Error-detected N-photon cluster state generation based on the controlledphase gate using a quantum dot in an optical microcavity

Abstract

We propose a scheme for error-detected generation of an N-photon cluster state with a quantum dot (QD) embedded in a single-sided optical microcavity (QD-cavity system). The basic structure of this scheme is an error-detected controlled-phase (C-phase) gate on the hybrid electron–photon system. In this scheme, the fidelity of N-photon cluster state generation can be reached unity even if low-Q cavity and cavity leakage are considered. By using error detecting, the generation of an N-photon cluster state can be performed by repeating until success, which also leads to a high success probability, compared with other schemes assisted by the QD-cavity system. The error-detected generation of an N-photon cluster state in the highly controllable way may benefit on the quantum network in the future.

Graphical abstract

Keywords

controlled-phase gate / cluster state / error-detected / QD-cavity system

1 Introduction

Quantum computing has attracted widespread attention because of its great superiority in speed and operation. Quantum computers can handle some tasks that are difficult for classic computers [1-6]. Quantum Entanglement plays an important role in quantum computation and quantum gates, including the gates based on nuclear magnetic resonance [7], holonomic quantum computation [8-11], quantum dots [12-15], diamond nitrogen vacancy centers [16, 17], superconducting qubits [18-20], photons based on polarization states [21-26], hyperparallel photonic quantum gates [27-33], atom systems [34-36], and one-way quantum computation [37]. It is also an important resource in quantum communication, such as quantum key distribution [38-40], quantum secret sharing [41], quantum secure direct communication [42-47], quantum teleportation [48], quantum dense coding [49, 50], and quantum repeater [51-56] in which there are five core techniques for its application, including entanglement generation [57-59], entanglement distribution [60-64], entanglement purification [65-76] and entanglement concentration [77-85], entanglement swapping with entangled-state analysis [86-95], and quantum state storage.

The quantum cluster state, a special type of N-qubit entangled state, was introduced by Briegel and Raussendorf [96] in 2001. The cluster state owns the advantages of Greenberger–Horne–Zeilinger (GHZ) and W-class entangled states and has been used for one-way quantum computation [97-100]. A series of theoretical and experimental schemes for generating the cluster state have been proposed [101-105]. In 2005, Borhani and Loss [101] proposed a scheme for creating a cluster state with Heisenberg interactions. In 2008, Tokunaga et al. [102] experimentally demonstrated a four-photon cluster state by using two single photons and an entangled photon pair with a high fidelity. In the same year, Yukama et al. [103] achieved four-mode continuous-variable cluster states experimentally. In 2009, Ceccarelli et al. [59] created a six-qubit linear cluster state with a two-photon hyperentangled state. In this hyperentangled cluster state, three qubits are encoded in each particle, one in the polarization and two in the linear momentum degrees of freedom. In 2011, Yu et al. [104] proposed schemes for generating photon and electron-spin cluster states based on C-phase gate by using QD-cavity system. In 2016, Zhang et al. [105] demonstrated a high-fidelity four-photon cluster state in experiment via a spontaneous parametric down-conversion process. The scheme can be directly extended to more photons to generate an N-qubit linear cluster state. To generate a 2N-qubit linear cluster state from N pairs of entangled photons, only (N–1) Hong–Ou–Mandel interferences are needed. The success probability for 2N-qubit cluster state can reach {\left(\frac{1}{4}\right)}^N. Recently, cluster states can be created by many different physical systems, such as nitrogen–vacancy (NV) center, superconducted circuits, quantum dots, and so on.

