Single-step multipartite entangled states generation from coupled circuit cavities

Abstract

Green–Horne–Zeilinger states are a typical type of multipartite entangled states, which plays a central role in quantum information processing. For the generation of multipartite entangled states, the single-step method is more preferable as the needed time will not increase with the increasing of the qubit number. However, this scenario has a strict requirement that all two-qubit interaction strengths should be the same, or the generated state will be of low quality. Here, we propose a scheme for generating multipartite entangled states of superconducting qubits, from a coupled circuit cavities scenario, where we rigorously achieve the requirement via adding an extra z-direction ac classical field for each qubit, leading the individual qubit-cavity coupling strength to be tunable in a wide range, and thus can be tuned to the same value. Meanwhile, in order to obtain our wanted multi-qubits interaction, x-direction ac classical field for each qubit is also introduced. By selecting the appropriate parameters, we numerically shown that high-fidelity multi-qubit GHZ states can be generated. In addition, we also show that the coupled cavities scenario is better than a single cavity case. Therefore, our proposal represents a promising alternative for multipartite entangled states generation.

Graphical abstract

Keywords

quantum information processing / quantum entanglement / quantum state engineering

1 Introduction

Entanglement is one of the most counterintuitive consequences of quantum physics, and nowadays it plays a central role in quantum computation and quantum communication [1]. Multipartite entangled states are entangled states of many qubits which are indispensable resource for research in large scale quantum computation [2], multipartite quantum communication [3], quantum simulation [4] and quantum-to-classical transition [5]. Therefore, generating entangled states of an increasing number of qubits is an important benchmark for modern quantum technology [6].

Green–Horne–Zeilinger (GHZ) states [7] are a typical type of maximally entangled states, and the generation of which have been paid much attention recently [8-21]. Conventional way of GHZ states is generated in a step by step way, based on high-fidelity quantum gates. In this way, the number of entangled qubits is only increased by one at a time, and thus the needed generation time will increase with the increasing of the number of the involved qubits. In addition, this method will also result in accumulation of individual gate operation errors when the number of entangled qubits increases. Alternatively, GHZ state can be generated in a single step [22-35], via the deliberately designed collective interaction, irrespective of the number of entangled qubits.

Meanwhile, superconducting transmon qubits, a kind of superconducting Josephson-junction qubit, are one of the promising solid-state processor for quantum state manipulation [36-38]. Recently, the setup of multipartite superconducting qubits connecting to a common bus resonator has been used for single-step entangled state generation. But, with the increasing of the number of qubits, the probability to generate a GHZ state decrease dramatically. This is because that the single-step method has a strict requirement that all the two-qubit interaction strengths should be the same. However, for each qubit, its coupling strength with the bus resonator is different so that the qubit–qubit interaction does not evolve synchronously.

Here, we present a scheme to solve the above-mentioned difficulty in a two coupled cavities scenario, where each superconducting qubit is biased by a z-direction magnetic flux in order to tune the qubit-resonator coupling. Specifically, this modulation can effectively make the qubit-resonator couplings to be tunable via the amplitudes of the external driving fields, so that the same coupling strength requirement for many qubits can be met. Meanwhile, in order to obtain our wanted multi-qubits interaction, different from that of in Refs. [22, 23, 33, 34], x-direction ac classical field for each qubit is also introduced. By selecting the appropriate parameters, we analytically proved that multi-qubit GHZ states can be generated and we also numerically simulated the obtained high fidelity. In addition, for the target GHZ generation task, we also numerically show that the coupled cavities scenario is better than a single cavity case. Therefore, our proposal represents a promising alternative for multipartite entangled GHZ states generation with superconducting qubits.

