Progress with stabilized cat qubits towards hardware-efficient fault-tolerance*

Cat qubits are bosonic encodings of quantum information that promise a significant reduction in hardware overhead towards fault-tolerance. By confining the state of a quantum harmonic oscillator — a superconducting cavity mode for instance — to the 2D manifold of the Schrödinger cat states, one obtains a biased noise qubit where one error component (bit-flips) is exponentially suppressed with the cat size. It is then possible to perform an extensive set of bias-preserving gates paving the way towards a hardware-efficient concatenation with another encoding for suppressing the other error component (phase-flips). So far, two types of confinements have been considered for such cat qubits each with its own advantages: a dissipative confinement based on an engineered nonlinear dissipation, and a Hamiltonian confinement based on a driven Kerr nonlinearity. In this work, we analyze the possibility of combining these two types of confinements and the benefits of such a combination. We also discuss the optimization of the phase-flip error correction process in view of simplifying the experimental requirements to reach the fault-tolerance threshold.