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

The accuracy of logical operations on quantum bits (qubits) must be improved for quantum computers to outperform classical ones in useful tasks. One method to achieve this is quantum error correction (QEC), which prevents noise in the underlying system from causing logical errors. This approach derives from the reasonable assumption that noise is local, that is, it does not act in a coordinated way on different parts of the physical system. Therefore, if a logical qubit is encoded non-locally, we can-for a limited time-detect and correct noise-induced evolution before it corrupts the encoded information(1). In 2001, Gottesman, Kitaev and Preskill (GKP) proposed a hardware-efficient instance of such a non-local qubit: a superposition of position eigenstates that forms grid states of a single oscillator(2). However, the implementation of measurements that reveal this noise-induced evolution of the oscillator while preserving the encoded information(3-7) has proved to be experimentally challenging, and the only realization reported so far relied on post-selection(8,9), which is incompatible with QEC. Here we experimentally prepare square and hexagonal GKP code states through a feedback protocol that incorporates non-destructive measurements that are implemented with a superconducting microwave cavity having the role of the oscillator. We demonstrate QEC of an encoded qubit with 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 to previous implementations of QEC(10-14), can mitigate all logical errors generated by a wide variety of noise processes and facilitate fault-tolerant quantum computation.