Quantum error correction of a qubit encoded in grid states of an oscillator

Quantum bits are more robust to noise when they are encoded non-locally. In such an encoding, errors affecting the underlying physical system can then be detected and corrected before they corrupt the encoded information. In 2001, Gottesman, Kitaev and Preskill (GKP) proposed a hardware-efficient instance of such a qubit, which is delocalised in the phase-space of a single oscillator. However, implementing measurements that reveal error syndromes of the oscillator while preserving the encoded information has proved experimentally challenging: the only realisation so far relied on post-selection, which is incompatible with quantum error correction (QEC). The novelty of our experiment is precisely that it implements these non-destructive error-syndrome measurements for a superconducting microwave cavity. We design and implement an original feedback protocol that incorporates such measurements to prepare square and hexagonal GKP code states. We then demonstrate QEC of an encoded qubit with unprecedented suppression of all logical errors, in quantitative agreement with a theoretical estimate based on the measured imperfections of the experiment. Our protocol is applicable to other continuous variable systems and, in contrast with previous implementations of QEC, can mitigate all logical errors generated by a wide variety of noise processes, and open a way towards fault-tolerant quantum computation.