Quantum group codes for non-Clifford logic: enhanced decoding, addressability and parallelizability
Abstract
We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-Z gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-Z gate parallelizability. More precisely, on input a good quantum AG code over the alphabet 𝔽q with transversal CmZ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet 𝔽q2 supporting a transversal CmZ gate as well as addressable and parallelizable Cm−1Z gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.
Related
- Hybrid Clifford Codes via Operator Algebra Quantum Error Correction and Projective Representation Theory
- Parallelizable and addressable transversal non-Clifford gates on good quantum LDPC codes
- Clifford Invariants and Three-Copy Quantum Tests
- Non-Clifford quantum cellular automata from invertible topological quantum field theories
- Minimax Quantum State Tomography with Periodic Clifford Measurements
BibTeX
@misc{gasnier2026quantum,
title = {{Quantum group codes for non-Clifford logic: enhanced decoding, addressability and parallelizability}},
author = {Jean Gasnier and Virgile Guémard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.27211},
eprint = {2606.27211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27211},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantum group codes for non-Clifford logic: enhanced decoding, addressability and parallelizability
AU - Jean Gasnier
AU - Virgile Guémard
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2606.27211
UR - https://arxiv.org/abs/2606.27211
AB - We introduce a framework based on classical quasi group codes to define a class of quantum CSS codes, called quantum group codes, supporting transversal multi-control-Z gates which are both addressable and parallelizable, thus allowing to efficiently implement circuits composed of non-Clifford gates at the logical level. Building on this, we use a lifting procedure of classical AG codes established from class field theory to construct good quantum group codes with improved decoding complexity and logical multi-control-Z gate parallelizability. More precisely, on input a good quantum AG code over the alphabet 𝔽_q with transversal C^mZ gate, we apply this lifting procedure to its underlying classical AG code and obtain a quantum group code over the alphabet 𝔽_q^2 supporting a transversal C^mZ gate as well as addressable and parallelizable C^m−1Z gates. In addition, this quantum code admits a quasi-quadratic time decoder with a linear decoding radius. This is to be compared with the previous quantum AG codes which have a cubic-time decoder. Hence, our work implies a decrease of the time complexity of state-of-the-art magic-state distillation protocols by an almost linear factor.
ER -