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 high-impedance <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>L</a:mi><a:mi>C</a:mi></a:math> 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 (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>T</c:mi><c:mn>1</c:mn></c:msub></c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>T</e:mi><e:mn>2</e:mn></e:msub></e:math>), even in the presence of extrinsic noise, imperfect control, and device-parameter variations. We show that the qubit supports exponentially robust single-qubit 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 <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>Q</g:mi></g:math> factors down to approximately <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>1000</i:mn></i:math> 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 <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>T</k:mi></k:math> gate.