From Syndromes to Computation: A General Framework for Logical Operations in Quantum Error-Correcting Codes
Abstract
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈ d/2⌉ for distance-d codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed T gate on the $[[16,4,4]]$ BB code, T-type rotations on the $[[64,8,8]]$ BB code, and CCZ-, CS-, and T-type operations on the $[[36,6,6]]$ Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields CrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.
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BibTeX
@misc{nelson2026from,
title = {{From Syndromes to Computation: A General Framework for Logical Operations in Quantum Error-Correcting Codes}},
author = {Hunter Nelson and Arshag Danageozian and ChunJun Cao and Sophia E. Economou and Edwin Barnes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17612},
eprint = {2610.17612},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17612},
}
RIS
TY - UNPB
TI - From Syndromes to Computation: A General Framework for Logical Operations in Quantum Error-Correcting Codes
AU - Hunter Nelson
AU - Arshag Danageozian
AU - ChunJun Cao
AU - Sophia E. Economou
AU - Edwin Barnes
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17612
UR - https://arxiv.org/abs/2610.17612
AB - Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈ d/2⌉ for distance-d codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed T gate on the $[[16,4,4]]$ BB code, T-type rotations on the $[[64,8,8]]$ BB code, and CCZ-, CS-, and T-type operations on the $[[36,6,6]]$ Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields C^rZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.
ER -