Recently, an electron spin in a single charged quantum dot (QD) has been increasingly widespread in quantum computation [106-109], quantum network [110], and quantum information processing [111]. QD-cavity system generally includes to two different types: one is a quantum dot embedded in a single-sided optical microcavity, and the other involves a double-sided optical microcavity. With its long electron-spin coherence time [112-116], fast manipulation, and easy initialization [117-119], in recent years, QD-cavity system has developed greatly in both theoretical and experimental field [104, 120-122]. In 2008, Hu et al. proposed two schemes to entangle two remote spins [120] and realise photon entangler [121]. In 2011, Yu et al. [104] proposed schemes for constructing a controlled-phase gate and cluster states with single charged QDs in the strong-coupling regime. In ideal condition, the fidelity of the generation of cluster states can reach unity one. However, in practical condition, the parameters of the cavity decay rate, exciton dipole decay rate, and the weak coupling strength may cause the error during the input-output process, so the fidelity and the success probability would be affected by these parameters. In 2011, Kastoryano et al. [123] prepared the generation of a maximally entangled state of two atoms in an optical cavity, in which the error can be heralded. In 2012, Li et al. [124] proposed a practical scheme for single emitter-single-photon interfacing, the fidelity of which is robust even in the weak coupling regime. In 2016, Li and Deng [14] presented a robust error-rejecting quantum entangling gate based on single-sided QD-cavity system. In this scheme, the fidelity of the entangling gate can reach unit one with a detectable error. In 2017, Li et al. [15] presented some high-fidelity quantum circuits for quantum computing on electron spins of QDs embedded in low-Q optical microcavities, in which the fidelities of the quantum gates can, in principle, be robust to imperfections involved in a practical input-output process of a single photon by converting the infidelity into a heralded error. Subsequently, this error-detected method [14, 15] was applied in other tasks of quantum information processing, such as universal hyperparallel photonic quantum gates [29, 31], quantum repeaters [56], and error-detected generation and complete analysis of hyperentangled Bell states for photons [94], and hyperentanglement purification [74].

In this paper, we propose a scheme for the error-detected generation of an N-photon cluster state with the quantum dot (QD) embedded in a single-sided optical microcavity. The basic structure of this scheme is an error-detected controlled-phase gate for the hybrid electron–photon system. In this scheme, the errors during the input–output process in practical condition can be detected, which brings a almost unity fidelity and a high success probability. Our scheme for constructing an error-detected controlled-phase gate and N-photon cluster state can be performed by repeating until success. It shows that our scheme is useful in quantum information process in the future.

2 Error-detected hybrid controlled phase gate

We consider a singly charged self-assembled Ga As or In(Ga) As quantum dot(QD) embedded in a single-sided micropillar cavity as shown in Fig. 1(a). The bottom reflector of this cavity is 100% reflective while the top one is partially reflective, so that light can be coupled into and out of the cavity through the top reflector. The optical transition properties of a singly charged QD are dominated by Pauli’s exclusion principle [125], as shown in Fig. 1(b). The optical excited states are the trion states X^{-} (|\uparrow \downarrow \Uparrow \rangle and |\downarrow \uparrow \Downarrow \rangle) consisting of two antiparallel electrons and one hole [126]. |\uparrow \rangle and |\downarrow \rangle represent electron-spin states |+\frac{1}{2}\rangle and |-\frac{1}{2}\rangle. |\Uparrow \rangle and |\Downarrow \rangle represent heavy-hole spin states |+\frac{3}{2}\rangle and |-\frac{3}{2}\rangle in the valence band, respectively. If the excess electron-spin state is |\uparrow \rangle, only the right-handed circularly polarized photon |R\rangle can be resonantly absorbed to create X^{-} in the state |\uparrow \downarrow \Uparrow \rangle. If the excess electron-spin state is |\downarrow \rangle, only the left-handed circularly polarized photon |L\rangle can be resonantly absorbed to create X^{-} in the state |\downarrow \uparrow \Downarrow \rangle. The transitions between states |\uparrow \downarrow \Uparrow \rangle and |\downarrow \rangle, |\downarrow \uparrow \Downarrow \rangle and |\uparrow \rangle are dipole forbidden.

The optical property of this QD-cavity system can be obtained by solving the Heisenberg equations of the cavity field operator \widehat{a} and the dipole operator {\widehat{\sigma }}_{-} in the interaction picture as formulated in the standard textbook [127]:

\begin{array}{}\text{(1)}& \begin{array}{l}\frac{\text{d}\widehat{a}}{\text{d}t}=-\left[\text{i}({\omega }_c-\omega )+\frac{\kappa }{2}+\frac{{\kappa }_s}{2}\right]\widehat{a}-g{\widehat{\sigma }}_{-}-\sqrt{\kappa }{\widehat{a}}_{in}\mathrm{,}\\ \frac{\text{d}{\widehat{\sigma }}_{-}}{\text{d}t}=-\left[\text{i}({\omega }_{X^{-}}-\omega )+\frac{\gamma }{2}\right]{\widehat{\sigma }}_{-}-g{\widehat{\sigma }}_z\widehat{a}\mathrm{,}\\ {\widehat{a}}_{out}={\widehat{a}}_{in}+\sqrt{\kappa }\widehat{a}\mathrm{,}\end{array}\end{array}

