PreprintCC BY 4.0arXiv
Geometric aspects of fault tolerance
Abstract
We highlight three geometric aspects of fault tolerance. First, we construct an operator-algebraic formulation of the fiber bundle framework for fault tolerance, and show that a generalization of the Knill-Laflamme conditions implies a flat connection for this framework. Next, we develop this framework for Gottesman-Kitaev-Preskill (GKP) codes. We show that, for each polarization type, their moduli space embeds smoothly into a Grassmannian of finite-dimensional subspaces of a regularized distribution space. Finally, we explain how GKP codes associated with Jacobians give rise to magic states that can be obtained by code space projections of Gaussian states and identify a CCZ-type magic state arising this way.
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BibTeX
@misc{conrad2026geometric,
title = {{Geometric aspects of fault tolerance}},
author = {Jonathan Conrad},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35513},
eprint = {2610.35513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35513},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric aspects of fault tolerance
AU - Jonathan Conrad
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35513
UR - https://arxiv.org/abs/2610.35513
AB - We highlight three geometric aspects of fault tolerance. First, we construct an operator-algebraic formulation of the fiber bundle framework for fault tolerance, and show that a generalization of the Knill-Laflamme conditions implies a flat connection for this framework. Next, we develop this framework for Gottesman-Kitaev-Preskill (GKP) codes. We show that, for each polarization type, their moduli space embeds smoothly into a Grassmannian of finite-dimensional subspaces of a regularized distribution space. Finally, we explain how GKP codes associated with Jacobians give rise to magic states that can be obtained by code space projections of Gaussian states and identify a CCZ-type magic state arising this way.
ER -