Critical Schrodinger Cat Qubit

Encoding quantum information onto bosonic systems is a promising route to quantum error correc-tion. In a cat code, this encoding relies on the confinement of the dynamics of the system onto the two-dimensional manifold spanned by Schrodinger cats of opposite parity. In dissipative cat qubits, an engineered dissipation scheme combining two-photon drive and two-photon loss has been used to autonomously stabilize this manifold, ensuring passive protection against, e.g., bit-flip errors regardless of their origin. Similarly, in Kerr-cat qubits, where highly performing gates can be engineered, two-photon drive and Kerr nonlinearity cooperate to confine the system to a twofold-degenerate ground-state manifold spanned by cat states of opposite parity. Dissipative, Hamiltonian, and hybrid confinement mechanisms have been investigated at resonance, i.e., for driving frequencies matching that of the cavity. Here, we pro-pose a critical cat code, where both two-photon loss and Kerr nonlinearity are present and the two-photon drive is allowed to be out of resonance. The performance of this code is assessed via the spectral theory of Liouvillians in all configurations ranging from the purely dissipative to the Kerr limit. We show that large detunings and small, but non-negligible, two-photon loss rates are fundamental to achieve optimal performance. We further demonstrate that the competition between nonlinearity and detuning results in a first-order dissipative phase transition, leading to a squeezed vacuum steady state. We show that to achieve the maximal suppression of the logical bit-flip rate requires initializing the system in the metastable state emerging from the first-order transition and we detail a protocol to do so. Efficiently operating over a broad range of detuning values, the critical cat code is particularly resistant to random frequency shifts charac-terizing multiple-qubit operations, opening avenues for the realization of reliable protocols for scalable and concatenated bosonic qubit architectures.