where {\omega }_c, ω and {\omega }_{X^{-}} describe the frequencies of the cavity mode, the input photon and the trion X^{-} transition, respectively. g is the coupling strength between the QD and the cavity mode, which vanishes if the cavity mode does not match the trion X^{-} transition of the QD (so-called empty cavity or cold cavity). \gamma /2 is the X^{-} dipole decay rate, while \kappa /2 and {\kappa }_s/2 represent the cavity field decay rates into the input/output modes and the leaky modes (side leakage), respectively. {\widehat{a}}_{in} and {\widehat{a}}_{out} are the input and output field operators. From these equations, one can get the reflection coefficient [120, 121, 128, 129]:

\begin{array}{}\text{(2)}& \begin{array}{ll}{r}_1(\omega )\hfill & =\frac{\langle {\widehat{a}}_{out}\rangle }{\langle {\widehat{a}}_{in}\rangle }\hfill \\ \hfill & =1-\frac{\kappa \left[\text{i}({\omega }_{X^{-}}-\omega )+\frac{\gamma }{2}\right]}{\left[\text{i}({\omega }_{X^{-}}-\omega )+\frac{\gamma }{2}\right]\left[\text{i}({\omega }_c-\omega )+\frac{\kappa }{2}+\frac{{\kappa }_s}{2}\right]+g^2}\mathrm{.}\hfill \end{array}\end{array}

For g=0, the reflection coefficient reduces to

\begin{array}{}\text{(3)}& r_0(\omega )=\frac{\text{i}({\omega }_c-\omega )-\frac{\kappa }{2}+\frac{{\kappa }_s}{2}}{\text{i}({\omega }_c-\omega )+\frac{\kappa }{2}+\frac{{\kappa }_s}{2}}\mathrm{.}\end{array}

Suppose initially the excess electron-spin state is |{\phi }_s \rangle =\frac{1}{\sqrt{2}}(|\uparrow \rangle +|\downarrow \rangle ), while the input photon lies in the superposition of the circularly polarized state |{\phi }_p \rangle =\frac{1}{\sqrt{2}}(|R\rangle +|L\rangle ). After the photon is reflected by the QD-cavity, the system composed of the photon and the excess electron spin evolves to the state

\begin{array}{}\text{(4)}& |\text{Φ}\rangle =\frac{1}{2}[(r_1|\uparrow \rangle +r_0|\downarrow \rangle )\otimes |R\rangle +(r_0|\uparrow \rangle +r_1|\downarrow \rangle )\otimes |L\rangle ]\mathrm{.}\end{array}

With the interaction process described above, we can construct an error-detected C-phase gate on the hybrid electron-photon system, as shown in Fig. 2. It contains a QD-cavity, three CPBSs (circularly polarized beam splitters) which transmit |R\rangle polarization of photons and reflect |L\rangle polarization of photons, a single-photon detector D, an attenuator whose decay rate is \frac{r_1-r_0}{2} and three HWPs (half-wave plates). Among these HWPs, HWP₁ and HWP₂ are half-wave plates set at 22.5° to complete the hadamard operation. HWP₃ is set at 45° to flip the state of the circularly polarized photon. Suppose the input photon state is |{\phi }_p\rangle =\frac{1}{\sqrt{2}}(|R\rangle +|L\rangle ), while the electron spin state is |{\phi }_s \rangle \text{}=\text{}\frac{1}{\sqrt{2}}( |\uparrow \rangle + |\downarrow \rangle ). After the photon passes through the CPBS₁ (which divides the photon into two paths a, b) and HWP₁ [which performs the hadamard operation |R\rangle \to \frac{1}{\sqrt{2}}(|R\rangle +|L\rangle )\mathrm{,}|L\rangle \to \frac{1}{\sqrt{2}}(|R\rangle -|L\rangle )], the state of the system composed of the QD and the input photon evolves into

\begin{array}{}\text{(5)}& |\text{Φ}{\rangle }_1=\frac{1}{2}(|\uparrow \rangle +|\downarrow \rangle )\otimes \left[\frac{1}{\sqrt{2}}(|R_a\rangle +|L_a\rangle )+|L_b\rangle \right]\mathrm{.}\end{array}

