  The unique features of quantum theory offer a powerful new paradigm for
information processing. Translating these mathematical abstractions into useful
algorithms and applications requires quantum systems with significant
complexity and sufficiently low error rates. Such quantum systems must be made
from robust hardware that can coherently store, process, and extract the
encoded information, as well as possess effective quantum error correction
(QEC) protocols to detect and correct errors. Circuit quantum electrodynamics
(cQED) provides a promising hardware platform for implementing robust quantum
devices. In particular, bosonic encodings in cQED that use multi-photon states
of superconducting cavities to encode information have shown success in
realizing hardware-efficient QEC. Here, we review recent developments in the
theory and implementation of quantum error correction with bosonic codes and
report the progress made towards realizing fault-tolerant quantum information
processing with cQED devices.