2 The theoretical scheme

The proposed setup for Generating GHZ state is illustrated in Fig. 1, which consists of a two coupled cavity in the circuit QED [37] scenario. Setting \hslash =1 hereafter, the Hamiltonian of the two coupled cavities is

\begin{array}{}\text{(1)}& \begin{array}{ll}{H}_c\hfill & ={\omega }_r{a}^{+}a+{\omega }_r{b}^{+}b+J(a{b}^{+}+a^{+}b)\hfill \\ \hfill & ={\omega }_{+}{P}_{+}^{+}{P}_{+}+{\omega }_{-}{P}_{-}^{+}{P}_{-}\mathrm{,}\hfill \end{array}\end{array}

where a^{†} (b^{†}) and a (b) are the creation and annihilation operators for the cavity A (B), respectively; J is the coupling strength between the two cavity modes. The two localized normal modes of this coupled system are P_{\pm }=(a\pm b)/\sqrt{2}, and the frequencies of them are {\omega }_{\pm }={\omega }_r\pm J, with {\omega }_r being chose to be the same for simplicity. Otherwise, the two modes and their frequency will be slightly modified.

In each cavity, there are N transmon qubits, and the free Hamiltonian of them are

\begin{array}{}\text{(2)}& H_q={\displaystyle \sum _{j=1}^{2N}}\frac{{\omega }_{q_j}}{2}{\sigma }_j^z\mathrm{,}\end{array}

where the static qubit frequencies are {\omega }_{q\mathrm{,}j}. The 2N qubits are simultaneously coupled to their corresponding cavities, and the coupling Hamiltonian can be generally written as

\begin{array}{}\text{(3)}& H_{int}={\displaystyle \sum _{j=1}^N}\frac{g_j}{2}{a}^{+}{\sigma }_j^{-}+{\displaystyle \sum _{j=N+1}^{2N}}\frac{g_j}{2}{b}^{+}{\sigma }_j^{-}+H\mathrm{.}c\mathrm{.,}\end{array}

where {\sigma }_j^z={|1\rangle }_j\langle 1|-{|0\rangle }_j\langle 0|, {\sigma }_j^{+}=|1{\rangle }_j\langle 0|, {\sigma }_j^{-}=|0{\rangle }_j\langle 1|, with |0{\rangle }_j and |1{\rangle }_j being the ground and excited states of jth qubit, g_j is the jth qubit-cavity coupling strength.

Meanwhile, all qubits are simultaneously driven by the classical field along x and z directions as

\begin{array}{}\text{(4)}& \begin{array}{l}{H}_{dz}={\displaystyle \sum _{j=1}^{2N}}\frac{A_j\sin ({\omega }_{j}t+\phi )}{2}{\sigma }_j^z\mathrm{,}\\ H_{dx}={\displaystyle \sum _{j=1}^{2N}}\frac{{\text{Ω}}_j}{2}\left({\text{e}}^{-\text{i}{\omega }_{d_j}t}{\sigma }_j^{+}+{\text{e}}^{\text{i}{\omega }_{d_j}t}{\sigma }_j^{-}\right)\mathrm{,}\end{array}\end{array}

where {\text{Ω}}_j is the Rabi frequency of the classical field along x direction and A_j is the amplitude of the classical field which can drive jth qubit along z direction.

In the interaction picture with respect to H_0=H_c+H_{dz}+H_q, the interaction Hamiltonian H_w=T\exp (\text{i}{\displaystyle \int }{H}_0\text{d}t)(H_{int}+H_{dx})T\exp (-\text{i}{\displaystyle \int }{H}_0\text{d}t), with T being the time-ordering operator, will be

\begin{array}{}\text{(5)}& \begin{array}{l}{H}_w={\displaystyle \sum _{j=1}^N}\frac{g_j}{2\sqrt{2}}(P_{+}^{+}{\text{e}}^{\text{i}{\omega }_{+}t}\text{}+\text{}{P}_{-}^{+}{\text{e}}^{\text{i}{\omega }_{-}t}){\sigma }_j^{-}{\text{e}}^{-\text{i}{\omega }_{q_j}t}{\text{e}}^{\text{i}{\alpha }_j\cos ({\omega }_{j}t+\phi )}\\ +{\displaystyle \sum _{j=N+1}^{2N}}\frac{g_j}{2\sqrt{2}}(P_{+}^{+}{\text{e}}^{\text{i}{\omega }_{+}t}\text{}-\text{}{P}_{-}^{+}{\text{e}}^{\text{i}{\omega }_{-}t}){\sigma }_j^{-}{\text{e}}^{-\text{i}{\omega }_{q_j}t}{\text{e}}^{\text{i}{\alpha }_j\cos ({\omega }_{j}t+\phi )}\\ +{\displaystyle \sum _{j=1}^{2N}}\frac{{\text{Ω}}_j}{2}{\text{e}}^{-\text{i}({\omega }_{d_j}t+{\phi }_d)}{\sigma }_j^{+}{\text{e}}^{\text{i}{\omega }_{q_j}t}{\text{e}}^{-\text{i}{\alpha }_j\cos ({\omega }_{j}t+\phi )}+H\mathrm{.}c\mathrm{.}\end{array}\end{array}