Here |R_a\rangle and |L_a\rangle denote the |R\rangle and |L\rangle polarizations of a photon through path a which is generated from |R\rangle polarization of the photon by HWP₁, while |L_b\rangle denotes the reflected component of the photon from CPBS₁ through path b. After the photon interacts with the QD-cavity, the state of the system composed of the QD and the input photon changes to

\begin{array}{}\text{(6)}& \begin{array}{ll}|\text{Φ}{\rangle }_2=\hfill & \frac{1}{2}[\frac{1}{\sqrt{2}}(r_1|\uparrow \rangle +r_0|\downarrow \rangle )\otimes |R_a\rangle \hfill \\ \hfill & +\frac{1}{\sqrt{2}}(r_0|\uparrow \rangle +r_1|\downarrow \rangle )\otimes |L_a\rangle \text{}+\text{}(|\uparrow \rangle \text{}+\text{}|\downarrow \rangle )\otimes |L_b\rangle ]\mathrm{.}\hfill \end{array}\end{array}

Let the photon pass through HWP₂, which applies the hadamard operation on the photon, then the state of the hybrid electron-photon system evolves into

\begin{array}{}\text{(7)}& \begin{array}{ll}|\text{Φ}{\rangle }_3\hfill & =\frac{1}{2}\left[\frac{1}{\sqrt{2}}\cdot \frac{1}{\sqrt{2}}(r_1|\uparrow \rangle +r_0|\downarrow \rangle )(|R_a\rangle +|L_a\rangle )+\frac{1}{\sqrt{2}}\cdot \frac{1}{\sqrt{2}}(r_0|\uparrow \rangle +r_1|\downarrow \rangle )(|R_a\rangle -|L_a\rangle )+(|\uparrow \rangle +|\downarrow \rangle )\otimes |L_b\rangle \right]\hfill \\ \hfill & =\frac{1}{2}\left[\frac{1}{2}(r_1+r_0)(|\uparrow \rangle +|\downarrow \rangle )\otimes |R_a\rangle +\frac{1}{2}(r_1-r_0)(|\uparrow \rangle -|\downarrow \rangle )\otimes |L_a\rangle +(|\uparrow \rangle +|\downarrow \rangle )\otimes |L_b\rangle \right]\mathrm{.}\hfill \end{array}\end{array}

After the photon passes through HWP₃ which performs the bit-flip operator \widehat{X}=|R\rangle \langle L|+|L\rangle \langle R| on the photon, the state of the hybrid electron–photon system becomes

\begin{array}{}\text{(8)}& \begin{array}{ll}|\text{Φ}{\rangle }_4\text{}=\hfill & \frac{1}{2}[\frac{1}{2}(r_1+r_0)(|\uparrow \rangle +|\downarrow \rangle )\otimes |L_a\rangle \hfill \\ \hfill & +\frac{1}{2}(r_1\text{}-\text{}{r}_0)(|\uparrow \rangle \text{}-\text{}|\downarrow \rangle )\text{}\otimes \text{}|R_a\rangle \text{}+\text{}(|\uparrow \rangle \text{}+\text{}|\downarrow \rangle )\otimes |L_b\rangle ]\mathrm{.}\hfill \end{array}\end{array}

Then let |L_b\rangle pass through the attenuator whose decay rate is \frac{r_1-r_0}{2}. In the mean time, |L_a\rangle will be reflected by CPBS₂. After that, the state of the hybrid electron-photon system evolves into

\begin{array}{}\text{(9)}& \begin{array}{ll}|\text{Φ}{\rangle }_5=\hfill & \frac{(r_1\text{}-\text{}{r}_0)}{4}\left[(|\uparrow \rangle +|\downarrow \rangle )\otimes |L_b\rangle +(|\uparrow \rangle \text{}-\text{}|\downarrow \rangle )\otimes |R_a\rangle \right]\hfill \\ \hfill & +\frac{(r_1+r_0)}{4}(|\uparrow \rangle +|\downarrow \rangle )\otimes |L_a\rangle \mathrm{.}\hfill \end{array}\end{array}

