Driving superconducting qubits into chaos

TL;DR

This work analyzes the stability of Kerr-cat qubits realized in driven Kerr parametric oscillators implemented with SNAIL transmon circuits. By combining Floquet theory for the driven quantum oscillator with classical phase-space methods (Poincaré sections and Lyapunov exponents), it identifies a boundary between regular dynamics and local chaos that can melt the Kerr-cat qubit. The onset of chaos is linked to the formation and subsequent disintegration of a Bernoulli lemniscate in phase space near the phase-space center, with a threshold near $\Gamma K/\omega_0 \approx 0.0187$ marking the first positive Lyapunov growth, followed by a broader chaotic region as nonlinearities increase. The authors propose practical chaos-detection metrics based on the Floquet state $|\mathcal{F}_{\text{min}}\rangle$, such as the growth of $n_{\text{min}} = \langle \mathcal{F}_{\text{min}}| \hat{n} |\mathcal{F}_{\text{min}}\rangle$ and the Shannon entropy $S_{\text{min}}$, and discuss implications for gate speed limits and for using chaos as a platform to study quantum dynamics in superconducting circuits.

Abstract

Kerr parametric oscillators are potential building blocks for fault-tolerant quantum computers. They can stabilize Kerr-cat qubits, which offer advantages toward the encoding and manipulation of error-protected quantum information. The recent realization of Kerr-cat qubits made use of the nonlinearity of the SNAIL transmon superconducting circuit and a squeezing drive. Increasing nonlinearities can enable faster gate times, but, as shown here, can also induce chaos and melt the qubit away. We determine the region of validity of the Kerr-cat qubit and discuss how its disintegration could be experimentally detected. The danger zone for parametric quantum computation is also a potential playground for investigating quantum chaos with driven superconducting circuits.

Figures (5)

• Figure 1: Regularity and chaos. Quantum and classical analysis in phase space and the quasienergy spectrum for $\omega_d/\omega_0=1.999866$. a - f, Phase space analysis of the parameters indicated in Table \ref{['tab:points6']}. The black dots give the classical Poincaré sections for many different initial conditions, the red line in a marks the separatrix that defines the Bernoulli lemniscate, colors from blue to orange indicate the values of the participation ratio of coherent states projected in the Floquet states. g, Measure of quantum chaos given by the average ratio $\overline{r}$ of consecutive quasienergies spacings as a function of $K/\omega_0$ and $\Gamma$. The six points A-F marked in g are the same ones chosen for the phase spaces in a-f. They were selected to illustrate the behavior in the regular, mixing, and chaotic regimes. The solid black curve in g corresponds to equation (\ref{['eq:LEinside']}) and indicates the parametric case, where the classical Lyapunov exponent becomes positive in the vicinity of the center of the lemniscate, while the black dashed line corresponds to equation (\ref{['eq:LEinout']}) and indicates the parameters for which chaos sets in both inside and outside the original lemniscate, which by then has disappeared. h - i, Lyapunov exponents for the same parameters used in (d)-(e). Zero Lyapunov exponent (dark blue) indicates regularity.
• Figure 2: Kerr-cat qubit disintegration. a, Expectation value of the number operator, $n_{\text{min}}$, and b, Shannon entropy, $S_{\text{min}}$, for the Floquet state $|\mathcal{F}_{\text{min}}\rangle$. The two quantities are shown as a function of the Kerr amplitude $K/\omega_0$ for $\Gamma=30$ (triangles) and $\Gamma=80$ (squares). In a: The blue background indicates regular region and the orange background indicates chaotic region; they are separated by the same black dashed line shown in Fig. \ref{['fig:1']}(g). Panels (I), (II), (III), and (IV) depict the Husimi functions for the Floquet state $|\mathcal{F}_{\text{min}}\rangle$ indicated in a and b as points (I), (II), (III), and (IV) with $K/\omega_0=\{0.33,3.66,8.66, 12\} \times 10^{-4}$.
• Figure 3: Comparison between full and expanded SNAIL potential in phase space. a, Poincaré sections from the classical Hamiltonian with the full SNAIL potential. b, Poincaré sections from classical Hamiltonian with the SNAIL potential expanded to fourth order. Both plots have the parameters corresponding to the point D in the Table 1 of the main text. Black points are used for regular orbits and red points are used for chaotic orbits. The blue solid line represents the separatrix for the non-expanded potential ( a) and the potential expanded to fourth order ( b).
• Figure 4: Emergence of the Kerr cat qubit. a, Phase space metapotential of the classical Hamiltonian $h_{cl}(t)$ in equation (\ref{['eq:SMh0']}) representing a large asymmetric double well. Black points are used for regular orbits. The red points indicate orbits with positive Lyapunov exponents (chaos). The two green symbols indicate the critical points: circle for $(q_0,p_0)=(0,0)$, and cross for $(q_2, p_2)=(d_{+},0)$. The blue line is the separatrix of the asymmetric double well. b, Enlarged image of panel a close to the point $(0,0)$, providing a view of the additional symmetric double well that emerges at the phase space center. The red line is the Bernoulli lemniscate. The distance between the two minima is $2 \sqrt{2 \Gamma}$. c, Enlarged image of panel b close to the point $(0,0)$. The distance between the phase space center $(0,0)$ and the hyperbolic point of the Bernoulli lemniscate is $\sqrt{2} \Pi$.
• Figure 5: Parameter selection and quantum chaos maps. a, Kerr amplitude $K^{(2)}$ of the second-order effective Hamiltonian (in color) as a function of $g_3/\omega_0$ and $g_4/\omega_0$. Red is used for $K^{(2)}>0$ and blue for $K^{(2)}<0$; the solid black lines mark constant values of $K^{(2)}$; the green line marked as (c) is for $K^{(2)}=g_4$, the cyan line (d) is for $K^{(2)}=10g_4$, the purple line (e) is for $K^{(2)}=100g_4$, and the orange line (f) is for $K^{(2)}=-0.6976g_4$. b, Absolute difference between $K^{(2)}$ and $K$ as a function of $|K/\omega_0|$ for different choices of $K^{(2)}=\mathcal{C}g_4$, as indicated. c-f, Measure of quantum chaos given by the average ratio $\overline{r}$ of consecutive quasienergies spacings as a function of $K/\omega_0$ and $\Gamma$, for $\omega_d/\omega_0=1.999866$ and ( c) $K^{(2)}/g_4=1$, ( d) $K^{(2)}/g_4=10$, ( e) $K^{(2)}/g_4=100$, and ( f) $K^{(2)}/g_4=-0.6976$. The black solid curve indicatesthe parametric case, where the classical Lyapunov exponent becomes positive in the vicinity of the center of the lemniscate, and the black dashed curve indicates the parameters for which chaos sets in both inside and outside the original lemniscate.