Good Quantum Locally Testable Codes from Lossless Cubical Complexes
Abstract
Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.
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BibTeX
@misc{cohen2026good,
title = {{Good Quantum Locally Testable Codes from Lossless Cubical Complexes}},
author = {Itay Cohen and Itai Leigh and Assaf Reiner and Amnon Ta-Shma and Elad Tzalik},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00525},
eprint = {2610.00525},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00525},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Good Quantum Locally Testable Codes from Lossless Cubical Complexes
AU - Itay Cohen
AU - Itai Leigh
AU - Assaf Reiner
AU - Amnon Ta-Shma
AU - Elad Tzalik
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.00525
UR - https://arxiv.org/abs/2610.00525
AB - Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.
ER -