If the photon passes through CPBS₃ without being detected by photon detector D, a single-qubit operation {\widehat{\sigma }}_z^s\left({\widehat{\sigma }}_z^s=|\uparrow \rangle \langle \uparrow |-|\downarrow \rangle \langle \downarrow |\right) on the electron spin will evolve the system into

\begin{array}{}\text{(10)}& \begin{array}{ll}|\text{Φ}{\rangle }_6\hfill & =\frac{r_1-r_0}{4}[(|\uparrow \rangle -|\downarrow \rangle )\otimes |L\rangle +(|\uparrow \rangle +|\downarrow \rangle )\otimes |R\rangle ]\hfill \\ \hfill & =\frac{r_1-r_0}{4}[|\uparrow \rangle |L\rangle -|\downarrow \rangle |L\rangle +|\uparrow \rangle |R\rangle +|\downarrow \rangle |R\rangle ]\hfill \\ \hfill & =\frac{r_1-r_0}{4}[(|\uparrow \rangle +|\downarrow \rangle {\widehat{\sigma }}_z^p)\otimes (|R\rangle +|L\rangle )]\mathrm{.}\hfill \end{array}\end{array}

Here {\widehat{\sigma }}_z^p=|R\rangle \langle R|-|L\rangle \langle L|. If the photon detector clicks, the state of the hybrid electron-photon system will project to |\text{Φ}{\rangle }_6^{\prime }=\frac{r_1+r_0}{4}\left(|\uparrow \rangle +|\downarrow \rangle \right)\otimes |L_a\rangle.

The Eq. (10) represents an error-detected C-phase gate on the hybrid electron-photon system with the electron as a controlling qubit while the photon as the targeting qubit. |L_a\rangle in the last term on the right-hand side of Eq. (9) (|\text{Φ}{\rangle }_6^{\prime }) is the left-handed polarization component which passed through the QD-cavity and triggered a click on a photon detector. The detection of the |L_a\rangle signals an error caused by the unbalance of the two reflection coefficients r₁ and r₀ that may originate from the low-Q cavity (or the mismatch between the incident photon field and the cavity mode). Fortunately, under this circumstance, the state of the electron spin is the same as the original state as if there was no interaction with the photon. If the C-phase gate fails, one can input another photon to repeat above process until the single photon detector D will not click.

3 N-photon cluster state generation

With the help of error-detected hybrid controlled phase gate, one can get an N-photon cluster state with high fidelity. A cluster state is a multipartite entangled state which has special features suitable for implementing a quantum computer on a network, and it is first defined by Briegel and Raussendorf [96] in 2001. An N-photon cluster state is defined as |\text{Φ}\rangle =(|R_N\rangle +|L_N\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-1})\otimes (|R_{N-1}\rangle +|L_{N-1}\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-2})\dots \otimes (|R_1\rangle +|L_1\rangle ). The schematic diagram of our error-detected N-photon cluster state generation is shown in Fig. 3, which consists of the error-detected C-phase gate and a HWP (half-wave plate). The electron spin lies in the state \frac{1}{\sqrt{2}}(|\uparrow \rangle +|\downarrow \rangle ). The N photons with the same frequency are initially in the linear polarized state \frac{1}{\sqrt{2}}(|R_i\rangle +|L_i\rangle ) (here i=\mathrm{1,2,}\dots \mathrm{,}N), respectively. They are injected in sequence into the QD-cavity. The time interval between two photons is \text{Δ}t, which is required to be less than the electron spin coherence time. The detailed process to generate N-photon cluster state is described as follows.

