A critical Schrodinger cat qubit*

Encoding quantum information onto bosonic systems is a promising route to quantum error correction. In a cat code, this encoding relies on the confinement of the system’s dynamics 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 dissipation has been used to autonomously stabilize this manifold, ensuring passive protection against, e.g., phase-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 Hamiltonianly confine the system onto the cat manifold.

Dissipative, Hamiltonian, and hybrid confinement mechanisms have been investigated at resonance, i.e., for driving frequencies matching that of the cavity. We propose a critical cat code, where both two-photon dissipation and Kerr nonlinearity operate in a detuned regime. The competition between nonlinearity and detuning triggers a first-order dissipative phase transition, making the encoding efficient over a wide range of parameters in the proximity of the critical point.

The performance of the code is benchmarked within the general framework of the Liouvillian spectral theory. We introduce a channel fidelity leakage rate, a measure allowing for a fair comparison between our critical stabilization mechanism and its Hamiltonian, dissipative, and resonant-hybrid counterparts in the presence of both photon loss and dephasing noise. We find that the critical cat outperforms the others, and we show that this enhanced performance lies within reach of current experimental setups. Efficiently operating over a broad range of detuning values, the critical cat code is particularly resistant to random frequency shifts characterizing multiple-qubit operations, opening venues for the realization of reliable protocols for scalable and concatenated bosonic qubit architectures.