Generalized cross-resonance scheme for maximally-entangling two-qutrit gates

Abstract

To utilize higher-dimensional quantum systems, in this Letter, we derive a generalized cross-resonance (GCR) scheme for realizing maximally entangling two-qutrit gates on fixed-frequency transmons beyond the 0-1 subspace. Our two-qutrit gates, namely, $U_{CR}^{01}$ and $U_{CR}^{12}$, acting on the $0{\text -}1$ and $1{\text -}2$ energy transitions of transmons, respectively, directly allow for entanglement on the $1{\text -}2$ levels. Unlike the known works, our gate is parametric in nature, enabling us to construct multiple entangling gates of interest. By performing simulations in Qiskit, we demonstrate two-qutrit generalized controlled-$X$ ($U_{CX}^{01}$ and $U_{CX}^{12}$) and controlled-$H$ ($U_{CH}^{01}$ and $U_{CH}^{12}$) gates, which are instances of the proposed $U_{CR}$ gates, with reported gate fidelities of $86.14\%~(99.73\%),~84.6\%~(97.88\%),~92.35\%~(99.39\%)$, and $91.99\%~(98.99\%)$, respectively with (and without) noise. We also reveal a two-qutrit Bell state with a fidelity of $99.06 \pm 0.01\%$, with a complete Bell state preparation in a $\sim514$ ns pulse sequence, which is less than the gate time of the known scheme by cross-Kerr-based entangling gates.

Figures (14)

• Figure 1: The schematic diagram of a coupled two transmon system. $\omega_1$ and $\omega_2$ correspond to the frequency of the transition $\ket{0}\rightarrow \ket{1}$ in Transmon $1$ and Transmon $2$, respectively, and $\delta_1$ and $\delta_2$ are their respective anharmonicities. The two transmons are coupled through a resonator of frequency $\omega_r$ with coupling strength $J$. The proposed Generalized-Cross Resonance scheme generalizes the qubit-qubit Cross Resonance scheme to the qutrit setting, where the control qutrit (Transmon 1) is irradiated with a microwave pulse of frequency $\omega_d$ the equals either $\omega_2$ corresponding to $\ket{0}\rightarrow \ket{1}$ transition or, $\omega_2 + \delta_2$ corresponding to $\ket{1} \rightarrow \ket{2}$ transition of the target qutrit (Transmon 2).
• Figure 2: The comparison between the process matrices ($\chi_{ideal}$, $\chi_{noisy}$) of the ideal gates and the gates simulated under the effects of decoherence. (a) $\chi_{ideal}$ obtained using QPT for the $\operatorname{U}_{\operatorname{CX}}^{01}$ gate as defined in \ref{['eqn:ucx01']}. (b) $\chi_{noisy}$ obtained using QPT for the simulated $\operatorname{U}_{\operatorname{CX}}^{01}$ gate. We obtain a process fidelity of $86.14\% \pm 0.01\%$ for the simulated $\operatorname{U}_{\operatorname{CX}}^{01}$ gate when the decoherence time T ($T_1=T_2=T$) = $1700\mu s$. (c) $\chi_{ideal}$ obtained using QPT for the $\operatorname{U}_{\operatorname{CH}}^{12}$ gate as defined in \ref{['eqn:uch12']}. (d) $\chi_{noisy}$ obtained using QPT for the simulated $\operatorname{U}_{\operatorname{CH}}^{12}$ gate. We obtain a process fidelity of $91.99 \% \pm 0.01\%$ for the simulated $\operatorname{U}_{\operatorname{CH}}^{12}$ gate when the decoherence time T = $1700\mu s$.
• Figure 3: The QST plots of the (a) two-qutrit Bell state as defined in \ref{['eqn:bell-state']}, (b) the two-qutrit Bell state prepared under the noiseless setting, and (c) the two-qutrit Bell state prepared under the noisy setting ($T_1 = T_2 = 1700\mu s$). We prepare the two-qutrit Bell state using the proposed $\operatorname{U}_{\operatorname{CX}}^{01}$ and $\operatorname{U}_{\operatorname{CH}}^{12}$ gates and obtain state fidelities of $\mathcal{F}_{noiseless}=99.06 \pm 0.01\%$ and $\mathcal{F}_{noisy}=78.73 \pm 0.01\%$, repectively. For our choice of parameters, the total time for the preparation of the qutrit Bell state is $\sim514$ ns.
• Figure 4: Step-by-step procedure for quantum process tomography.
• Figure 5: The action of the proposed $\operatorname{U}_{\operatorname{CR}}^{01}$ and $\operatorname{U}_{\operatorname{CR}}^{12}$ parametric gates over time.