Efficient generation of Greenberger–Horne–Zeilinger states of N driven qubits mediated by a cavity

Abstract

In this paper, we propose a protocol for generating N-qubit Greenberger–Horne–Zeilinger (GHZ) states in a superconducting system. The physical model contains N superconducting flux qubits and a cavity, where the N superconducting flux qubits are coupled to the cavity simultaneously. The preparation of GHZ states can be realized in one step by applying a pair of classical fields to each qubit. The protocol is promising for generating N-qubit GHZ states because the operation time is independent of the number of the qubits in the strong drive regime. Furthermore, we consider the effects of random noise and decoherence, and numerical simulations show that the protocol is insensitive to these disturbing factors. Therefore, the protocol may be useful in the generation of GHZ states.

Appendices

Appendix A: The calculation of the system evolution

Assuming that

$$\begin{aligned} |+\rangle _n=\frac{1}{\sqrt{2}}\left( |e\rangle _n+|g\rangle _n\right) ,\nonumber \\ |-\rangle _n=\frac{1}{\sqrt{2}}\left( |e\rangle _n-|g\rangle _n\right) , \end{aligned}$$

(A.1)

then, we get

$$\begin{aligned} |g\rangle _n=\frac{1}{\sqrt{2}}\left( |+\rangle _n-|-\rangle _n\right) ,\nonumber \\ |e\rangle _n=\frac{1}{\sqrt{2}}\left( |+\rangle _n+|-\rangle _n\right) . \end{aligned}$$

(A.2)

We define the symmetric Dicke state with k qubits being in the $|+\rangle$ as

$$\begin{aligned} \left| \frac{N}{2},-\frac{N}{2}+k\right\rangle =\left( C_N^k\right) ^{-1/2}\sum _jP_j\left( |+_1,+_2,\ldots ,+_k,-_{k+1},-_{k+2},\ldots ,-_{N}\rangle \right) ,\nonumber \\ \end{aligned}$$

(A.3)

where $\{P_j\}$ denotes the set of all distinct permutation of the qubits. Therefore, the tensor product states of N qubits are

$$\begin{aligned} \bigotimes _{n=1}^{N}|g\rangle _n= & {} \bigotimes _{n=1}^{N}\frac{1}{\sqrt{2}}\left( |+\rangle _n-|-\rangle _n\right) , \nonumber \\= & {} \frac{1}{\sqrt{2^N}}\sum _{k=0}^{N}\sqrt{C_N^k}(-1)^k\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle , \nonumber \\= & {} \frac{1}{\sqrt{2^N}}\left( \sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle -\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) ,\nonumber \\ \bigotimes _{n=1}^{N}|e\rangle _n= & {} \bigotimes _{n=1}^{N}\frac{1}{\sqrt{2}}\left( |+\rangle _n+|-\rangle _n\right) ,\nonumber \\= & {} \frac{1}{\sqrt{2^N}}\sum _{k=0}^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle ,\nonumber \\= & {} \frac{1}{\sqrt{2^N}}\left( \sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle +\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) ,\nonumber \\ \end{aligned}$$

(A.4)

respectively. According to Eq. (A.4), when the evolution operator $U(\tau )$ acts on the initial state $|\psi _0\rangle$, for $N=2\eta$, the final state of the system $\vert \psi _f\rangle$ can be written as

$$\begin{aligned} \vert \psi _f\rangle= & {} U(\tau )|\psi _0\rangle \nonumber \\= & {} \exp \left[ -i \phi S_x\right] \exp \left[ i \frac{\pi }{2}S_x^2\right] \left( \bigotimes _{n=1}^{N}\left| g\rangle _n\right) |0\right\rangle _c\nonumber \\= & {} \exp \left[ -i \phi S_x\right] \exp \left[ i \frac{\pi }{2}S_x^2\right] \nonumber \\{} & {} \frac{1}{\sqrt{2^N}}\left( \sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right. \nonumber \\{} & {} \left. -\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c,\nonumber \\= & {} \exp \left[ -i \phi S_x\right] \frac{1}{\sqrt{2^N}}\nonumber \\{} & {} \left( \sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}e^{i\frac{\pi }{2}(k-\eta )^2}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right. \nonumber \\{} & {} \left. -\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}e^{i\frac{\pi }{2}(k-\eta )^2}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c.\ \ \ \ \end{aligned}$$

