Theory of spacetime quantum fault tolerance
Abstract
We present a unified algebraic theory of Clifford spacetime quantum fault tolerance, which underpins universal fault-tolerant quantum computation. We construct a spacetime chain complex from a tensor network representation of a circuit that encodes fault propagation, detector syndromes, and logical actions through the associated gauge and stabilizer structure. We show that this theory (1) unifies various formulations of quantum fault tolerance including spacetime codes, Gottesman's gadget framework, fault complexes, and ZX calculus; (2) connects spacetime distances to the gadget-level correctness properties in Gottesman's formalism; (3) naturally accommodates a range of fault models, in particular phenomenological and circuit-level models, bringing these often separately formulated descriptions into a common algebraic framework through different tensor decompositions and fault assignments; and (4) recovers many known fault tolerance constructions and results as special cases, including repeated syndrome extraction, correlated detectors for transversal and fold-transversal gates, Steane- and Knill-type syndrome extraction gadgets, and single-shot error correction.
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BibTeX
@misc{xu2026theory,
title = {{Theory of spacetime quantum fault tolerance}},
author = {Yijia Xu and Yixu Wang and Zi-Wen Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39758},
eprint = {2609.39758},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39758},
}
RIS
TY - UNPB
TI - Theory of spacetime quantum fault tolerance
AU - Yijia Xu
AU - Yixu Wang
AU - Zi-Wen Liu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.39758
UR - https://arxiv.org/abs/2609.39758
AB - We present a unified algebraic theory of Clifford spacetime quantum fault tolerance, which underpins universal fault-tolerant quantum computation. We construct a spacetime chain complex from a tensor network representation of a circuit that encodes fault propagation, detector syndromes, and logical actions through the associated gauge and stabilizer structure. We show that this theory (1) unifies various formulations of quantum fault tolerance including spacetime codes, Gottesman's gadget framework, fault complexes, and ZX calculus; (2) connects spacetime distances to the gadget-level correctness properties in Gottesman's formalism; (3) naturally accommodates a range of fault models, in particular phenomenological and circuit-level models, bringing these often separately formulated descriptions into a common algebraic framework through different tensor decompositions and fault assignments; and (4) recovers many known fault tolerance constructions and results as special cases, including repeated syndrome extraction, correlated detectors for transversal and fold-transversal gates, Steane- and Knill-type syndrome extraction gadgets, and single-shot error correction.
ER -