Exploration of Fluxonium Parameters for Capacitive Cross-Resonance Gates

Abstract

We study the cross-resonance effect in capacitively-coupled fluxonium qubits and devise a simple formula for their maximum ZX interaction strength. By going beyond the perturbative regime, we find that a CNOT gate can generally be realized in under 200 ns with residual ZZ limited to 50 kHz, for fluxonium qubits with frequencies below 1 GHz. Our analysis relies on a semi-analytical method: we first numerically diagonalize the Floquet Hamiltonian of the strongly-driven control qubit and then perturbatively incorporate the weak qubit-qubit coupling to obtain an effective Hamiltonian. We also derive frequency collision windows around harmful control-target and control-spectator transitions. For large fluxonium devices, we predict a collision-free yield that is considerably less sensitive to junction variability compared to transmons in the same layout. These results support the viability of an all-fluxonium cross-resonance architecture with only capacitive couplings.

Figures (22)

• Figure 1: Overview of the cross-resonance effect between two capacitively-coupled fluxoniums. (\ref{['fig:cross-resonance-circuit']}) Circuit diagram showing the control qubit (left, blue) driven by a pulse modulated at the frequency of the target qubit (right, orange) with peak amplitude $\Omega$. Both qubits are half-flux ($\Phi_0/2$) biased and have individual charging energies $E_{C,q}$, Josephson energies $E_{J,q}$ and inductive energies $E_{L,q}$, and are coupled via their charge operators $\hat{n}_q$ with coupling coefficient $J$. (\ref{['fig:cross-resonance-example-calibration']}) A CNOT gate is calibrated by first preparing the control qubit in $|\psi\rangle_{\mathrm{c}}=|0\rangle_{\mathrm{c}}$ or $|1\rangle_{\mathrm{c}}$ and the target qubit in $|0\rangle_{\mathrm{t}}$, and then applying the cross-resonance pulse at a fixed amplitude for varying durations, followed by a measurement of the target qubit along the $Y$ axis. (\ref{['fig:cross-resonance-example-graph']}) The control qubit ideally remains in its initial eigenstate while the target qubit rotates about the $X$ axis (in the $ZY$ plane) at an angular rate conditioned on the control-qubit eigenstate, $|\psi\rangle_{\mathrm{c}}$. When the pulse duration is $\tau_{\mathrm{CNOT}}$, a net phase difference of $\pi$ accumulates between the two conditional rotations, realizing the CNOT gate up to single-qubit phases and an additional target-qubit rotation.
• Figure 2: ( \ref{['fig:optimal-drive-amplitude-delta-p-typical']}, \ref{['fig:optimal-drive-amplitude-delta-p-small-ac']}, \ref{['fig:optimal-drive-amplitude-delta-p-large-ac']} ) Conditional polarization $\Delta p$ (blue) and incoherent error rate $\varepsilon^\mathrm{incoh}_{\mathrm{c}}/t_g$ (orange) as a function of the dimensionless drive amplitude $\Omega|n_{\mathrm{c}}^{10}|/\omega_{\mathrm{c}}^{10}$ for representative drive frequencies (top left label). The numerically obtained $\Delta p$ (solid blue) is compared with the third-order perturbative result (dashed blue). ( \ref{['fig:optimal-drive-amplitude-spectrum-typical']}, \ref{['fig:optimal-drive-amplitude-spectrum-small-ac']}, \ref{['fig:optimal-drive-amplitude-spectrum-large-ac']} ) Floquet quasienergy spectrum of the off-resonantly-driven control fluxonium, also as a function of $\Omega|n_{\mathrm{c}}^{10}|/\omega_{\mathrm{c}}^{10}$ with the same drive frequency as the top panel. Levels are labeled adiabatically, ignoring "small" avoided crossings, see main text. The inset of (\ref{['fig:optimal-drive-amplitude-spectrum-small-ac']}) shows the avoided crossing between the 0 and 2 modes, where the color of each level is based on its cycle-averaged energy. The optimal amplitude (star$\bigstar$) minimizes $\varepsilon^\mathrm{incoh}_{\mathrm{c}} \propto \varepsilon^\mathrm{incoh}_{\mathrm{c}}/(t_g \Delta p)$. The control fluxonium has parameters $E_{J,{\mathrm{c}}}/2\pi = \qty{4}{\GHz}$, $E_{C,{\mathrm{c}}}/2\pi = \qty{1}{\GHz}$ and $E_{L,{\mathrm{c}}}/2\pi = \qty{1}{\GHz}$.
• Figure 3: (\ref{['fig:polarization-spectrum-delta-p']}) Optimal conditional polarization magnitude $|\Delta p^\star|$ (blue) and corresponding optimal drive amplitude $\Omega^\star$ (green), as a function of the drive frequency $\omega_{\mathrm{d}}$. The absolute value of $\Delta p^\star$ is plotted since its sign flips at $\omega_{\mathrm{d}} = \omega^{10}_{\mathrm{c}}$. The amplitude is optimized to minimize the error contribution $\varepsilon^\mathrm{incoh}_{\mathrm{c}}(\Omega^\star)$, shown in (\ref{['fig:polarization-spectrum-error']}). Resonances with bare transition frequencies (vertical lines) are annotated with $f_{ij} \coloneqq (E_{\mathrm{c}}^i - E_{\mathrm{c}}^j)/2\pi$. The control fluxonium has the same parameters as in \ref{['fig:optimal-drive-amplitude']}.
• Figure 4: Optimal conditional polarization $|\Delta p^\star|$ as a function of the drive frequency $\omega_{\mathrm{d}}$ and the control fluxonium Josephson-to-charging energy ratio $E_{J,{\mathrm{c}}}/E_{C,{\mathrm{c}}}$, while fixing $E_{L,{\mathrm{c}}}/E_{J,{\mathrm{c}}} = (0.25, 0.10)$ and $E_{C,{\mathrm{c}}}/2\pi = (1.0, 1.3)\,\unit{\GHz}$ in panels (\ref{['fig:polarization-spectrum2d-0.25']}, \ref{['fig:polarization-spectrum2d-0.10']}), respectively. The lower frequency cutoff is equal to one third of the control-qubit frequency. Resonances with bare transition (black curves) are annotated with $f_{ij} \coloneqq (E_{\mathrm{c}}^i - E_{\mathrm{c}}^j)/2\pi$.
• Figure 5: Estimated CNOT time for two capacitively-coupled fluxonium qubits with frequencies $\omega^{10}_{({\mathrm{c}}, {\mathrm{t}})}/2\pi = (500, 800)\,\unit{\MHz}$ and a coupling strength $J$ determined by $|\mu_{ZZ}|/{2\pi} = \qty{50}{\kHz}$, for varying Josephson-to-charging energy ratios $E_{J,q}/E_{C,q}$. The inductive-to-Josephson energy ratio of the target fluxonium is $E_{L,{\mathrm{t}}}/E_{J,{\mathrm{t}}}=0.25$ in both panels and that of the control is (\ref{['fig:cnot-duration-equal']}) $E_{L,{\mathrm{c}}}/E_{J,{\mathrm{c}}}=0.25$ and (\ref{['fig:cnot-duration-small-inductance']}) $0.10$. The round marker in (\ref{['fig:cnot-duration-small-inductance']}) corresponds to the system parameters used in \ref{['sec:time-domain-simulation']}.