Self-Correcting Gottesman-Kitaev-Preskill Qubit and Gates in a Driven-Dissipative Circuit

We show that a self-correcting Gottesman-Kitaev-Preskill (GKP) qubit can be realized with a highimpedance 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 the coherence time (T1 and T2), even in the presence of extrinsic noise, imperfect control, and device-parameter variations. We show that the qubit supports exponentially robust singlequbit Clifford gates, implemented via appropriate control of the switch, and readout and/or initialization via supercurrent measurement. The self-correcting properties of the qubit allow it to operate at approximately 1-K temperatures and resonator Q factors down to approximately 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.