Enhancing Kerr-Cat Qubit Coherence with Controlled Dissipation

TL;DR

This work identifies leakage out of the Kerr-cat qubit manifold as a key limiter of bit-flip protection and demonstrates a hybrid stabilization strategy that merges Hamiltonian confinement with frequency-selective engineered dissipation. By coherently controlling transitions to leakage manifolds and coupling to a dissipative channel, the authors quantify leakage $p_1$ (≈9% without dissipation) and show that engineered dissipation reduces $p_1$ to a few percent, concurrently increasing the bit-flip time $T_Z$ from hundreds of microseconds to as high as 3.6 ms. Importantly, the dissipation is frequency-selective, leaving the X and Y coherence times $T_X$ and $T_Y$ largely intact (~2.5 μs), thereby enhancing noise bias without sacrificing KCQ operations. The results illuminate the interplay between Hamiltonian confinement and dissipative stabilization, offering a practical route toward robust bosonic quantum error correction with Kerr-cat qubits and guiding principles for dissipative control in driven nonlinear oscillators.

Abstract

Quantum computing crucially relies on maintaining quantum coherence for the duration of a calculation. Bosonic quantum error correction protects this coherence by encoding qubits into superpositions of noise-resilient oscillator states. In the case of the Kerr-cat qubit (KCQ), these states derive their stability from being the quasi-degenerate ground states of an engineered Hamiltonian in a driven nonlinear oscillator. KCQs are experimentally compatible with on-chip architectures and high-fidelity operations, making them promising candidates for a scalable bosonic quantum processor. However, their bit-flip time must increase further to fully leverage these advantages. Here, we present direct evidence that the bit-flip time in a KCQ is limited by leakage out of the qubit manifold and experimentally mitigate this process. We coherently control the leakage population and measure it to be > 9%, twelve times higher than in the undriven system. We then cool this population back into the KCQ manifold with engineered dissipation, identify conditions under which this suppresses bit-flips, and demonstrate increased bit-flip times up to 3.6 milliseconds. By employing both Hamiltonian confinement and engineered dissipation, our experiment combines two paradigms for Schrödinger-cat qubit stabilization. Our results elucidate the interplay between these stabilization processes and indicate a path towards fully realizing the potential of these qubits for quantum error correction.

Figures (18)

