Error-transparent operations on a logical qubit protected by quantum error correction

Universal quantum computation is striking for its unprecedented capability in processing information, but its scalability is challenging in practice because of the inevitable environment noise. Although quantum error correction (QEC) techniques have been developed to protect stored quantum information from leading orders of errors, the noise-resilient processing of the QEC-protected quantum information is highly demanded but remains elusive. Here, we demonstrate phase gate operations on a logical qubit encoded in a bosonic oscillator in an error-transparent (ET) manner. Inspired by Refs. [9,10], the ET gates are extended to the bosonic code and are able to tolerate errors during the gate operations, regardless of the random occurrence time of the error. With precisely designed gate Hamiltonians through photon-number-resolved AC-Stark shifts, the ET condition is fulfilled experimentally. We verify that the ET gates outperform the non-ET gates with a substantial improvement of the gate fidelity after an occurrence of the single-photon-loss error. Our ET gates in the superconducting quantum circuits are readily for extending to multiple encoded qubits and a universal gate set is within reach, paving the way towards fault-tolerant quantum computation.