When the first photon passes through the error-detected C-phase gate block until CPBS₃ without being detected by photon detector D, the state of the hybrid electron-photon system becomes

\begin{array}{}\text{(11)}& |\text{Φ}{\rangle }_7=\frac{r_1-r_0}{4}[(|\uparrow \rangle +|\downarrow \rangle {\widehat{\sigma }}_z^{p\mathrm{,1}})\otimes (|R_1\rangle +|L_1\rangle )]\mathrm{.}\end{array}

Then the second photon is put into the setup. After it is reflected by the QD-cavity and passes through these linear optical elements until CPBS₃ without being detected by photon detector D, the output state of the system contains two photons and one electron changes to

\begin{array}{}\text{(12)}& \begin{array}{ll}|\text{Φ}{\rangle }_8=\hfill & \frac{1}{\sqrt{2}}\cdot {\left(\frac{r_1\text{}-\text{}{r}_0}{2\sqrt{2}}\right)}^2[|\uparrow \rangle \text{}\otimes \left(|R_2\rangle \text{}+\text{}|L_2\rangle )\text{}\otimes \text{}(|R_1\rangle \text{}+\text{}|L_1\rangle \right)\hfill \\ \hfill & +|\downarrow \rangle \otimes ((|R_2\rangle -|L_2\rangle )\otimes {\widehat{\sigma }}_z^{p\mathrm{,1}}(|R_1\rangle +|L_1\rangle ]\mathrm{.}\hfill \end{array}\end{array}

After the second photon passes through the HWP₄ which performs the hadamard operation on this photon, a \pi /2 microwave pulse is added on the electron spin which performs a hadamard operation \left[|\uparrow \rangle \to \frac{1}{\sqrt{2}}\left(|\uparrow \rangle +|\downarrow \rangle \right)\mathrm{,}|\downarrow \rangle \to \frac{1}{\sqrt{2}}\left(|\uparrow \rangle -|\downarrow \rangle \right)\right] on the electron, and the state of the system contains two photons and one electron becomes

\begin{array}{}\text{(13)}& \begin{array}{ll}|\text{Φ}{\rangle }_9\hfill & =\frac{1}{\sqrt{2}}\cdot {\left(\frac{r_1-r_0}{2\sqrt{2}}\right)}^2[\left(|\uparrow \rangle +|\downarrow \rangle {\widehat{\sigma }}_z^{p\mathrm{,2}}\right)\hfill \\ \hfill & \otimes \left(|R_2\rangle +|L_2\rangle {\widehat{\sigma }}_z^{p\mathrm{,1}}\right)\otimes \left(|R_1\rangle +|L_1\rangle \right)]\mathrm{.}\hfill \end{array}\end{array}

Here the term on the right-handed side of the Eq. (13) contains two C-phase gates. One is the C-phase gate on the hybrid electron-photon system, and another one is the C-phase gate between two photons. If there is a click on photon detector D during the second photon passing through the setup, the system state will project to

\begin{array}{}\text{(14)}& \begin{array}{ll}|\text{Φ}{\rangle }_9^{\prime }=\hfill & \frac{1}{\sqrt{2}}\cdot \frac{(r_1-r_0)(r_1+r_0)}{{\left(2\sqrt{2}\right)}^2}[\left(|\uparrow \rangle +|\downarrow \rangle {\widehat{\sigma }}_z^{p\mathrm{,1}}\right)\hfill \\ \hfill & \otimes |L_{2a}\rangle \otimes \left(|R_1\rangle +|L_1\rangle \right)]\hfill \end{array}\end{array}

The click of the photon detector D signals there is an error during the interaction between the second photon and the electron. After the detection on the second photon, the state of the system contains the first photon and the electron will be projected to the state in Eq. (11), so one can put another photon as the second one into this setup without initiating the electron-spin state.

The remaining photons are injected into the setup and repeat the above process as shown in Fig. 3. After the N photons pass through the setup shown in Fig. 3, if there is no click on the photon detector D, the state of the system contains N photons and one electron becomes

\begin{array}{}\text{(15)}& \begin{array}{ll}|\text{Φ}{\rangle }_{10}=\hfill & \frac{1}{\sqrt{2}}\cdot {\left(\frac{r_1-r_0}{2\sqrt{2}}\right)}^N|\uparrow \rangle [(|R_N\rangle +|L_N\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-1})\hfill \\ \hfill & \otimes (|R_{N-1}\rangle \text{}+\text{}|L_{N-1}\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-2})\dots \otimes (|R_1\rangle \text{}+\text{}|L_1\rangle )]\hfill \\ \hfill & +\frac{1}{\sqrt{2}}\cdot {\left(\frac{r_1-r_0}{2\sqrt{2}}\right)}^N|\downarrow \rangle [(|R_N\rangle -|L_N\rangle {\widehat{\sigma }}_z^{p\mathrm{,}N-1})\hfill \\ \hfill & \otimes (|R_{N-1}\rangle +|L_{N-1}\rangle {\widehat{\sigma }}_z^{p\mathrm{,}N-2})\dots \otimes (|R_1\rangle +|L_1\rangle )]\mathrm{.}\hfill \end{array}\end{array}