where {\alpha }_j=A_j/{\omega }_j. Note that

\begin{array}{}\text{(6)}& {\text{e}}^{\text{i}{\alpha }_{j}cos({\omega }_{j}t+\phi )}\equiv {\displaystyle \sum _{m=-\infty }^{\infty }}{\text{i}}^m{J}_m({\alpha }_j){\text{e}}^{\text{i}m({\omega }_{j}t+\phi )}\mathrm{,}\end{array}

thus setting {\omega }_{d_j}={\omega }_{q_j}, \phi =\pi /2, and

\begin{array}{}\text{(7)}& \begin{array}{l}|{\omega }_{-}-{\omega }_{q_j}+m{\omega }_j|\gg \{|\delta |\mathrm{,}{g}_j{J}_1({\alpha }_j)\}\mathrm{,}m\ne -1\\ |{\omega }_{+}-{\omega }_{q_j}+m{\omega }_j|\gg \{|\delta |\mathrm{,}{g}_j{J}_1({\alpha }_j)\}\mathrm{,}\end{array}\end{array}

with \delta ={\omega }_{-}-{\omega }_{q_j}-{\omega }_j, we can neglect the terms oscillating fast, thus Eq. (5) reduces to

\begin{array}{}\text{(8)}& H_w^{\prime }={\displaystyle \sum _{j=1}^{2N}}\frac{J_0({\alpha }_j){\text{Ω}}_j}{2}{\sigma }_j^x+\frac{1}{2\sqrt{2}}\left[{\displaystyle \sum _{j=1}^N}{g}_j{J}_1({\alpha }_j){\text{e}}^{\text{i}\delta t}{P}_{-}^{†}{\sigma }_j^{-}-{\displaystyle \sum _{j=N+1}^{2N}}{g}_j{J}_1({\alpha }_j){\text{e}}^{\text{i}\delta t}{P}_{-}^{†}{\sigma }_j^{-}+H\mathrm{.}c\mathrm{.}\right]\mathrm{.}\end{array}

Defining H_{w_0}={\displaystyle {\sum }_{j=1}^{2N}}\frac{J_0({\alpha }_j){\text{Ω}}_j}{2}{\sigma }_j^x. In the interaction picture, the interacting Hamiltonian will be [39]

\begin{array}{}\text{(9)}& \begin{array}{ll}{H}_{int}^{\prime }=\hfill & {\text{e}}^{\text{i}\delta t}{P}_{-}^{†}{\displaystyle \sum _{j=1}^N}\frac{g_j}{4\sqrt{2}}{J}_1\left({\alpha }_j\right)\left({\sigma }_j^x+{|-\rangle }_j\langle +|{\text{e}}^{-\text{i}{J}_0({\alpha }_j){\text{Ω}}_{j}t}-{|+\rangle }_j\langle -|{\text{e}}^{\text{i}{J}_0({\alpha }_j){\text{Ω}}_{j}t}\right)\hfill \\ \hfill & -{\text{e}}^{\text{i}\delta t}{P}_{-}^{†}{\displaystyle \sum _{j=N+1}^{2N}}\frac{g_j}{4\sqrt{2}}{J}_1\left({\alpha }_j\right)\left({\sigma }_j^x+{|-\rangle }_j\langle +|{\text{e}}^{-\text{i}{J}_0({\alpha }_j){\text{Ω}}_{j}t}-{|+\rangle }_j\langle -|{\text{e}}^{\text{i}{J}_0({\alpha }_j){\text{Ω}}_{j}t}\right)+H\mathrm{.}c\mathrm{.}\hfill \end{array}\end{array}