• Figure 1: Experiment concept and implementation. a, Photograph of the superconducting nonlinear oscillator chip ($a$, blue) placed inside one half of the microwave cavity ($b$, orange). b, Circuit schematic of the nonlinear oscillator and microwave cavity system, with their respective frequencies $\omega_{\mathrm{a},\mathrm{b}}$ and loss rates $\kappa_{\mathrm{a},\mathrm{b}}$ indicated. The nonlinear oscillator consists of an array of three SNAILs frattini_3-wave_2017 in parallel with a shunting capacitance. Microwave drives applied to the oscillator-cavity system, as well as parametric interactions, are respectively represented by colored pulses and two-way arrows (see text for description). c, Sketch of the quasi-potential energy $E$ of the Hamiltonian (Eq. \ref{['eq:hamiltonian']}) as a function of the complex phase-space coordinate $\beta$ (blue). Black contour lines correspond to energies of the eigenstates $\left|\psi_{i}^{\pm}\right>$ of Eq. \ref{['eq:hamiltonian']}, with transition frequencies in the rotating frame, $\omega_{ij}$, indicated. An energy splitting $\Delta E_{1}$ ($\Delta E_{2}$) between the $\left|\psi_{\mathrm{1}}^{\pm}\right>$ ($\left|\psi_{\mathrm{2}}^{\pm}\right>$) states is schematically indicated. Engineered dissipation (red arrows) brings population from $\left|\psi_{\mathrm{1}}^{\pm}\right>$ to $\left|\psi_{\mathrm{0}}^{\pm}\right>$. d, Wigner function representations of the states $\left|\pm Z\right>$ and $\left|+X\right>$ in the KCQ manifold, for $\varepsilon_{2}={2.4} K$ and $\Delta={8} K$.
• Figure 2: Coherent control and population measurement of leakage manifolds. a, Pulse sequence to perform the following functions: (1) initialize a steady-state population across qubit and leakage manifolds, with wait time $\tau_{\mathrm{wait}}$ indicated; (2) coherently drive transitions between $\left|\psi_{i}^{\pm}\right>$ and $\left|\psi_{j}^{\pm}\right>$ manifolds ($i\neq j$), where the blue dotted box is a placeholder for pulses depicted in the panel insets; and (3) measure the $\left|\psi_{i}^{\pm}\right>$-manifold-dependent readout (RO) cavity response. b, Phase of reflected readout signal with the oscillator in $\left|\psi_{\mathrm{0}}^{\pm}\right>$ (black), $\left|\psi_{\mathrm{1}}^{\pm}\right>$ (light blue) and $\left|\psi_{\mathrm{2}}^{\pm}\right>$ (dark blue). The horizontal axis shows the detuning $\delta \omega_{\mathrm{b}} = \omega_{\mathrm{probe}} - \omega_{\mathrm{b}}$, where $\omega_{\mathrm{probe}}$ is the readout signal frequency and $\omega_{\mathrm{b}}$ is the cavity resonance in the absence of the squeezing drive. c, Rabi oscillations for the $\left|\psi_{\mathrm{1}}^{\pm}\right> \leftrightarrow \left|\psi_{\mathrm{2}}^{\mp}\right>$ transition as a function of pulse amplitude $A$ in units of the arbitrary waveform generator output voltage. Measurements performed with (without) an initial $\pi_{\mathrm{01}}$-pulse are shown by light (dark) blue markers. The conditional $\pi_{\mathrm{01}}$-pulse is indicated by a white fill in the pulse sequence. Solid lines are fits to extract the leakage population $p_{1}$ (see Supplementary Information Section IV.F). d, Schematic representation of the relaxation times $T_{1}^{ij}$, and pure dephasing times $T_{\phi}^{ij}$ between $\left|\psi_{i}^{\pm}\right>$ and $\left|\psi_{j}^{\pm}\right>$ manifolds. e--h Coherence measurements of $\left|\psi_{\mathrm{1}}^{\pm}\right>$ (e,f) and $\left|\psi_{\mathrm{2}}^{\pm}\right>$ (g,h) manifolds. Experimental data (dots) are plotted alongside an analytical fit (solid line). Measurements were performed for $\varepsilon_{2}={2.4} K$ and $\Delta={8} K$. The y-axis in panels c,e,f,g and h is normalized with respect to the $\left|\psi_{\mathrm{0}}^{\pm}\right> \leftrightarrow \left|\psi_{\mathrm{1}}^{\mp}\right>$ Rabi contrast.
