Fig. 1.

Kerr-cat qubit concepts. (A) SNAIL nonlinear oscillator circuit with frequency ${\omega }_Q$, nonlinear factors $g_3,g_4$ and Kerr nonlinearity K, and its energy levels with wave-functions. (B) SNAIL oscillator pumped by a two-photon stabilization drive at ${\omega }_S=2{\omega }_Q$, and the engineered pseudopotential profile with its energy levels and wave-functions. (C) Pseudopotential profile and its energy levels engineered by a red-detuned two-photon stabilization drive, where the excited states become degenerate. (D) Bloch sphere defined by the coherent states and their superpositions, where $|\pm Z⟩=|\pm \alpha ⟩$, $|\pm X⟩=|C_{\alpha }^{\pm }⟩\propto |\alpha ⟩\pm |-\alpha ⟩$, $|\pm Y⟩=|C_{\alpha }^{\pm i}⟩\propto |\alpha ⟩\pm i|-\alpha ⟩$. (E) Pseudopotential profile under a single-photon drive Ω at ${\omega }_S/2$ for $Z(\theta )$ gates. (F) Pseudopotential profile under a blue-detuned two-photon stabilization drive for the $X(\pi /2)$ gate. The detuning was chosen to be $5K$ as an example.