Combined Dissipative and Hamiltonian Cat Qubit Confinement*

Quantum error correction with biased-noised qubits can drastically reduce the hardware overhead for universal and fault-tolerant quantum computation. In particular, cat qubits feature an exponential error bias thanks to their non-local encoding in the phase space of a quantum harmonic oscillator, and are thus a promising kind of biased noise qubits. To confine the state of an oscillator to the cat qubit manifold, two main approaches exist: a Kerr-based conservative confinement with high gate performances, and a dissipative confinement with robust protection to thermal and dephasing noise. Here, we introduce a combined dissipative and Hamiltonian to benefit from the best of both worlds.