(A.5)

When $\eta$ is odd, Eq. (A.5) can be reduced to

$$\begin{aligned} \vert \psi _f\rangle= & {} U(\tau )|\psi _0\rangle \nonumber \\= & {} \exp \left[ -i \phi S_x\right] \frac{1}{\sqrt{2^N}}\left( i\sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}|\frac{N}{2},\right. \nonumber \\{} & {} \left. -\frac{N}{2}+k\rangle -\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c,\nonumber \\= & {} \exp \left[ -i \phi S_x\right] \left[ \frac{i}{2}\big (\bigotimes _{n=1}^{N}|g\rangle _n+\bigotimes _{n=1}^{N}|e\rangle _n\big ) -\frac{1}{2}\left( \bigotimes _{n=1}^{N}|e\rangle _n-\bigotimes _{n=1}^{N}|g\rangle _n\right) \right] |0\rangle _c,\nonumber \\= & {} \exp \left[ -i \phi S_x\right] \left[ \frac{i+1}{2}\bigotimes _{n=1}^{N}|g\rangle _n+\frac{i-1}{2}\bigotimes _{n=1}^{N}|e\rangle _n\right] |0\rangle _c,\nonumber \\{} & {} \Rightarrow \exp \left[ -i \phi S_x\right] \frac{1}{\sqrt{2}}\left( \bigotimes _{n=1}^{N}|g\rangle _n+i \bigotimes _{n=1}^{N}|e\rangle _n\right) |0\rangle _c. \end{aligned}$$

(A.6)

When $\eta$ is even, Eq. (A.5) can be reduced to

$$\begin{aligned} \vert \psi _f\rangle= & {} U(\tau )|\psi _0\rangle \nonumber \\= & {} \exp \left[ -i \phi S_x\right] \frac{1}{\sqrt{2^N}}\left( \sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}\bigg |\frac{N}{2},\right. \nonumber \\{} & {} \left. -\frac{N}{2}+k\bigg \rangle -i\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}|\frac{N}{2},-\frac{N}{2}+k\rangle \right) |0\rangle _c,\nonumber \\= & {} \exp \left[ -i \phi S_x\right] \left[ \frac{1}{2}\left( \bigotimes _{n=1}^{N}|g\rangle _n+\bigotimes _{n=1}^{N}|e\rangle _n\right) -\frac{i}{2}\left( \bigotimes _{n=1}^{N}|e\rangle _n-\bigotimes _{n=1}^{N}|g\rangle _n\right) \right] |0\rangle _c, \nonumber \\= & {} \exp \left[ -i \phi S_x\right] \left[ \frac{i+1}{2}\bigotimes _{n=1}^{N}|g\rangle _n+\frac{1-i}{2}\bigotimes _{n=1}^{N}|e\rangle _n\right] |0\rangle _c,\nonumber \\{} & {} \Rightarrow \exp \left[ -i \phi S_x\right] \frac{1}{\sqrt{2}}\left( \bigotimes _{n=1}^{N}\left| g\rangle _n-i\bigotimes _{n=1}^{N}|e\rangle _n\right) |0\right\rangle _c. \end{aligned}$$

(A.7)

Therefore, for the case that N is even, setting $\phi =0$, the N-qubit GHZ states can be obtained.