Assuming that J_0({\alpha }_j){\text{Ω}}_j\gg \{\delta \mathrm{,}{J}_1({\alpha }_j)g_j\}, and eliminate the oscillate with high frequencies, H_{int}^{\prime } reduces to

\begin{array}{}\text{(10)}& H_{int}=\frac{{\text{e}}^{\text{i}\delta t}}{4\sqrt{2}}\left[{\displaystyle \sum _{j=1}^N}{g}_j{J}_1({\alpha }_j)P_{-}^{†}{\sigma }_j^x-{\displaystyle \sum _{j=N+1}^{2N}}{g}_j{J}_1({\alpha }_j)P_{-}^{†}{\sigma }_j^x\right]+H\mathrm{.}c\mathrm{..}\end{array}

Note that, in the case, we can control the effective coupling strength g_j by varying the externally the classical field, i.e., by controlling the amplitude {\alpha }_j, so that

\begin{array}{}\text{(11)}& g_{j}J({\alpha }_j)/\sqrt{2}=g\end{array}

can be met. Then H_{int} can be written in the form of

\begin{array}{}\text{(12)}& H_{int}=\frac{g}{2}{P}_{-}^{†}{J}_x{\text{e}}^{\text{i}\delta t}+H\mathrm{.}c\mathrm{.,}\end{array}

where we have set J_x=J_1^x-J_2^x with J_1^x=\frac{1}{2}{\displaystyle {\sum }_{j=1}^N}{\sigma }_j^x and J_2^x=\frac{1}{2}{\displaystyle {\sum }_{j=N+1}^{2N}}{\sigma }_j^x.

The evolution operator of the effective Hamiltonian reads

\begin{array}{}\text{(13)}& U(\gamma )\text{}=\text{}\exp \left(\text{i}\gamma J_x^2\right)\exp \left(\text{i}B{P}_{-}^{†}{J}_x\right)\exp \left(\text{i}{B}^{\ast }{P}_{-}{J}_x\right)\mathrm{,}\end{array}

where

\begin{array}{}\text{(14)}& \begin{array}{l}\gamma =\frac{g^2}{4\delta }\left[t+\frac{1}{\text{i}\delta }\left({\text{e}}^{-\text{i}\delta t}-1\right)\right]\mathrm{,}\\ B=\frac{g}{2\text{i}\delta }\left({\text{e}}^{\text{i}\delta t}-1\right)\mathrm{.}\end{array}\end{array}

It is obvious that B(t) is a periodic function of time and vanishes at \delta \tau =2k\pi where k=\mathrm{1,2,3,}\cdots. At those time intervals, the evolution operator in Eq. (13) reduces to

\begin{array}{}\text{(15)}& U(\tau )=\exp \left[\text{i}\gamma (\tau )J_x^2\right]\mathrm{,}\end{array}

which can be directly used to generate GHZ states when \gamma (\tau )=\pi /2. In this case, \delta =\sqrt{k}g\mathrm{,}\tau =2\pi \sqrt{k}/g. For a fast scheme, we can set k=1.

For an initial state |\text{Ψ}(0)\rangle =|00\cdots 0{\rangle }_a{\displaystyle \otimes }|00\cdots 0{\rangle }_b for 2N qubits in two cavities a and b, the final state is found to be a GHZ state of

\begin{array}{}\text{(16)}& \begin{array}{ll}|\text{Ψ}(\tau ){\rangle }_N\hfill & \text{}=\text{}U\left(\frac{\pi }{2}\right)|\text{Ψ}(0)\rangle \hfill \\ \hfill & =\text{}\frac{1}{\sqrt{2}}\left[|00\cdots 0{\rangle }_a|00\cdots 0{\rangle }_b\text{}-\text{}\text{i}|11\cdots 1{\rangle }_a|11\cdots 1{\rangle }_b\right]\mathrm{,}\hfill \end{array}\end{array}

when N is even; the detail derivation are presented in Appendix A.

