Haah's 3D cubic code thermalizes rapidly
Abstract
Haah's 3D cubic code was introduced as a candidate for a self-correcting quantum memory in three dimensions, using the absence of string logical operators to obstruct thermal errors. It has been a long-standing open question whether it is a self-correcting quantum memory. Even though the memory time was explored numerically, no analytical upper bound was known. Here, we settle this question rigorously and show that the cubic code thermalizes rapidly at every positive temperature. Our proof relies on the framework of [Commun. Math. Phys. 407, 195 (2026)] to turn a decay-of-correlations condition into a modified logarithmic Sobolev inequality for the Davies generator. To the best of our knowledge, this is the first analytic proof of rapid mixing for all positive temperatures for any fracton code.
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BibTeX
@misc{stengele2026haahs,
title = {{Haah's 3D cubic code thermalizes rapidly}},
author = {Sebastian Stengele and Libor Caha and Ángela Capel and Simone Warzel},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39959},
eprint = {2609.39959},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39959},
}
RIS
TY - UNPB
TI - Haah's 3D cubic code thermalizes rapidly
AU - Sebastian Stengele
AU - Libor Caha
AU - Ángela Capel
AU - Simone Warzel
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39959
UR - https://arxiv.org/abs/2609.39959
AB - Haah's 3D cubic code was introduced as a candidate for a self-correcting quantum memory in three dimensions, using the absence of string logical operators to obstruct thermal errors. It has been a long-standing open question whether it is a self-correcting quantum memory. Even though the memory time was explored numerically, no analytical upper bound was known. Here, we settle this question rigorously and show that the cubic code thermalizes rapidly at every positive temperature. Our proof relies on the framework of [Commun. Math. Phys. 407, 195 (2026)] to turn a decay-of-correlations condition into a modified logarithmic Sobolev inequality for the Davies generator. To the best of our knowledge, this is the first analytic proof of rapid mixing for all positive temperatures for any fracton code.
ER -