Noise redistribution by stabilizer codes: an organizing view of concatenated three-qubit codes
Abstract
We present an organizing framework for concatenated quantum error correction based on the observation that the syndrome projection of a stabilizer code reshapes the noise channel in a code-determined manner. For any stabilizer code, one level of QEC maps the physical noise to a block-diagonal logical channel whose diagonal Pauli weights are fixed by the code geometry. Coherences between different syndrome classes vanish, and no physical Pauli twirl is required to reach this block-diagonal form. Concatenation can then be viewed as the iteration of the same redistribution step: the dominant logical error after one level selects the code used at the next. We instantiate the framework with the two smallest stabilizer codes, the bit-flip and phase-flip codes, and apply it to amplitude damping. The redistribution rules give the concatenation protocol C2(σ̂y) C1(σ̂x), whose ideal-recovery fidelity scales as fl=1−𝒪(γ2). Solving the crossover condition Fl>Fl−1 in the ideal-gate limit yields a critical damping parameter γc≈0.30; in terms of the initial channel fidelity, f=1−1.72755γ2. Under the same ideal-recovery assumption, we compare the overall gate accuracy rate with that of the five-qubit code. The scope of the framework, in particular for coherent and correlated noise, is stated explicitly.
Related
- Fault-tolerant quantum error correction for constant-excitation stabilizer codes under coherent noise
- Learning to Decode Concatenated Quantum Codes with Hierarchical Message Passing
- Circuit-level benchmarks of GKP-concatenated qLDPC Codes
- Injection is all you need: parallel universal logic in high-rate concatenated self-dual codes
- High-Rate Concatenated Quantum Error-Correcting Codes for Qudits
BibTeX
@misc{huang2026noise,
title = {{Noise redistribution by stabilizer codes: an organizing view of concatenated three-qubit codes}},
author = {Long Huang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31007},
eprint = {2610.31007},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31007},
}
RIS
TY - UNPB
TI - Noise redistribution by stabilizer codes: an organizing view of concatenated three-qubit codes
AU - Long Huang
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31007
UR - https://arxiv.org/abs/2610.31007
AB - We present an organizing framework for concatenated quantum error correction based on the observation that the syndrome projection of a stabilizer code reshapes the noise channel in a code-determined manner. For any stabilizer code, one level of QEC maps the physical noise to a block-diagonal logical channel whose diagonal Pauli weights are fixed by the code geometry. Coherences between different syndrome classes vanish, and no physical Pauli twirl is required to reach this block-diagonal form. Concatenation can then be viewed as the iteration of the same redistribution step: the dominant logical error after one level selects the code used at the next. We instantiate the framework with the two smallest stabilizer codes, the bit-flip and phase-flip codes, and apply it to amplitude damping. The redistribution rules give the concatenation protocol C_2(σ̂_y) C_1(σ̂_x), whose ideal-recovery fidelity scales as f_l=1−𝒪(γ^2). Solving the crossover condition F_l>F_l−1 in the ideal-gate limit yields a critical damping parameter γ_c≈0.30; in terms of the initial channel fidelity, f=1−1.72755γ^2. Under the same ideal-recovery assumption, we compare the overall gate accuracy rate with that of the five-qubit code. The scope of the framework, in particular for coherent and correlated noise, is stated explicitly.
ER -