• Figure 3: Effect of engineered dissipation on leakage population.a, Schematic of the engineered dissipation process acting on the oscillator. The state of the oscillator-cavity system is indicated with $\left|\psi_i^\pm,n\right>$, where $\left|\psi_i^\pm\right>$ labels an oscillator manifold and $\left|n\right>$ the cavity photon number. The states $\left|\psi_{\mathrm{1}}^{\pm}\right>$ have an energy splitting $\Delta E_1$. A coherent photon exchange interaction with rate $g_{\mathrm{diss}}$ (pink) resonantly couples $\left|\psi_{1}^{\mp},0\right>$ and $\left|\psi_{\mathrm{0}}^{\pm},1\right>$ when $\delta\omega_{\mathrm{diss}} = 0$. The cavity relaxation, at rate $\kappa_{\mathrm{b}}$, transfers population from $\left|\psi_{\mathrm{0}}^{\pm},1\right>$ to $\left|\psi_{\mathrm{0}}^{\pm},0\right>$. b, Leakage population $p_{1}$ as a function of $g_{\mathrm{diss}}$ and $\kappa_{\mathrm{diss}}$. Experimental data (markers) for different values of $p_{2}$ are compared with simulation results (shaded regions), which include quantum heating (red) as well as thermal noise in the oscillator (yellow) and cavity (green). Error bars correspond to an uncertainty of one standard deviation obtained from the fitting procedure (see Supplementary Information Section IV.F). Measurements were performed for $\varepsilon_{2}={2.4} K$ and $\Delta={8} K$.
• Figure 4: Impact of engineered dissipation on bit-flip time, $T_{\mathrm{Z}}$. a, Pulse sequence to measure the change in $T_{\mathrm{Z}}$ due to engineered dissipation. Variables inside pulses indicate parameters swept to obtain the data shown in panels b, c and e. b, Relative change in $Z$-state readout contrast, $\delta\langle Z\rangle$, as a function of $\delta\omega_{\mathrm{diss}}$ and $\varepsilon_{2}$, for $\Delta={7} K$. The orange dotted line identifies $\varepsilon_{\mathrm{2,th}}$. Black and gray dots indicate the parameters used in c. c, Average $Z$-state readout contrast as a function of delay time $\Delta t$, for $\varepsilon_{2}=2.26K$ and $\Delta={7} K$. Experimental data for on- (off-) resonant engineered dissipation are shown as black (gray) dots. Solid lines show exponential fits to the data. d, Simulation of the experiment in b, as explained in the text. The blue dotted line identifies the simulated $\varepsilon_{\mathrm{2,th}}$ value. e, Threshold values $\varepsilon_{\mathrm{2,th}}$ for different $\Delta$, extracted from measurements as in b (orange circles), or simulations as in d (blue diamonds). Filled markers denote a transition from negative to positive $\delta\langle Z\rangle$ (as in b,d), indicating a transition from decreased ($\downarrow$) to increased ($\uparrow$) $T_{\mathrm{Z}}$. Open markers denote measurements for which $\delta\langle Z\rangle$ saturates to zero above the transition ($T_{\mathrm{Z}}\leftrightarrow$). Error bars on measured $\varepsilon_{\mathrm{2,th}}$ correspond to one standard deviation and account for both the uncertainty in the $\varepsilon_{2}$ calibration and statistical noise in the measurement data. Solid (dashed) lines show isolines of the energy splitting $\Delta E_{1}$ for $\left|\psi_{\mathrm{1}}^{\pm}\right>$ ($\Delta E_{2}$ for $\left|\psi_{\mathrm{2}}^{\pm}\right>$) with values given in frequency units ($\Delta E_{i}/h$). The values of $\varepsilon_{2}$ and $\Delta$ used for b are indicated by the vertical black dotted line, and measurements in Figs. \ref{['fig2']}, \ref{['fig3']} and \ref{['fig5']} are performed for parameters indicated by the star.
• Figure 5: Dependence of KCQ coherence times on dissipation rate.a, Pulse sequence to measure bit-flip time, $T_{\mathrm{Z}}$, with engineered dissipation. The swept delay time $\Delta t$ and engineered interaction rate $g_{\mathrm{diss}}$ are indicated. b, Pulse sequence to measure $X$- and $Y$-state coherence times, $T_{\mathrm{X,Y}}$. We initialize the KCQ in $\left|+X\right>$ or $\left|+Y\right>$ by ramping on the squeezing drive followed by a conditional gate $Z_{1}$. After the delay time, the gates $Z_{2}$ and $X(\pi/2)$ rotate the decayed state to the $Z$-axis for readout (see Methods). c, Measured $T_{\mathrm{Z}}$ as a function of $g_{\mathrm{diss}}$, and the corresponding rate $\kappa_{\mathrm{diss}}$. d, Measured $T_{\mathrm{X}}$ ($T_{\mathrm{Y}}$) as a function of $g_{\mathrm{diss}}$ and $\kappa_{\mathrm{diss}}$ indicated by light blue diamonds (dark blue dots). Measurements were performed for $\varepsilon_{2}={2.4} K$ and $\Delta={8} K$.