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Abstract
We propose a scheme for quantum geometric computation on a fiber-cavity-fiber system, in which two atoms are located in two single-mode cavities, respectively, connected with each other by optical fiber. This scheme not only has the feature of virtual excitation of photons in the cavity quantum electrodynamics (CQED) that can reduce the effect of decay effectively but also has the advantage of geometric phase to withstand noises due to its built-in noise-resilience feature and robust merit. Specifically, our proposal combined with optimized-control-technology (OCT) can reduce gate operation error by adjusting the time-dependent amplitude and phase of the resonant field which further enhances the robustness of the quantum operation. The robustness against decoherence is demonstrated numerically and the scheme may be applied in the remote quantum information processing tasks and quantum computation.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recent years, quantum computation (QC) has attracted a lot of attention due to its intrinsic quantum characters of qubits, such as entanglement and superposition principle, which provide the ability to solve some intractable problems for classic computation [1]. An essential precondition for implementing QC is to construct the high fidelity quantum logical gate. Meanwhile, the geometric phase determined by the closed path of quantum operation [2–4] can avoid some local noise [2,5–14] and can be used to construct robust quantum gates to realize fault-tolerant quantum computers. Therefore, in the past decades, the geometric phases have been proposed [4,15,16] and achieved in a great deal of quantum systems, for instance, the cavity quantum electrodynamics (CQED) [17,18], ion trap [19,20], nuclear magnetic resonance (NMR) [21–23] and nitrogen vacancy (NV)-center [24], etc. We can see that most of the proposed quantum geometric computing schemes are localized, so distributed quantum computing is therefore worth exploring and has important value for distributed quantum information processing tasks.
As a promising candidate to implement distributed quantum computation task, a great deal of robust schemes have been proposed [25–31] and demonstrated in CQED systems [32–38]. Particularly, Zheng in ref. [39] puts forward a scheme to achieve the quantum gate without excitation in the distant cavities connected by optical fibers that since the virtual-photon-induced quantum gates is insensitive to the environment the gate error caused by unnecessary decoherence can be greatly reduced.
However, the unavoidable interaction between the quantum system and the surrounding environment induces the decoherence of the quantum systems [40,41] which tremendously degrades the fidelity of geometric quantum gate due to the slower implementation than its dynamical counterpart. In particular, the early proposed geometric quantum computation (GQC) is based on adiabatic condition [42–45], and its long evolution time will lead to more decoherence and error inevitably. To overcome this drawback, the nonadiabatic geometric quantum computation (NGQC) based on non-Abelian and Abelian geometric phases [40,41,46–56] has been proposed in theory and also demonstrated by experiments [57–70]. If the optimized-control-technology (OCT) [71–76] can be combined to further reduce the system error, the robustness of quantum logic gates will be improved undoubtedly.
In the present work, we propose a robust scheme to construct distributed quantum logic gate in atom-cavity-fiber configuration. The operational error in the scheme is minimized by employing the nonadiabatic geometric quantum operation as well as the OTC. Meanwhile, the atomic spontaneous emission and the leakage of cavity as well as fiber in the system can be overcome through virtual excitation of excited state and cavity modes, respectively.
The paper is organized as follow. First, we derived the two-level effective Hamiltonian in the atom-cavity-fiber system in Sec. 2. Then, a general two-qubit quantum logic gate can be constructed with by applying the NGQC to the CQED system in Sec. 3, we use iSWAP gate as an example to show the feasibility of the scheme. Next, we improve the fidelity of the gate and enhance the robustness of this scheme by combining it with OCT in Sec. 4. Finally, the robustness of the scheme is discussed by the master equation.
2. Model and its effective Hamiltonian
Here, we consider two identical cavities connected by an optical fiber, as shown in Fig. 1, and each cavity traps an atom with three energy levels [see Fig. 1(a)]. The optical fiber satisfying the short optical fiber limit [77–79], i.e. the number of fiber modes significantly interacting with the cavities modes $n=l \bar {\nu } / 2\pi c \lesssim 1$ with the cavity decay rate $\bar {\nu }$, the length of each fiber $l$ and the speed of light $c$, essentially has only one resonant mode interacting with the cavity modes. In the Interaction picture, the Hamiltonian of the atom-cavity-fiber is written as
Fig. 1. The schematic for the physical model where the two atoms are trapped in two cavities, respectively and the two cavities are connected by an optical fiber. (a) The atoms in a $\Lambda$-level structure owns one excited state $\left | e\right \rangle _{j}$ and two ground states $\left | 0\right \rangle _{j}$ and $\left | 1\right \rangle _{j}$. The lasers drive the transition $\left | e\right \rangle _{j}\leftrightarrow \left | 1\right \rangle _{j}$ with the detuning $\Delta _1$ and Rabi frequency $\Omega _{j}$ ( j=1, 2), respectively, and the states $\left | e\right \rangle _{j}$ and $\left | 1\right \rangle _{j}$ of atom are couplied to the corresponding cavity with the coupling strength $g_{j}$. (b) Geometric illustration of the iSWAP gate on a Bloch sphere where the single-loop evolution path is divided into four symmetrical parts ①, ②, ③ and ④.
