  Realizing the potential of quantum computing will require achieving
sufficiently low logical error rates. Many applications call for error rates in
the $10^{-15}$ regime, but state-of-the-art quantum platforms typically have
physical error rates near $10^{-3}$. Quantum error correction (QEC) promises to
bridge this divide by distributing quantum logical information across many
physical qubits so that errors can be detected and corrected. Logical errors
are then exponentially suppressed as the number of physical qubits grows,
provided that the physical error rates are below a certain threshold. QEC also
requires that the errors are local and that performance is maintained over many
rounds of error correction, two major outstanding experimental challenges.
Here, we implement 1D repetition codes embedded in a 2D grid of superconducting
qubits which demonstrate exponential suppression of bit or phase-flip errors,
reducing logical error per round by more than $100\times$ when increasing the
number of qubits from 5 to 21. Crucially, this error suppression is stable over
50 rounds of error correction. We also introduce a method for analyzing error
correlations with high precision, and characterize the locality of errors in a
device performing QEC for the first time. Finally, we perform error detection
using a small 2D surface code logical qubit on the same device, and show that
the results from both 1D and 2D codes agree with numerical simulations using a
simple depolarizing error model. These findings demonstrate that
superconducting qubits are on a viable path towards fault tolerant quantum
computing.