3 Numerical simulations

In order to obtain the effective Hamiltonian, several approximations have been made here According to the postulated conditions in Eq. (7), we need to choose a suitable value of the frequency {\omega }_j of z-direction magnetic. {\omega }_j is set to around a fixed value, which should be adjusted with respect to the corresponding qubit frequency {\omega }_{qj}, in order to ensure δ to be equal for different qubits. For simplicity, we set {\omega }_j/(2\pi )=2\pi \times 600 MHz in our numerical simulation. Meanwhile, suitable driving amplitudes A_js of the z-direction classical fields can be chosen to make sure that the effective qubit-cavity coupling strengths to be the same, as shown in Eq. (11), in spite of the fact that the original qubit-cavity coupling strength g_js are not identical. Here, A_js are used to tune g_j to a same value g/(2\pi )=15 (10) MHz in the two (four) qubits case. In addition, since {\alpha }_j becomes a certain value, we also choose the amplitudes of each x-direction magnetic flux {\text{Ω}}_j to meet the condition of J_0({\alpha }_j){\text{Ω}}_j=\text{Ω}. For our simulation, the used master equation is

\begin{array}{}\text{(17)}& \begin{array}{ll}\frac{\text{d}\rho (t)}{\text{d}t}=\hfill & -\text{i}\left[H_w\mathrm{,}\rho (t)\right]+\kappa L(a)+\kappa L(b)\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^{2N}}[\beta L({\sigma }_j^{-})+\gamma L({\sigma }_j^z)]\mathrm{,}\hfill \end{array}\end{array}

where L\left(B\right)=B\rho (t)B^{†}-B^{†}B\rho (t)/2-\rho (t)B^{†}B/2 is the Lindblad operator with B\in \{a\mathrm{,}b\mathrm{,}{\sigma }_j^z\mathrm{,}{\sigma }_j^{-}\}. Here, the decay and dephasing rates for all the qubits and the decay for both cavities are all set to be equal as \kappa =\beta =\gamma =2\pi \times 4 kHz.

As shown in Fig. 2, the fidelities of two qubits entangled state generation are plotted with respective to time, for both the one and two cavities cases and the other parameters are list in Table 1. Obviously in Fig. 2, when the \delta \tau =2\pi, the fidelity of red solid line reaches up to 99.02% which is larger than the same data of the blue dashed line. This proves that the performance of generating GHZ state by a z-direction biasing magnetic flux in the two-cavities case is better than that of the one cavity case. Note that, the quantity of inter-cavity coupling J exists only in the two-cavities case, and thus in the one cavity group we compensate this to the qubits’ frequency, so that the z-direction driving frequency {\omega }_j/(2\pi ) is still equal to 600 MHz, be consistent with that of the two-cavity case. Besides the number of cavities, decoherence and the oscillating terms are non-negligible factors, which decrease the fidelity of the generated GHZ state. In order to make all qubit-cavity coupling strength g_j to be a same value, each qubit is driven by the classical field along z directions which bring in the oscillating terms that decreases the fidelity about 0.5\%. Theoretically, with the increasing of the frequency {\omega }_j, the affect of the oscillating terms would decrease.

As shown in Fig. 3, the fidelities of four qubits entangled state generation are plotted with respective to time, for both the one and two cavities cases and the other parameters are list in Table 2. When the \delta \tau =2\pi, the value of fidelity is 96.1% and 90.5% for the two- and one-cavity cases, respectively. The same conclusion we can get is that the two-cavity case shows better performance. Therefore, keeping the qubits apart, e.g., in different cavities, will be beneficial to the increase of fidelity. This is because that the cross talk effect among the oscillating terms will be separated into to cavities and thus some of them have been suppressed.

4 Conclusion

In summary, we have put forward a project to generate GHZ state in two coupled cavities scenario. In this method, each qubit is driven simultaneously by the classical fields. Obviously, the role of the x-direction classical field is manipulating the qubit state. Therefore, we mainly adjust the qubit-cavity strength g_j by controlling the amplitude and frequency of the z-direction classical field. Because of deviation of superconducting manufacturing technique or the other environmental factor, we need a way to make all qubit-cavity coupling strength g_j to be a same value, which can be achieved here by selecting the appropriate parameters. In addition, we also show that the coupled cavities scenario is better than a single cavity case. Therefore, our proposal represents a promising alternative for multipartite entangled states generation with superconducting qubits.

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