Local Quantum Fault Tolerance Without Expansion
Abstract
We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical (d,d')-Toric code with d',d−d'≥ 2, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays. Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results: 1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical (d,d')-Toric complexes, resolving the local-existence question of Kubica and Preskill. 2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead. 3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate p below threshold, nd+1 noiseless sweep cycles recover the encoded state on a side-length-n (d,d')-Toric code with probability at least 1−O(poly(n))pΩ(n), extending earlier guarantees for independent errors.
Related
- Theory of spacetime quantum fault tolerance
- Engineering dissipation and control pulses for high-fidelity fault-tolerance quantum computing
- Fault tolerance of quantum circuits with tensor networks and symplectic geometry
- Local decoders for fault-tolerant quantum computation and translation-invariant stabilizer codes
- Classical Active-Space Hybrid Quantum Subspace Expansion (CASH-QSE): Quantum Corrections without Remeasuring the Classically Calculable Energy
BibTeX
@misc{ponarovsky2026local,
title = {{Local Quantum Fault Tolerance Without Expansion}},
author = {David Ponarovsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38847},
eprint = {2610.38847},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38847},
}
RIS
TY - UNPB
TI - Local Quantum Fault Tolerance Without Expansion
AU - David Ponarovsky
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38847
UR - https://arxiv.org/abs/2610.38847
AB - We study fault-tolerant quantum computation using a fixed, local error-correction rule and a geometrically simple code: the cubical (d,d')-Toric code with d',d−d'≥ 2, including the 4D Toric code. Under sufficiently weak local-stochastic Pauli noise, our construction uses noiseless, bounded-depth classical processing per physical time step. Its correction cycle acts only on the current syndrome, has constant depth, and is repeated unchanged during idle logical computation; it requires neither multiscale decoder states nor scale-dependent communication delays. Our proof treats errors and syndromes as chains in a cubical complex, combining geometric shrinking under the sweep rule with a space--time explanation-tree argument inspired by Toom's theorem. It does not rely on expansion. We establish three results: 1.Explicit higher-dimensional sweep rule. We give a local algorithm for choosing sweep corrections on cubical (d,d')-Toric complexes, resolving the local-existence question of Kubica and Preskill. 2. Fault-tolerant computation. We prove a circuit-level threshold theorem for noisy sweep cycles interleaved with logical gates. In particular, the 4D Toric code admits a fixed, constant-depth syndrome-based correction cycle without a multiscale decoder or expansion-based encoding. We do not claim constant total-depth overhead. 3. Ideal decoding under local-stochastic noise. For an initial local-stochastic Pauli error of rate p below threshold, n^d+1 noiseless sweep cycles recover the encoded state on a side-length-n (d,d')-Toric code with probability at least 1−O(poly(n))p^Ω(n), extending earlier guarantees for independent errors.
ER -