Self-correcting GKP qubit and gates in a driven-dissipative circuit

We show that a self-correcting GKP qubit can be realized with a high-impedance LC circuit coupled to a resistor and a Josephson junction via a controllable switch. When activating the switch in a particular stepwise pattern, the resonator relaxes into a subspace of GKP states that encode a protected qubit. Under continued operation, the resistor dissipatively error-corrects the qubit against bit flips and decoherence by absorbing noise-induced entropy. We show that this leads to an exponential enhancement of coherence time (T1 and T2), even in the presence of extrinsic noise, imperfect control, and device parameter variations. We show the qubit supports exponentially robust single-qubit Clifford gates, implemented via appropriate control of the switch, and readout/initialization via supercurrent measurement. The qubit's self-correcting properties allows it to operate at ~1K temperatures and resonator Q factors down to ~1000 for realistic parameters, and make it amenable to parallel control through global control signals. We discuss how the effects of quasiparticle poisoning -- potentially, though not necessarily, a limiting factor -- might be mitigated. We finally demonstrate that a related device supports a self-correcting magic T gate.