Then one can obtain the N-photon cluster state from above equation by measuring the electron spin state in two orthogonal bases (|\uparrow \rangle, |\downarrow \rangle). If the electron spin state lies in the state |\uparrow \rangle, the N-photon state will collapse to

\begin{array}{}\text{(16)}& \begin{array}{ll}|{\psi }_N\rangle =\hfill & {\left(\frac{r_1-r_0}{2\sqrt{2}}\right)}^N[(|R_N\rangle +|L_N\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-1})\hfill \\ \hfill & \otimes (|R_{N-1}\rangle \text{}+\text{}|L_{N-1}\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-2})\dots \otimes (|R_1\rangle \text{}+\text{}|L_1\rangle )]\mathrm{.}\hfill \end{array}\end{array}

If the measurement result is |\downarrow \rangle, the N-photon state can be written as

\begin{array}{}\text{(17)}& \begin{array}{ll}|{\psi }_N^{\prime }\rangle =\hfill & {\left(\frac{r_1-r_0}{2\sqrt{2}}\right)}^N[(|R_N\rangle -|L_N\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-1})\hfill \\ \hfill & \otimes (|R_{N-1}\rangle \text{}+\text{}|L_{N-1}\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}N-2})\dots \otimes (|R_1\rangle \text{}+\text{}|L_1\rangle )]\mathrm{.}\hfill \end{array}\end{array}

For the latter situation in Eq. (17), one can perform a single-qubit operation {\widehat{\sigma }}_z (the phase-flip operation {\widehat{\sigma }}_z=|R\rangle \langle R|-|L\rangle \langle L|) on the Nth photon. This phase-flip operation can be achieved with 3 HWPs, two of which are set at 22.5° to perform a hadamard operation on the photon, while the last one is set at 45° to perform a bit-flip operation X=|R\rangle \langle L|+|L\rangle \langle R| on the photon. Then the above state |{\psi }_N^{\prime }\rangle can be changed to the state |{\psi }_N\rangle, which is the result aimed to achieve. If the photon detector D clicks during above process, the system state will project into

\begin{array}{}\text{(18)}& \begin{array}{ll}|\text{Φ}{\rangle }_{11}=\hfill & \frac{1}{\sqrt{2}}\cdot \frac{r_1+r_0}{2\sqrt{2}}{\left(\frac{r_1-r_0}{2\sqrt{2}}\right)}^{j-1}(|\uparrow \rangle +|\downarrow \rangle )\hfill \\ \hfill & \otimes |L_{ja}\rangle \otimes (|R_{j-1}\rangle +|L_{j-1}\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}j-2})\hfill \\ \hfill & \otimes (|R_{j-2}\rangle +|L_{j-2}\rangle \cdot {\widehat{\sigma }}_z^{p\mathrm{,}j-3})\otimes \dots (|R_1\rangle \text{}+\text{}|L_1\rangle )\mathrm{,}\hfill \end{array}\end{array}

here j\in N.

Eq. (16) describes an N-photon cluster state. With this scheme, one can get an N-photon cluster state no matter which state the electron state is measured on. While the Eq. (18) (|\text{Φ}{\rangle }_{11}) contains |L_{ja}\rangle which is the left-handed polarization component passed through the QD-cavity. It leads to click on the photon detector D which means there are errors during the interaction between the photon and the electron. In this condition, one can put another photon which lies in the state \frac{1}{\sqrt{2}}\left(|R\rangle +|L\rangle \right) to repeat the above process until the cluster state is generated successfully.

