Symmetries of bosonic codes

Most popular bosonic codes obey either rotation symmetry or translation symmetry in phase space. In this talk I will discuss how the symmetries dictate which logical operations are fault tolerant, and some of the pros and cons of rotation symmetric vs. translation symmetric codes for scalable, fault-tolerant quantum computing. I will also introduce a new rotation symmetric code family called the Minimum Uncertainty Codes, which performs particularly well under loss and dephasing. Finally, I will show how the translation symmetry of Gottesman-Kitaev-Preskill codes allows for a particularly efficient gate implementation where all single-qubit Clifford gates are done in software and two-qubit gates are implemented with a three-wave mixing element.