Researchers have developed a new family of quantum error correction codes, called bosonic cyclic codes, addressing a key limitation of rotation-symmetric bosonic codes. The latter, while efficient in protecting idle quantum information against loss and dephasing, naturally allow only a single logical Pauli gate. Other logical operations require non-linear operations, hindering their application in complex quantum algorithms. Cyclic codes propose a balance between error protection and controllability, enabling the implementation of fault-tolerant logical phase gates.

The general construction of these codes reveals that sacrificing the detectability of a single photon loss, compared to a rotation-symmetric code, can yield a number of logical phase gates commensurate with the code's original rotation symmetry order. These gates are achieved via passive Gaussian rotations, simplifying their implementation. The researchers have applied this generalization to the well-known cat and binomial codes, creating cyclic cat and Vandermonde codes, respectively, and have confirmed that many of their desirable properties are preserved.

This advance also explores the SU(2) symmetry and rotation gates of the codes, which provide additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest binomial code, known as 'kitten'. Furthermore, a new error detection protocol is introduced. Finally, the research establishes a general paradigm for converting higher-order stabilizers into logical gates, a technique successfully applied to several multimode bosonic codes, opening new avenues for fault-tolerant quantum computing.