To penetrate into the significant nature of the system, we first introduce a Bosonic modes group of annihilation operators as [77]
Supposing the cavities and the fiber initially in the vacuum state, the initial state of the system will be restricted in the space {$\left | 0\right \rangle _{1}\left | 0\right \rangle _{2}\left | 0\right \rangle _{c}$, $\left | 0\right \rangle _{1}\left | 1\right \rangle _{2}\left | 0\right \rangle _{c}$, $\left |1\right \rangle _{1}\left | 0\right \rangle _{2}\left | 0\right \rangle _{c}$, $\left | 1\right \rangle _{1}\left | 1\right \rangle _{2}\left | 0\right \rangle _{c}$}, where the subscripts 1, 2, c denotes atom1, atom 2 and the bosonic mode $\hat {c}_{+}$, respectively, the $\left | \ \right \rangle _{c}$ signifies the Fork state of bosonic mode $\hat {c}_{+}$. According to the different evolution paths of the initial state, we can be divided into four evolution subspaces: When initial state is prepared in $\left | \phi _{1}\right \rangle =\left | 0\right \rangle _{1}\left | 0\right \rangle _{2}\left | 0\right \rangle _{c}$, the system keeps unchanged based on Hamiltonian in Eq. (1); if the initial state is $\left | \phi _{2}\right \rangle =\left |1\right \rangle _{1}\left | 1\right \rangle _{2}\left | 0\right \rangle _{c}$, the evolution will occur in the subspace $P_{110}$={$\left | \phi _{2}\right \rangle ,\left | \phi _{3}\right \rangle ,\left | \phi _{4}\right \rangle ,\ldots ,\left | \phi _{10}\right \rangle$} with
When the initial state is $\left | \phi _{2}\right \rangle =\left |1\right \rangle _{1}\left | 1\right \rangle _{2}\left | 0\right \rangle _{c}$, the Hamiltonian can be expanded as $\hat {\mathcal {H}_{I}}=\hat {\mathcal {H}_{g}}+\hat {\mathcal {H}_{\Omega }}+\hat {\mathcal {H}_{\Delta }}$ with
Considering the quantum Zeno dynamical process [80–82] with the condition $\hat {\mathcal {H}}_{g}\gg \ \hat {\mathcal {H}}_{\Omega }$ satisfied, i.e., $g\gg \Omega _{1,2}$, the system will evolve along the subspace of the initial eigenstates and the bosonic modes have three zero-eigenvalue eigenstates { $\left | \psi _{d1}\right \rangle =(\left | \phi _{7}\right \rangle +\left | \phi _{10}\right \rangle )/\sqrt {2}$, $\left | \psi _{d2}\right \rangle =(\left | \phi _{8}\right \rangle +\left | \phi _{9}\right \rangle )/\sqrt {2}$, $\left | \phi _{2}\right \rangle$}. Therefore, there is no effective coupling among $\left | \psi _{d1}\right \rangle$, $\left | \psi _{d2}\right \rangle$ and $\left | \phi _{2}\right \rangle$ and the system will be trapped in the initial state $\left | \phi _{2}\right \rangle =\left |1\right \rangle _{1}\left | 1\right \rangle _{2}\left | 0\right \rangle _{c}$ all the time.
