle \langle {\psi }_{{\rm{ID}}}^{1}|{\psi }_{{\rm{PR}}}\rangle }|=\frac{1}{\sqrt{2{\rm{N}}}}|\sqrt{{|({R}_{{\rm{h}}}+{N}_{{\rm{h}}}+{S}_{{\rm{h}}})-i({R}_{{\rm{0}}}+{S}_{{\rm{0}}})|}^{2}}|,\\ {{\rm{F}}}_{{\rm{2}}}\equiv |\sqrt{\langle {\psi }_{{\rm{PR}}}|{\psi }_{{\rm{ID}}}^{2}\rangle \langle {\psi }_{{\rm{ID}}}^{2}|{\psi }_{{\rm{PR}}}\rangle }|=\frac{1}{\sqrt{2{\rm{N}}}}|\sqrt{{|({R}_{{\rm{h}}}+{N}_{{\rm{h}}}+{S}_{{\rm{h}}})-({R}_{{\rm{0}}}+{S}_{{\rm{0}}})|}^{2}}|.\end{array}$$
(21)

The graphs and tables in Fig. 6 show the distributions and values of fidelities, F1 (QD1 gate: \(|{\psi }_{{\rm{ID}}}^{1}\rangle \leftrightarrow |{\psi }_{{\rm{PR}}}\rangle \)) and F2 (QD2 gate: \(|{\psi }_{{\rm{ID}}}^{{\rm{2}}}\rangle \leftrightarrow |{\psi }_{{\rm{PR}}}\rangle \)), according to the differences in the side-leakage rates, κs/κ, as well as the coupling strength, g/κ, with γ/κ = 0.1 under vacuum noise, N(ω), and sideband leakage and absorption, S(ω). As described in Fig. 6, when we take the experimental parameters as the strong coupling strength, g ≫ (κ, γ), and the small side-leakage rate, κs ≪ κ, with fixed γ/κ = 0.1, two fidelities (F1 and F2) are approaching 1 despite the affection of vacuum noise, N(ω), and sideband leakage, S(ω). Thus, we can conclude that the influences of the noise rate, leakage rate, and phase shifts in Eq. 18 will be ignored if the magnitude of coupling strength (g/κ) increases, and if the side-leakage rate (κs/κ), decreases in the QD-cavity system.

Figure 6
figure 6

Fidelities, F1 (QD1 gate: ω − ωc = κ/2) and F2 (QD2 gate: ω − ωc = 0) of the output states for the side-leakage rate κs/κ and coupling strength g/κ with γ/κ = 0.1 and \({\omega }_{{{\rm{X}}}^{-}}={\omega }_{c}\) under vacuum noise, N(ω), for the operation of the QD-dipole and leaky modes, S(ω) (sideband leakage and absorption). In the tables, the values of fidelities F1 and F2 are listed for the differences in κs/κ with g/κ = 2.5, and also the differences in g/κ with κs/κ = 1.0.

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Consequently, by analysis of the efficiency and performance of the QD-cavity system in terms of the fidelities, F1 and F2, from the reflection operator, \({\hat{{\rm{R}}}}_{{\rm{p}}}(\omega )\), in Eq. 19, we demonstrate that the experimental implementation of our CRk gate using QD1 and QD2 gates is feasible when there is a strong coupling strength, g ≫ (κ, γ), and the small side-leakage rate, κs ≪ κ.

Conclusions

Thus far, we have designed an optical CRk gate which can realize the controlled-rotation k (CRk) operation between two photons, using QD confined in a single-sided optical cavity. The theoretical circuit of DQFT can be directly implemented using the proposed CRk gates. In addition, we can achieve scalability in the multi-photon DQFT scheme to simply arrange the CRk (k = 2, ‥, t) gates and HWPs (i.e. three-photon DQFT scheme in Sec. 3.2) for applicable multi-qubit DQFT. Thus, in order to successfully design a practical CRk gate in the structure of DQFT, we also analyzed the efficiency and performance of the interaction in the QD-cavity system to quantify the fidelities of the QD1 and QD2 gates (components of a CRk gate), and also demonstrated a method, using strong coupling strength and small side-leakage rate as g ≫ (κ, γ) and κs ≪ κ, to improve the robustness against the affection of the vacuum noise, N(ω), for the operation of the QD-dipole and leaky modes, S(ω) (sideband leakage and absorption)40,42,53,54,55,56, as described in Sec. 4.

