Bosonic Cyclic Codes: Trading Stabilizers for Gaussian Non-Clifford Phase Gates

• URL: http://arxiv.org/abs/2606.11010v1
• Date: Tue, 09 Jun 2026 15:51:47 GMT
• Title: Bosonic Cyclic Codes: Trading Stabilizers for Gaussian Non-Clifford Phase Gates
• Abstract summary: We introduce bosonic cyclic codes: a generalization of rotation-symmetric codes.<n>We find that sacrificing the detectability of a single photon loss can yield a number of logical phase gates.<n>We discuss the larger $SU(2)$ symmetry and rotation gates of the codes, as well as new non-Clifford gates for the smallest kitten' binomial code.
• Score: 0.15534429177540246
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Bosonic codes offer hardware-efficient approaches to quantum error correction, with the best encodings offering effective protection of idle quantum information against loss and dephasing - particularly rotation-symmetric codes, which include the cat and binomial code families. However, rotation-symmetric codes are only naturally endowed with a single logical Pauli gate, while other logical gates require the use of non-linear operations, obstructing the utility of these codes for realizing quantum algorithms. Here, we balance error protection with controllability by introducing bosonic cyclic codes: a generalization of rotation-symmetric codes that enable the measured tradeoff of error protection properties for fault-tolerant logical phase gates. Through our general construction, we find that sacrificing the detectability of a single photon loss relative to a rotation-symmetric code can yield a number of logical phase gates commensurate with the original rotation symmetry order of the code, all achievable via passive Gaussian rotations. Giving the corresponding generalizations of cat and binomial codes - which we dub cyclic cat and Vandermonde codes, respectively - we further find that many of the desirable properties of these codes transfer to the bosonic cyclic code setting. We go on to discuss the larger $SU(2)$ symmetry and rotation gates of the codes, which yield additional stabilizers and logical Pauli gates, as well as new non-Clifford gates for the smallest `kitten' binomial code, and provide a new error detection protocol. Finally, we introduce a general paradigm for converting higher-order stabilizers to logical gates, as in our generalization of rotation-symmetric codes, and apply it to several multimode bosonic codes.

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