Note on Logical Gates by Gauge Field Formalism of Quantum Error Correction

Abstract:The gauge field formalism, or operator-valued cochain formalism, has recently emerged as a powerful framework for describing quantum Calderbank-Shor-Steane (CSS) codes. In this work, we extend this framework to construct a broad class of logical gates for general CSS codes, including the S, Hadamard, T, and (multi)controlled-Z gates, under the condition where fault-tolerance or circuit-depth optimality is not necessarily imposed. We show that these logical gates can be expressed as exponentials of polynomial functions of the electric and magnetic gauge fields, which allows us to derive explicit decompositions into physical gates. We further prove that their logical action depends only on the (co)homology classes of the corresponding logical qubits, establishing consistency as logical operations. Our results provide a systematic method for formulating logical gates for general CSS codes, offering new insights into the interplay between quantum error correction, algebraic topology, and quantum field theory.

arXiv:2511.15224 [hep-th]

(or arXiv:2511.15224v2 [hep-th] for this version)

https://doi.org/10.48550/arXiv.2511.15224 Focus to learn more arXiv-issued DOI via DataCite

Prog. Theor. Exp. Phys. 2026, 073B12

https://doi.org/10.1093/ptep/ptag105 Focus to learn more DOI(s) linking to related resources

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.