How the Kerr-Cat Qubit Dies -- And How to Rescue It

Abstract

Kerr-cat qubits have been experimentally shown to exhibit a large noise bias, with one decay channel suppressed by several orders of magnitude. In superconducting implementations, increasing the microwave drive on the nonlinear oscillator that hosts the Kerr-cat qubit should, in principle, further enhance this bias. Instead, experiments reveal that above a critical drive amplitude the tunneling time, which is the less dominant decay channel, ceases to increase and even decreases. Here, we show that this breakdown arises from the multimode nature of the circuit implementation. Specifically, additional modes, including the buffer mode used to deliver the stabilizing drive and higher modes of the Josephson junction array, can induce multiphoton resonances that sharply degrade Kerr-cat coherence. We uncover this mechanism by retaining the full circuit nonlinearities and treating the strong drive exactly within a Floquet-Markov framework that accounts for quasidegeneracies of the Kerr-cat spectrum. Our results not only provide an explanation for the sudden reduction of the tunneling time but also demonstrate that the Kerr-cat qubit can be very robust in the presence of a carefully engineered electromagnetic environment. Beyond the Kerr-cat qubit, the tools developed here apply broadly to strongly driven dissipative systems with quasidegenerate spectra, including superconducting devices under subharmonic driving (e.g., parametric amplifiers and couplers) and protected qubits where quasidegeneracies similarly govern coherence.

Figures (17)

• Figure 1: (a) Cut along $\text{Re}[\alpha]$ of the metapotential associated with the shifted squeezed Kerr Hamiltonian $\hat{H}_\text{SK} - \varepsilon_2^2/K$ for $\varepsilon_2 = 10$ and $K = 1$. Horizontal gray lines indicate the eigenenergies, and the colored curves the marginals of the Wigner distributions of the corresponding eigenstates. For the quasidegenerate level pairs 0--1, 2--3, and 4--5, dissipative tunneling across the double well is suppressed due to destructive interference between two relaxation pathways. Population transfer between wells can instead occur by climbing the energy ladder—via photon emission and absorption—until nondegenerate levels (here 6 and 7) are populated. The number of quasidegenerate level pairs within the well is controlled by the photon number $|\alpha|^2 = \varepsilon_2/K = 10$. (b) Floquet quasienergy spectrum of the Kerr-cat qubit showing spectral kissings (circles) where the quasienergy differences between pairs of excited states asymptotically approach $\omega_d/2$. Dashed lines indicate drive amplitudes at which these differences become comparable to the decay rate. The circuit parameters are chosen close to those of Ref. Frattini2024, with $E_J/2\pi = 272.436$ GHz, $E_C/2\pi = 107.8$ MHz, $\alpha = 0.046$, and $\varphi_\text{ext} = 0.33 \times 2\pi$, yielding a 0--1 transition frequency $\omega_{01}/2\pi = 6.094$ GHz and a self-Kerr nonlinearity $K/2\pi = 1.18$ MHz.
• Figure 2: (a) Circuit implementation of the Kerr-cat qubit, consisting of an array of SNAILs shunted by a large capacitor (green) and driven through a buffer mode (blue) whose resonance frequency is close to the drive frequency. (b) Circuit diagram of a single SNAIL composed of an array of large Josephson junctions with energy $E_J$ in parallel with a smaller junction of energy $\alpha E_J$, forming a loop threaded by an external magnetic flux (gray). (c) Semiclassical representation of the circuit in (a), the driven buffer mode is replaced by a classical drive (gray) and a dissipative environment modeled by an admittance $\mathrm{Y}(\omega)$ (blue). (d) Typical spectral density $J(\omega) \propto \textrm{Re}[\mathrm{Y}(\omega)]/C$ featuring a broad peak around the drive frequency $\omega_d$ reflecting the low quality factor of the buffer mode. The values around $J(m\omega_d/2)$ for $m\in \mathbb{Z}^+$ are taken from Ref. Venkatraman:2024.
• Figure 3: Wigner distributions of the first two Floquet modes, $\ket{\phi_0}$ and $\ket{\phi_1}$, at a drive amplitude $\varepsilon_d/2\pi = 2.5~\text{GHz}$ corresponding to approximately nine photons in the cat manifold. These two modes are the logical cat states.
• Figure 4: Matrix elements of the SNAIL charge operator in the Floquet mode basis as a function of drive amplitude, responsible for (a) direct tunneling and (b) intrawell leakage. In (b), we highlight three types of heating processes: solid lines indicate photon absorption at $\omega_d/2$, dashed lines at $\omega_d$, and dotted lines denote drive-induced processes that persist at zero temperature. Before the kissing between $\epsilon_2$ and $\epsilon_3$ (left-most vertical dashed line), tunneling is dominated by the intermediate transitions to these modes outside of the metapotential double well, with a rate proportional to $|X_{210}|^2 + |X_{301}|^2$, with additional contributions from other transitions whose matrix elements are shown in (b). After the kissing, these dominant processes interfere, and their contribution $|X_{210} - X_{301}|^2$ to the tunneling rate becomes suppressed. Furthermore, leakage to these modes does not lead to direct tunneling because the rate $|X_{230} - X_{321}|^2$ also becomes suppressed. This pattern repeats for higher modes at each of the other two spectral kissings indicated by the vertical dashed lines.
• Figure 5: Wigner distributions illustrating how the action of the dissipator affects the initially localized coherent state $\ket{\beta+\alpha}$ (indicated by the black circle). (a) At $\varepsilon_d/2\pi = 1.5GHz$, before the spectral kissing, the jump operators $\hat{L}_1 = X_{210}\ket{\phi_2}\bra{\phi_1}$ and $\hat{L}_2 = X_{301}\ket{\phi_3}\bra{\phi_0}$ act incoherently. The resulting state is $p_1 \hat{\rho}_1 + p_2 \hat{\rho}_2$, where $\hat{\rho}_{1,2} = \hat{L}_{1,2} \ket{\beta+\alpha}\bra{\beta+\alpha} \hat{L}_{1,2}^\dag /p_{1,2}$ and $p_{1,2} = \bra{\beta+\alpha} \hat{L}_{1,2}^\dag \hat{L}_{1,2} \ket{\beta+\alpha}$. The state becomes delocalized across the double-well. (b) At $\varepsilon_d/2\pi = 2.5GHz$, after the kissing, the jump operators interfere coherently. The resulting state is $\hat{L} \ket{\beta+\alpha} / \sqrt{\bra{\beta+\alpha} \hat{L}^\dag \hat{L} \ket{\beta+\alpha}}$ with $\hat{L} = \hat{L}_1 + \hat{L}_2$, preserving localization in the original well.