If the initial state is prepared in $\left | \phi _{11}\right \rangle =\left |1\right \rangle _{1}\left | 0\right \rangle _{2}\left | 0\right \rangle _{c}$ or $\left | \phi _{15}\right \rangle =\left |0\right \rangle _{1}\left | 1\right \rangle _{2}\left | 0\right \rangle _{c}$, the Hamiltonian in the subspace $P_{100}$ can be expanded as $\hat {H_{I}}=\hat {H}_{g}+\hat {H}_{\Omega }+\hat {H}_{\Delta }$ with
Then, in the picture of the eigenstates $\left | \Psi _{\pm }\right \rangle =(\left | \phi _{12}\right \rangle \pm \sqrt {2}\left | \phi _{13}\right \rangle +\left | \phi _{14}\right \rangle )/2$ and $\left | \Psi _{d}\right \rangle =(\left | \phi _{12}\right \rangle -\left | \phi _{14}\right \rangle )/\sqrt {2}$ corresponding to the eigenvalues $\pm \sqrt {2}g$ and 0 of $\hat {H_{g}}$, the Hamiltonian can be represented as
Fig. 2. (a): The comparison of the population of $\phi _{11}$ from original Hamiltonian with the effective Hamiltonian, where $\varphi _{1}=0, \ \varphi _{2}=\pi , \ \Delta =10\Omega ,\ g=15\Omega , \ \nu =10g$ and the end time of evolution is $100/\Omega$. The red and black curves represent the population of $\phi _{11}$ evolved under the original and effective Hamiltonian, respectively. (b): Population of excited state of cavities; (c): The total population of excited cavity and fiber modes; (d): Population of ground state of atom 1.
3. Two-qubit geometric quantum logic gate
From the effective Hamiltonian of the whole system in Eq. (10), one can see that $\left |\phi _{1}\right \rangle =|0\rangle _{1}|0\rangle _{2}|0\rangle _{c}$ and $\left |\phi _{2}\right \rangle =|1\rangle _{1}|1\rangle _{2}|0\rangle _{c}$ are decoupled from the effective systematic dynamics, which means that these two states would keep invariant during the evolution process. The effective Hamiltonian consisting of $\left |\phi _{11}\right \rangle =|1\rangle _{1}|0\rangle _{2}|0\rangle _{c}$ and $\left |\phi _{15}\right \rangle =|0\rangle _{1}|1\rangle _{2}|0\rangle _{c}$ can be seen as an two-level structure. Obviously, if the whole system initial states are in the ground state computational subspace $\{\left |\phi _{1}\right \rangle , \left |\phi _{15}\right \rangle , \left |\phi _{11}\right \rangle , \left |\phi _{2}\right \rangle \}$, the effective Hamiltonian in Eq. (10) can be re-written as
The reduced form of the evolution operator
The dynamic evolution is governed by the time-dependent Schrödinger equation as
The eigenstate of $H(t)$ can be generally defined as
Then, we can get the final geometric evolution operation,
Then, the evolution operator in the whole computational subspace $\{\left |\phi _{1}\right \rangle , \left |\phi _{15}\right \rangle , \left |\phi _{11}\right \rangle , \left |\phi _{2}\right \rangle \}$ is given by
After taking the Hamiltonian Eq. (13), i.e., $\hat {H}(t)$ into the time-dependent Schödinger equation, we can get the following constraint equation
Thus, the evolution path of the system can be determined by designing the shapes of $\chi (t)$ and $\beta (t)$.
According to Eq. (17), one can design the pulses to construct different kinds of universal gates. As a concrete example, here we consider to construct the pure geometric iSWAP gate to verify the performance of the scheme. Reference [87] has shown that iSWAP gate can be used to construct two-qubit controlled-phase and CNOT gates easily with the help of single-qubit rotation operations, which means that iSWAP gate is an universal gate for quantum computation.
In order to obtain a pure geometric gate, we eliminate the dynamical phase in the cyclic evolution by separating a single-loop evolution path into four equal parts by (see Fig. 1(b))
To construct a iSWAP gate, one can choose $\gamma =\pi /2$, and employ an arbitrary initial state $\left | \phi (0)\right \rangle =(\cos \theta _{1}\left |0\right \rangle _{1}+\sin \theta _{1}\left |1\right \rangle _{1})\otimes (\cos \theta _{2}\left | 0 \right \rangle _{2}+\sin \theta _{2}\left |1\right \rangle _{2})$ to calculate the average fidelity. The cyclic evolution time for the gate is finished within the duration $\tau =\pi ^2\sqrt {41}/5\Omega _\textrm {max}$ for the maximum amplitude $\Omega _\textrm {max}$ of the Rabi frequency and the ideal final states is given by $\left |\Phi (\tau )\right \rangle =\hat {U}_\textrm {whole}\left |\Phi (0)\right \rangle$. The average fidelity of the gate is defined as [88]
Fig. 3. The numerical simulation of the iSWAP gate. (a): The shape of $\widetilde {\Omega }$ and $\phi$ for the iSWAP gate. (b) The evolution of iSWAP gate fidelity for the initial state $\left | \phi (0)\right \rangle =(\cos \theta _{1}\left |0\right \rangle _{1}+\sin \theta _{1}\left |1\right \rangle _{1})\otimes (\cos \theta _{2}\left | 0 \right \rangle _{2}+\sin \theta _{2}\left |1\right \rangle _{2})$ with the original and the effective Hamiltonian, where the subscripts 1, 2 denote atom1 and atom2. The fidelity is defined as $\mathcal {F}= \frac {1}{4\pi ^2}\int ^{2\pi }_{0}\int ^{2\pi }_{0}\langle \phi (\tau )|\rho |\phi (\tau )\rangle \textrm {d}\theta _{1}\textrm {d}\theta _{2}$.