For $N=2\eta -1$, the final state of the system $\vert \psi _f\rangle$ can be written as

$$\begin{aligned} \vert \psi _f\rangle= & {} U(\tau )|\psi _0\rangle \nonumber \\= & {} \exp \left[ -i \phi S_x\right] \exp \left[ i \frac{\pi }{2}S_x^2\right] \left( \bigotimes _{n=1}^{N}|g\rangle _n\right) |0\rangle _c \nonumber \\= & {} \exp \left[ -i \phi S_x\right] \exp \left[ i \frac{\pi }{2}S_x^2\right] \frac{1}{\sqrt{2^N}}\nonumber \\{} & {} \left( \sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle -\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c,\nonumber \\= & {} \exp \left[ -i \phi S_x\right] \frac{1}{\sqrt{2^N}}\nonumber \\{} & {} \left( \sum _{k=0,2,4\ldots }^{N}\sqrt{C_N^k}e^{i\frac{\pi }{2}\left( k-\eta -\frac{1}{2}\right) ^2}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right. \nonumber \\{} & {} \left. -\sum _{k=1,3,5\ldots }^{N}\sqrt{C_N^k}e^{i\frac{\pi }{2}\left( k-\eta -\frac{1}{2}\right) ^2}\left| \frac{N}{2}, -\frac{N}{2}+k\right\rangle \right) |0\rangle _c,\nonumber \\= & {} \exp \left[ -i \phi S_x\right] \exp \left[ -i \frac{\pi }{2}S_x\right] \frac{1}{\sqrt{2^N}}\nonumber \\{} & {} \left( \sum _{k=0,2,4\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}e^{i\frac{\pi }{2}(k-\eta )^2}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right. \nonumber \\{} & {} \left. -\sum _{k=1,3,5\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}e^{i\frac{\pi }{2}(k-\eta )^2}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c. \end{aligned}$$

(A.8)

Thus, for the case that N is odd, setting $\phi =-\frac{\pi }{2}$, Eq. (A.8) is reduced to

$$\begin{aligned} \vert \psi _f\rangle= & {} \frac{1}{\sqrt{2^N}}\left( \sum _{k=0,2,4\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}e^{i\frac{\pi }{2}(k-\eta )^2}\bigg |\frac{N}{2},\right. \nonumber \\{} & {} \left. -\frac{N}{2}+k\bigg \rangle \ -\sum _{k=1,3,5\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}e^{i\frac{\pi }{2}(k-\eta )^2}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c. \end{aligned}$$

(A.9)

When $\eta$ is odd, Eq. (A.9) can be reduced to

$$\begin{aligned} \vert \psi _f\rangle= & {} \frac{1}{\sqrt{2^N}}\left( i\sum _{k=0,2,4\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \ -\sum _{k=1,3,5\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c, \nonumber \\= & {} \left[ \frac{1}{2}\left( \bigotimes _{n=1}^{N}|g\rangle _n+\bigotimes _{n=1}^{N}|e\rangle _n\right) -\frac{i}{2}\left( \bigotimes _{n=1}^{N}|e\rangle _n-\bigotimes _{n=1}^{N}|g\rangle _n\right) \right] |0\rangle _c, \nonumber \\= & {} \left[ \frac{i+1}{2}\bigotimes _{n=1}^{N}|g\rangle _n+\frac{1-i}{2}\bigotimes _{n=1}^{N}|e\rangle _n\right] |0\rangle _c,\nonumber \\{} & {} \Rightarrow \frac{1}{\sqrt{2}}\left( \bigotimes _{n=1}^{N}|g\rangle _n-i\bigotimes _{n=1}^{N}|e\rangle _n\right) |0\rangle _c. \end{aligned}$$

(A.10)