4 Discussion and summary

The fidelity for the generation of N-photon cluster state is defined as F=|\langle {\psi }_r|{\psi }_i\rangle {|}^2. |{\psi }_i\rangle and |{\psi }_r\rangle are the final states in the ideal condition and realistic condition, respectively. In our scheme, the N-photon cluster state generation can be achieved with the unity fidelity in both the ideal condition and the realistic condition. Therefore, the fidelity is independent of the side leakage {\kappa }_s/\kappa, the detuning ({\omega }_c-\omega )/\kappa, the number of photons N, and the cooperativity C\equiv g^2/{\kappa }_T, here {\kappa }_T={\kappa }_s+\kappa. It benefits from our error-detected procedure. Specifically, when there are errors caused by imperfect input-output process, these errors can be detected by photon detector D without affecting the output state. The output state means the the state of the system contains N photons without detected and one electron in the QD cavity. That means the fidelity is not affected by these errors caused by realistic input-output process such as the cavity decay, nonzero bandwidth, photon loss, and the finite coupling, which is far different from other schemes for generating cluster state [130]. If the photon detector D clicks, one can input another photon to repeat the process without reinitialization of the QD since the electron spin still lies in the initial states.

In our scheme, the probability to get the N-photon cluster state successfully is

\begin{array}{}\text{(19)}& {\eta }_s={\left(\frac{r_1-r_0}{2}\right)}^{2N}\mathrm{.}\end{array}

While the heralded error probability of this scheme is

\begin{array}{}\text{(20)}& {\eta }_e={\displaystyle \sum _{j=1}^N}\frac{{\left(r_1+r_0\right)}^2}{8}{\left(\frac{r_1-r_0}{2}\right)}^{2j-2}\mathrm{.}\end{array}

In this case, the electron spin can be directly used in the recycling procedure. By taking the recycling procedure into account, the success probability η of our error-rejecting N-photon cluster state is defined as

\begin{array}{}\text{(21)}& \eta =\frac{{\eta }_s}{1-{\eta }_e}\mathrm{.}\end{array}

The success probability of above N-photon cluster state varies with the side leakage {\kappa }_s/\kappa, the error rate of the detuning ({\omega }_c-\omega )/\kappa, the number of photons N and the cooperativity C\equiv {\text{g}}^2/\gamma {\kappa }_T, here {\kappa }_T={\kappa }_s+\kappa, as shown in Fig. 3. In the ideal condition when the QD is strongly coupled to the cavity, C\gg 1. For the low-Q cavity (resonantly scattering regime), the typical value of C is 1/4. Here we let the cooperativity C=1 as a normal value of the Purcell regime. We set \gamma /\kappa =0.1 where γ is typically \ll 0.1\kappa for most QD-cavity experiments [131-133]. As for the spin coherent time, it can reach 0.44 μs in GaAs QDs, which is much longer than the time scale for generating entanglement [134], so the error-detected cluster state generation scheme can work many times before the decoherence time. And we let the number of photon N be 2 (black solid line), 3 (red dotted line) and 4 (blue dashed line), respectively. Figure 4 shows that the success probability decreases as the side leakage {\kappa }_s/\kappa and the particle number N increasing. When the detuning ({\omega }_c-\omega )/\kappa =0.1, the success probability to get the 3-photon cluster state can reach about 0.24. When the detuning ({\omega }_c-\omega )/\kappa =0, the success probability to get the 3-photon cluster state can get 0.27.

The total success probability of our N-photon cluster state generation scheme is affected by error items during practical input–output process. Luckily, error items lead to clicks on the photon detector D, which signal the restart of our setup. By repeating interaction procedure until cluster state is prepared successfully, the success probability can be improved largely especially in weak coupling situation. More than that, our scheme only use single-sided optical microcavities, which only have unilateral input–output processes. Compared with other existing double-sided QD-cavity systems or other physical systems [130, 135], our scheme has a higher success probability.

In summary, we have proposed a scheme for generating an N-photon cluster state with error-detected method. The basic structure of this scheme is an error-detected controlled-phase gate for the hybrid electron-photon system. Our scheme considers the errors that may occur during the interaction process, and these errors lead to a click on the photon detector. It results a higher success probability compared with the other existing schemes by introducing the recycling procedure. More than that, when there are errors, the electron spin state has been not changed, one can put another photon without initialize the electron of the QD-cavity. These advantages make our schemes for generating N-photon cluster state more useful in the quantum computation and quantum network in the future.

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