Closed form logical error rate approximations for surface codes

@misc{regev2026closed,
  title = {{Closed form logical error rate approximations for surface codes}},
  author = {Shaked Regev and Daniel Dilley and Ryan Bennink},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.03054},
  eprint = {2605.03054},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.03054},
}

TY  - UNPB
TI  - Closed form logical error rate approximations for surface codes
AU  - Shaked Regev
AU  - Daniel Dilley
AU  - Ryan Bennink
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2605.03054
UR  - https://arxiv.org/abs/2605.03054
AB  - We propose a novel method to calculate logical error rates in surface codes, assuming independent and identically distributed physical errors. These results fit well known scaling laws for studied physical error rates and code distances, but break down in some other regimes. Increasing the code distance still reduces logical error rates, but returns diminish quickly beyond a certain point. We show how to use our method to analyze hypothetical quantum computers with various configurations and select designs with lower error rates. Currently, this requires expensive classical simulations of quantum decoders for various distances and physical error rates or inaccurate extrapolation from minimal experimental data. Instead, we use the symmetry of the problem to count the configurations that result in a logical error with our novel software. Given a physical error rate, we can deduce the probability of a logical error to provably good accuracy quickly. We include an analysis of measurement errors to allow for a more complete comparison of different surface code implementations. We extend this counting method to a particular locally-correlated error model and another globally-correlated error model.
ER  -