When $\eta$ is even, Eq. (A.9) can be reduced to

$$\begin{aligned} \vert \psi _f\rangle= & {} \frac{1}{\sqrt{2^N}}\left( \sum _{k=0,2,4\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \ -i\sum _{k=1,3,5\ldots }^{N}\!\!\!\!\!\sqrt{C_N^k}\left| \frac{N}{2},-\frac{N}{2}+k\right\rangle \right) |0\rangle _c,\nonumber \\= & {} \left[ \frac{i}{2}\left( \bigotimes _{n=1}^{N}|g\rangle _n+\bigotimes _{n=1}^{N}|e\rangle _n\right) -\frac{1}{2}\left( \bigotimes _{n=1}^{N}|e\rangle _n-\bigotimes _{n=1}^{N}|g\rangle _n\right) \right] |0\rangle _c, \nonumber \\= & {} \left[ \frac{i+1}{2}\bigotimes _{n=1}^{N}|g\rangle _n+\frac{i-1}{2}\bigotimes _{n=1}^{N}|e\rangle _n\right] |0\rangle _c,\nonumber \\{} & {} \Rightarrow \frac{1}{\sqrt{2}}\left( \bigotimes _{n=1}^{N}|g\rangle _n+i \bigotimes _{n=1}^{N}|e\rangle _n\right) |0\rangle _c. \end{aligned}$$

(A.11)

Therefore, the N-qubit GHZ states can be obtained. The calculations are the same as that in the case where N is even.

Appendix B: The derivation of the evolution operator

Due to

$$\begin{aligned} R_2=\exp \left[ \frac{-gS_x\left( a^{\dag }-a\right) }{\Delta }\right] ,\ \ \ U'_R(t)=\exp \left[ -i\left( \Delta t a^{\dag }a-\frac{g^2t}{\Delta }S_x^2+\Omega 'tS_x\right) \right] ,\nonumber \\ \end{aligned}$$

(B1)

we have

$$\begin{aligned} U(t)= & {} R_2U'_RR_2^{\dag } \nonumber \\= & {} \exp \left[ \frac{-gS_x\left( a^{\dag }-a\right) }{\Delta }\right] \exp \left[ -i\left( \Delta t a^{\dag }a-\frac{g^2t}{\Delta }S_x^2+\Omega 'tS_x\right) \right] \nonumber \\{} & {} \exp \left[ \frac{-gS_x\left( a-a^{\dag }\right) }{\Delta }\right] \nonumber \\= & {} \exp \left[ \frac{-gS_x\left( a^{\dag }-a\right) }{\Delta }\right] \exp \left[ -i \Delta a^{\dag }at\right] \exp \left[ \frac{-gS_x\left( a-a^{\dag }\right) }{\Delta }\right] \nonumber \\{} & {} \exp \left[ \frac{i g^2t S_x^2}{\Delta }\right] \exp \left[ -i \Omega ' t S_x\right] \nonumber \\= & {} \exp \left[ -i \Delta a^{\dag }at\right] \nonumber \\{} & {} \exp \left[ \frac{-gS_x}{\Delta }\left( a^{\dag }e^{i \Delta t}-ae^{-i \Delta t}\right) \right] \nonumber \\{} & {} \exp \left[ \frac{-gS_x\left( a-a^{\dag }\right) }{\Delta }\right] \exp \left[ \frac{i g^2t S_x^2}{\Delta }\right] \exp \left[ -i \Omega ' t S_x\right] \nonumber \\= & {} \exp \left[ -i \Delta a^{\dag }at\right] \exp \left[ \frac{-gS_x}{\Delta }\left[ a^{\dag }\left( e^{i \Delta t}-1\right) -a\left( e^{-i \Delta t}-1\right) \right] \right] \nonumber \\{} & {} \exp \left[ \frac{g^2S_x^2}{2\Delta }\left( e^{-i \Delta t}-e^{i \Delta t}\right) \right] \exp \left[ \frac{i g^2t S_x^2}{\Delta }\right] \nonumber \\{} & {} \times \exp \left[ -i \Omega ' t S_x\right] .\ \ \ \ \end{aligned}$$

(B2)

Here we used the relations

$$\begin{aligned}{} & {} \exp \left[ \frac{-gS_x(a^{\dag }-a)}{\Delta }\right] \exp \left[ -i \Delta a^{\dag }at\right] \nonumber \\{} & {} \quad =\exp \left[ -i \Delta a^{\dag }at\right] \exp \left[ \frac{-gS_x}{\Delta }\left( a^{\dag }e^{i \Delta t}-ae^{-i \Delta t}\right) \right] . \end{aligned}$$

(B3)