Protected and controllable GKP qubits in high-impedance Josephson circuits.

Dynamics of an oscillator governed by two commuting modular operators can stabilize the GKP code. We propose to effectively engineer these in the rotating frame of a high-impedance Josephson circuit coupled to a cold reservoir. Numerical simulations show that logical errors triggered by single-photon dissipation are then exponentially suppressed when increasing the engineered dissipation strength. Our scheme also allows for protected initialization and readout of the encoded qubits, and for a set of exact Clifford gates.