Universal controlled-phase gate with cat-state qubits in circuit QED

Cat-state qubits (qubits encoded with cat states) have recently drawn intensive attention due to their enhanced lifetimes with quantum error correction. We here propose a method to implement a universal controlled-phase gate of two cat-state qubits via two microwave resonators coupled to a superconducting transmon qutrit. During the gate operation, the qutrit remains in the ground state; thus decoherence from the qutrit is greatly suppressed. This proposal requires only two basic operations and neither classical pulse nor measurement is needed; therefore the gate realization is simple. Numerical simulations show that high-fidelity implementation of this gate is feasible with current circuit QED technology. The proposal is quite general and can be applied to implement the proposed gate with two microwave resonators or two optical cavities coupled to a single three-level natural or artificial atom.

Figure 1

(a) Diagram of two microwave resonators $a$ and $b$ coupled to a transmon qutrit $\left(T_q\right)$. Each resonator can be a one-dimensional or three-dimensional resonator. The qutrit is capacitively or inductively coupled to each resonator. (b) Electronic circuit of a transmon qutrit, which consists of two Josephson junctions and a capacitor.

Figure 2

Resonator $a$ is far-off resonant with the $|g\rangle \leftrightarrow |e\rangle$ transition of the qutrit with coupling strength $g$ and detuning $\left|{\delta }_a\right|$, while resonator $b$ is far-off resonant with the $|e\rangle \leftrightarrow |f\rangle$ transition of the qutrit with coupling strength $\mu$ and detuning $\left|{\delta }_b\right|$. Here, $\left|{\delta }_a\right|={\omega }_a-{\omega }_{eg},\left|{\delta }_b\right|={\omega }_{fe}-{\omega }_b$, with ${\omega }_{eg} \left({\omega }_{fe}\right)$ being the $|g\rangle \leftrightarrow |e\rangle$ ($|e\rangle \leftrightarrow |f\rangle )$ transition frequency of the qutrit, with ${\omega }_a \left({\omega }_b\right)$ being the frequency of resonator $a \left(b\right)$. In addition, $\mathrm{\Delta }=\left|{\delta }_b\right|-\left|{\delta }_a\right|$. Note that the red vertical line represents the frequency ${\omega }_a$ of resonator $a$ while the blue vertical line represents the frequency ${\omega }_b$ of resonator $b$.

Figure 3

Illustration of resonator $a \left(b\right)$ is highly detuned (decoupled) from the $|e\rangle \leftrightarrow |f\rangle \left(\right|g\rangle \leftrightarrow |e\rangle )$ transition of the qutrit. The high detuning (or decoupling) can be made by prior adjustment of the level spacings of the transmon qutrit and/or the frequency of resonator $a \left(b\right)$, such that $\left|{\delta }_a^{\prime }\right|⋙g^{\prime }$ and $\left|{\delta }_b^{\prime }\right|⋙{\mu }^{\prime }$. Here, $g^{\prime }$ is the coupling constant between resonator $a$ and the $|e\rangle \leftrightarrow |f\rangle$ transition, ${\mu }^{\prime }$ is the coupling constant between resonator $b$ and the $|g\rangle \leftrightarrow |e\rangle$ transition, $\left|{\delta }_a^{\prime }\right|={\omega }_a-{\omega }_{fe}$ is the detuning between the frequency of resonator $a$ and the $|e\rangle \leftrightarrow |f\rangle$ transition frequency, and $\left|{\delta }_b^{\prime }\right|={\omega }_{eg}-{\omega }_b$ is the detuning between the frequency of resonator $b$ and the $|g\rangle \leftrightarrow |e\rangle$ transition frequency. Note that the coupling of both resonators with the $|g\rangle \leftrightarrow |f\rangle$ transition of the qutrit is negligible because of the forbidden or very weak $|g\rangle \leftrightarrow |f\rangle$ transition [49, 50].

Figure 4

Resonator $a$ is far-off resonant with the $|g\rangle \leftrightarrow |e\rangle$ transition of the qutrit with coupling strength $\tilde{g}$ and detuning $\left|{\tilde{\delta }}_a\right|$. Here, $\left|{\tilde{\delta }}_a\right|={\omega }_{eg}-{\tilde{\omega }}_a$, with ${\tilde{\omega }}_a$ being the adjusted frequency of resonator $a$ (labeled by the vertical line). The frequency of resonator $b$ is adjusted such that resonator $b$ is decoupled from the qutrit. Note that the dispersive qutrit-cavity coupling with a detuning $\left|{\tilde{\delta }}_a\right|$ illustrated here can also be obtained by adjusting the level spacings of the qutrit but the cavity frequency being fixed.

Figure 5

The illustration of resonator $a$ is highly detuned (decoupled) from the $|e\rangle \leftrightarrow |f\rangle$ transition of the qutrit, and the illustration of resonator $b$ is highly detuned (decoupled) from the $|g\rangle \leftrightarrow |e\rangle$ and $|e\rangle \leftrightarrow |f\rangle$ transitions of the qutrit. The decoupling can be made as long as the conditions $\left|{\tilde{\delta }}_a^{\prime }\right|⋙{\tilde{g}}^{\prime },\left|{\tilde{\delta }}_b\right|⋙\tilde{\mu }$, and $\left|{\tilde{\delta }}_b^{\prime }\right|⋙{\tilde{\mu }}^{\prime }$ can be satisfied. Here, ${\tilde{g}}^{\prime }$ is the coupling constant between resonator $a$ and the $|e\rangle \leftrightarrow |f\rangle$ transition, $\tilde{\mu }$ is the coupling constant between resonator $b$ and the $|e\rangle \leftrightarrow |f\rangle$ transition, and ${\tilde{\mu }}^{\prime }$ is the coupling constant between resonator $b$ and the $|g\rangle \leftrightarrow |e\rangle$ transition. In addition, $\left|{\tilde{\delta }}_a^{\prime }\right|={\tilde{\omega }}_a-{\omega }_{fe}$ is the detuning between the frequency of resonator $a$ and the $|e\rangle \leftrightarrow |f\rangle$ transition frequency, $\left|{\tilde{\delta }}_b\right|={\omega }_{fe}-{\tilde{\omega }}_b$ is the detuning between the frequency of resonator $b$ and the $|e\rangle \leftrightarrow |f\rangle$ transition frequency, and $\left|{\tilde{\delta }}_b^{\prime }\right|={\omega }_{eg}-{\tilde{\omega }}_b$ is the detuning between the frequency of resonator $b$ and the $|g\rangle \leftrightarrow |e\rangle$ transition frequency. Note that ${\tilde{\omega }}_a \left({\tilde{\omega }}_b\right)$ is the adjusted frequency of resonator $a \left(b\right)$.

Figure 6

Fidelity versus ${\kappa }^{-1}$. The plots are drawn for $\alpha =0.5$. Other parameters used in the numerical simulation are referred to in the text. Green curves are based on the effective Hamiltonians (4) and (11) but not considering decoherence and the inter-resonator crosstalk; blue curves are based on the effective Hamiltonians (4) and (11) and considering decoherence and the inter-resonator crosstalk; while the red curves are based on the full Hamiltonians (16) and (17) and taking decoherence and the inter-resonator crosstalk into account: (a) is plotted for $\theta =\phi =\pi /4$; (b) is for $\theta =\phi =\pi /3$; (c) is for $\theta =\pi /4,\phi =\pi /3$; while (d) is for $\theta =\pi /3,\phi =\pi /4$.