Quantum control and noise protection of a Floquet $0\text{−}\pi$ qubit

Time-periodic systems allow engineering new effective Hamiltonians from limited physical interactions. For example, the inverted position of the Kapitza pendulum emerges as a stable equilibrium with rapid drive of its pivot point. In this work we propose the Kapitzonium: a Floquet qubit that is the superconducting circuit analog of a mechanical Kapitza pendulum. Under periodic driving, the bit- and phase-flip rates of the emerging qubit states are exponentially suppressed with respect to the ratio of the effective Josephson energy to charging energy. However, we find that dissipation causes leakage out of the Floquet qubit subspace. We engineer a passive cooling scheme to stabilize the qubit subspace, which is crucial for high-fidelity quantum control under dissipation. Furthermore, we introduce a hardware-efficient fluorescence-based method for qubit measurement and discuss the experimental implementation of the Floquet qubit. Our work provides the fundamental steps to develop more complex Floquet quantum systems from the ground up to realize large-scale protected engineered dynamics.

Figure 1

Schematics of the Kapitzonium. (a) Kapitza pendulum. (b) Superconducting circuit implementing the Kapitzonium. (c) Effective double-well potential showing the disjoint wave functions and degeneracy of $|0\rangle$ and $|\pi \rangle$.

Figure 2

(a) Floquet eigenenergies of ${\widehat{H}}_0\left(t\right)$ compared with the spectrum of ${\widehat{H}}_{\text{eff}}$. The energy splitting in the ground-state manifold is $({\varepsilon }_1-{\varepsilon }_0)/2\pi \approx 4.7\mathrm{kHz}$, which is much smaller than the gap between the ground and first excited manifolds. (b) Localized Floquet modes $|{\mathrm{\Phi }}_k^0\left(t\right)\rangle$ and $|{\mathrm{\Phi }}_k^{\pi }\left(t\right)\rangle$ at $t=0$ for $k=0,1,2$. (c) Probability distributions of the localized Floquet modes $|{\mathrm{\Phi }}_0^0\left(t\right)\rangle$ and $|{\mathrm{\Phi }}_0^{\pi }\left(t\right)\rangle$ from 0 to $T$ in the $\varphi$ space. (d) Effective potential when performing different gates.

Figure 3

(a) Emission spectrum of the Kapitzonium under charge noise. The upper (lower) $m\to n$ label within each rounded rectangle corresponds to the cross (dot). Only transitions involving $|{\mathrm{\Psi }}_{0–3}\rangle$ with emission frequencies less than 25 GHz are shown here. (b) Schematic for the Kapitzonium coupled to a bandpass filter. (c) Two different regimes of the emission spectrum, which determines whether the qubit can be protected or not. (d) Idling and gate fidelity for different intrinsic loss rate $\kappa$, with (crosses and diamonds) and without (dots) the filter.

Figure 4

(a) Kapitzonium measurement by measuring the frequency of the emitted photon. (b) The measurement basis depends on how the degeneracy is lifted between ${\omega }_{20}$ and ${\omega }_{31}$. (c) Quantum trajectory simulation of the Kapitzonium $X$ measurement. (d) Single trajectory corresponding to the red dashed line in (c), where $S\left(\omega \right)$ is plotted in the rotating frame of ${\omega }_f$. (e) Charge sensitivity of the Floquet eigenenergies and the emission spectrum.

Figure 5

(a) The $Z$ gate simulation with a filter for different initial states. (b) The $Z$ measurement results.

Figure 6

Kapitzonium transition rates for $n_g=0.3$. The junction energies are $E_J/2\pi =60,80,100,120,140,160\mathrm{GHz}$, with $\omega /2\pi =10\text{GHz}$ and $E_C/2\pi =0.01\text{GHz}$ fixed.