4. Robustness under the optimal control
In order to enhance the robustness of this scheme and improve the fidelity of the gate against systematic error, we combine the schemes with the optimal control technology (OCT) [76,89]. As discussed in Sec. 3, the states $\{\left |\phi _{1}\right \rangle , \left |\phi _{15}\right \rangle , \left |\phi _{11}\right \rangle , \left |\phi _{2}\right \rangle \}$, $\{\left |\phi _{1}\right \rangle , \left |\phi _{2}\right \rangle \}$ are not evolved in the computational subspace, we only consider the effective dynamics in the reduced Hilbert subspace C ={$\left |\phi _{15}\right \rangle$, $\left |\phi _{11}\right \rangle$}. Nevertheless, when we make numerical simulations for the two-qubit gate, we consider the full Hilbert space. The Rabi frequency $\widetilde {\Omega } e^{i\varphi (t)}$ in Hamiltonian Eq. (13) can be rewritten as $\Omega _{x}(t)-i\Omega _{y}(t)$. The general solution of the time-dependent Schrödinger equation with $\psi (t)$ in the reduced computational space C ={$\left |\phi _{15}\right \rangle$, $\left |\phi _{11}\right \rangle$} can be written as
And the constrain equations are
The phase $f(t)$ and the azimuth angle $\beta (t)$ can be expressed as modified Fourier series satisfying the constrain equations Eq. (22)
One group of the optimized coefficients are shown in Table 1. With these parameters, the evolution operator in the reduced Hilbert subspace C ={$\left |\phi _{15}\right \rangle$, $\left |\phi _{11}\right \rangle$} can be written as [76]
which means that the evolution operator in the whole computational space $\{\left |\phi _{1}\right \rangle , \left |\phi _{15}\right \rangle , \left |\phi _{11}\right \rangle , \left |\phi _{2}\right \rangle \}$ can be written asThis is a standard iSWAP gate. Considering the static systematic error situation, i.e., $\widetilde {\Omega }\rightarrow (1+\epsilon )\widetilde {\Omega }$, substituting the parameters in Table 1 into Hamiltonian Eq. (1), the average fidelity $\mathcal {F}$ versus the static systematic error is shown in Fig. 4. For comparison, we also plot the average fidelity of the unoptimized iSWAP gate constructed in Sec. 3.
Fig. 4. The average fidelity of the iSWAP gate where the black line represents average fidelity without OCT[the evolution path satisfy Eq. (19) and the following], and the red line denotes the average fidelity combine with OCT[the conditions and parameters are given in Eq. (23) Eq. (24) and Table 1]. The $\Omega _\textrm {{max}}$ means the maximum value of the $\Omega$ is set to be $\Omega _\textrm {{max}}=2\pi \times 16$MHz.
In the atom-cavity-fiber system, since both the dissipation induced by atomic spontaneous emission and the cavity-fiber photon leakage need be taken into account, we discuss the robustness of the iSWAP gate by the Lindblad master equation as
Fig. 5. The numerical simulation of the iSWAP gate combined with OTC under different dissipation where we have set $\Gamma =\kappa$, where the red line and the blue line mean the fluctuation parameters of Rabi frequency $\epsilon =0.1$ and 0.4, respectively. The other parameters are the same as that used for the OTC-based curve in Fig. 4.
5. Discussion
In summary, we proposed a practical scheme in the atom-cavity-fiber system to achieve quantum geometric computation with the cavity mode and fiber mode virtually excited. Then we applied it to NGQC, constructing a two-qubit iSWAP gate. For improving the fidelity of the gate, we combine it with OCT to enhance its robustness. Finally, the robustness of the scheme is discussed by master equation. We argue that this scheme may have some concrete applications in distributed quantum computation in the future since it combines the advantages of geometric phase as well as optimized control.
Funding
National Key Research and Development Program of China (2017YFA0304503); National Natural Science Foundation of China (11674360, 11734018, 11804308, 11804375, 11835011, 12074346, 91421111); Natural Science Foundation of Henan Province (202300410481); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB21010100); Key R&D Project of Guangdong Province (2020B0303300001).
Acknowledgments
We would like to thank B.-J. Liu for useful discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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