Erasure Thresholds for Hyperbolic and Semi-Hyperbolic Surface Codes
Aygul Azatovna Galimova
TL;DR
This work constructs and analyzes a suite of hyperbolic and semi-hyperbolic CSS surface codes derived from {8,3}, {10,3}, and {12,3} tessellations, plus 11 fine-grained variants, and evaluates their circuit-level erasure and Pauli thresholds. Using Wythoff constructions, LINS subgroup enumeration, and a fine-graining protocol that preserves topology, the authors report Pauli pseudothresholds that rise with code size within each family and erasure thresholds often exceeding 5%, with measurable erasure pseudothresholds in several codes. Per-observable thresholds reveal erasure-to-Pauli ratios around 2.7×–3.9× for base families and 4.5×–5.3× for fine-grained scaling, aligning with planar code benchmarks for the latter. Phenomenological noise results provide complementary any-logical and per-logical rates, showing nontrivial resilience under realistic error models. Overall, fine-grained codes deliver higher thresholds at the expense of encoding rate, offering a practical route to robust fault-tolerant operation in architectures with erasure-detecting capabilities.
Abstract
We construct 14 hyperbolic CSS surface codes from $\{8,3\}$, $\{10,3\}$, and $\{12,3\}$ tessellations and 11 semi-hyperbolic (fine-grained) codes. We simulate all 25 codes under circuit-level erasure and Pauli noise. Under circuit-level Pauli noise, pseudothresholds increase with code size within each family ($0.24$--$0.49\%$ for $\{8,3\}$, $0.11$--$0.43\%$ for $\{10,3\}$, $0.07$--$0.13\%$ for $\{12,3\}$). For erasure noise, most codes have $p^*_{\mathrm{E}} > 5\%$. Per-observable family thresholds give erasure-to-Pauli ratios of $2.7$--$3.9\times$ for the base code families. Fine-grained scaling families achieve higher thresholds in both Pauli ($0.67$--$0.68\%$) and erasure ($3.0$--$3.5\%$), with ratios of $4.5$--$5.2\times$. Under phenomenological noise, per-logical $Z$-channel thresholds are ${\sim}2\%$ for $\{8,3\}$ and ${\sim}1\%$ for $\{10,3\}$; the $\{12,3\}$ threshold lies below $0.5\%$.