In the previous works of QFT (including DQFT) schemes, the various physical resources have been employed for the operation of QFT. Though the QFT schemes21,22, which are implemented using the linearly optical devices, has the advantage to realize the operation of QFT with the experimental simplicity, the performances (accuracies) of QFT schemes are probabilistic (as less than 35%)21,22. Also, the affection of the inaccuracy influences the entire performance of quantum algorithms because QFT or DQFT is a main subroutine of quantum algorithms. Therefore, many researchers have proposed the QFT schemes via the nonlinearly optical devices, such as cross-Kerr nonlinearities (XKNLs)17,25, and optical cavities15,27,31,32,33. However, the experimental realization of strong XKNL is still a big challenge75. And the decoherence effect, which can occurs the evolution of the mixed state after homodyne measurement, is also unavoidable when a coherent state is transmitted through a fiber in practice17,25,36,69. From this point of view, well-isolated qubits might not necessarily suffer from the same decoherence effects in Kerr medium. The optical cavity system, which is one of the candidates for the QFT schemes, has recently attracted extensive attention. Weinstein et al.17 and Scully et al.31 have designed to implement the operations of QFT using the Ion trap and cavity QED. But the drawback of these schemes is to require a large number of qubits for the operation of QFT. And the QFT schemes15,32,33 based on the distant atoms trapped in separate cavities could be restrictively utilized for the one-way quantum computing, due to the trapped atoms used as qubits.

Compared with the previous schemes15,17,21,22,25,27,31,32,33, our DQFT scheme consists of the QD-cavity systems (nonlinearly optical devices) to implement the CRk operations as the basic modules for the deterministic performance and simple expansion into multi-qubit DQFT. Moreover, in our scheme, the flying photons play the roles of qubits transferring quantum information, and QDs are the ancillary systems to efficiently perform the CRk operations for the applications of various quantum computations and algorithms.

Also, In view of the experimental realization, for the high fidelity (condition of g ≫ (κ, γ) and κs ≪ κ) of the interaction between a photon and the QD-cavity system, researchers have proposed a variety of experimental techniques, including the following: Rosenblum et al.68 obtained the coupling strength as g/(κ + κs) ≈ 0.5 in a micropillar cavity (d = 1.5 μm) with a quality factor Q = 8800, and for Q = 40000, the coupling strength could be increased to g/(κ + κs) ≈ 2.474. Furthermore, Arnold et al.76 enhanced the quality factor, Q = 215000 (κ ≈ 6.2 μeV), in order to acquire a small side-leakage rate. Reitzensteina et al.77 demonstrated two methods (the etching process and improving the sample growth) to reduce the side-leakage rate, κs/κ, in an In0.6Ga0.4As optical cavity having g/(κ + κs) ≈ 2.4 and Q = 40000. Thus, the QD-cavity systems in our CRk gate can be reliably operated with high fidelity, according to the results of our analysis presented in Sec. 4. Moreover, in order to ensure the reliable interaction between a flying photon and stationary qubit (electron spin in QD), the initialization (electron spin-superposition state) and manipulation of the spin state can be prepared by optical pumping or optical cooling78, and can be acquired using pulsed magnetic resonance techniques, nanosecond microwave pulses, or picosecond/femtosecond optical pulses79,80,81,82.

Therefore, we obtain the interaction of the CRk gate, using the QD-cavity system, with high fidelity for the DQFT scheme. Consequently, according to our analysis results, in practice (under vacuum noise and sideband leakage), we can obtain high efficiency and reliable performance of the interaction between a photon and QD within a single-sided optical cavity (CRk gate). Our results also indicate that our DQFT scheme using CRk gates can be experimentally feasible, as well as scalable for multi-qubit DQFT.