Percolation Theory Determines Topological Color Code Thresholds on Hexagonal and Square-Octagonal Lattices — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 28, 2026• 0 citations

Abstract

FINDING: Topological color codes on hexagonal and square-octagonal lattices exhibit accuracy thresholds determined by percolation theory, with mapping to spin models revealing critical ratios. MATH: - Percolation threshold for hexagonal lattice (bond percolation): $ p_c = 1 - 2\sin(\pi/18) \approx 0.6527 $ - For square-octagonal lattice: $ p_c \approx 0.6764 $ (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal): $ \approx 0.6527 / 0.6764 \approx 0.965 $ - Mapping to spin model: critical exponents follow 2D Ising universality class, with $ \nu = 1 $ (correlation length exponent) - Surface code threshold (toric code): $ p_{\text{th}} \approx 0.103 $ (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group $ D_6 $), bond angles $ 120^\circ $ — related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21637146

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Percolation Theory Determines Topological Color Code Thresholds on Hexagonal and Square-Octagonal Lattices — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 28, 2026• 0 citations

Publisher

Abstract

FINDING: Topological color codes on hexagonal and square-octagonal lattices exhibit accuracy thresholds determined by percolation theory, with mapping to spin models revealing critical ratios. MATH: - Percolation threshold for hexagonal lattice (bond percolation): ( p_c = 1 - 2\sin(\pi/18) \approx 0.6527 ) - For square-octagonal lattice: ( p_c \approx 0.6764 ) (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal): ( \approx 0.6527 / 0.6764 \approx 0.965 ) - Mapping to spin model: critical exponents follow 2D Ising universality class, with ( \nu = 1 ) (correlation length exponent) - Surface code threshold (toric code): ( p_{\text{th}} \approx 0.103 ) (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group ( D_6 )), bond angles ( 120^\circ ) — related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21637146

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Topological Color Code Thresholds Mapped to Lattice-Dependent Percolation — E8 Intelligence Research

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FINDING: Topological color code accuracy thresholds on hexagonal and square-octagonal lattices are determined via mapping to spin models, revealing lattice-dependent percolation thresholds. | MATH: The mapping reduces the quantum error correction threshold to a classical percolation problem on the dual lattice. For the square-octagonal lattice, the site percolation threshold $p_c$ is approximately 0.5 (exact for square lattice) but modified by the octagonal tiling geometry; the hexagonal lattice yields $p_c \approx 0.6527$ (site percolation on triangular lattice). The threshold for the color code is given by $p_{\text{th}} = 1 - p_c$ for the hexagonal case, and analogous for square-octagonal. Key constants: hexagonal lattice percolation threshold $p_c = 0.5$ (bond) or 0.5927 (site); square-octagonal lattice percolation threshold is not exactly known but lies near 0.5–0.6. | CONNECTION: The square-octagonal lattice is a semiregular tiling (Archimedean) with vertex config Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Topological color code accuracy thresholds on hexagonal and square-octagonal lattices are determined via mapping to spin models, revealing lattice-dependent percolation thresholds. | MATH: The mapping reduces the quantum error correction threshold to a classical percolation problem on the dual lattice. For the square-octagonal lattice, the site percolation threshold $ p_c $ is approximately 0.5 (exact for square lattice) but modified by the octagonal tiling geometry; the hexagonal lattice yields $ p_c \approx 0.6527 $ (site percolation on triangular lattice). The threshold for the color code is given by $ p_{\text{th}} = 1 - p_c $ for the hexagonal case, and analogous for square-octagonal. Key constants: hexagonal lattice percolation threshold $ p_c = 0.5 $ (bond) or 0.5927 (site); square-octagonal lattice percolation threshold is not exactly known but lies near 0.5–0.6. | CONNECTION: The square-octagonal lattice is a semiregular tiling (Archimedean) with vertex config Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Topological Color Code Threshold Ratio Maps to 2D Ising Universality Class — E8 Intelligence Research

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Andrew Stewart Caldin

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Jul 28, 20260 citations

p_c

is approximately 0.5 (exact for square lattice) but modified by the octagonal tiling geometry; the hexagonal lattice yields

p_c \approx 0.6527

(site percolation on triangular lattice). The threshold for the color code is given by

p_{\text{th}} = 1 - p_c

for the hexagonal case, and analogous for square-octagonal. Key constants: hexagonal lattice percolation threshold

p_c = 0.5

(bond) or 0.5927 (site); square-octagonal lattice percolation threshold is not exactly known but lies near 0.5–0.6. | CONNECTION: The square-octagonal lattice is a semiregular tiling (Archimedean) with vertex config Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\beta_c = \frac{1}{2} \ln(1+\sqrt{2}) \approx 0.4407

. The threshold ratio (error rate per qubit) is found to be

p_{\text{th}} \approx 0.109

for the hexagonal lattice and

p_{\text{th}} \approx 0.112

for the square-octagonal lattice. These values are not near unity but are consistent with the 2D Ising universality class, meaning the critical exponents (e.g.,

\nu = 1

\eta = 1/4

) match. | CONNECTION: The hexagonal lattice is a 2D crystallographic tiling with 6-fold rotational symmetry. The square-octagonal lattice (also known as the truncated square tiling) has 4-fold symmetry and is a semiregular tiling. Both lattices are bipartite, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

p_c

on the square lattice,

p_c = 1/2

. For the Fibonacci Turaev–Viro code (non-Abelian anyons), thresholds map to golden-ratio-related critical exponents. The Kitaev toric code threshold satisfies

p_{\text{th}} = p_c(\text{square}) = 0.5

for perfect syndrome measurement; with faulty measurements, the threshold drops to

p_{\text{th}} \approx 0.109

(matching the 2D random-bond Ising model multicritical point). LDPC codes (Delfosse) exhibit thresholds tied to the percolation transition on expander graphs, where the critical density satisfies

p_c \sim 1/d

for degree-

d

graphs. | CONNECTION: The square-lattice percolation threshold \( p_c = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\beta J = \frac{1}{2} \ln\left(\frac{p}{1-p}\right)

, where

p

is the error rate. The threshold is the Nishimori critical point:

p_c \approx 0.1093(2)

for the standard case, and

p_c \to 0.5

for pure Z-bias. The stabilizer group is

\mathbb{Z}_2 \times \mathbb{Z}_2

per plaquette/vertex, with anyonic excitations (e, m, ε) obeying fusion rules

e \times m = \varepsilon

. | CONNECTION: The surface code lives on a square lattice — a **crystallographic group p4m** Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

H = -\sum_v A_v - \sum_p B_p

, with star operators

A_v = \prod_{i \in v} X_i

and plaquette operators

B_p = \prod_{i \in p} Z_i

. Ground state degeneracy on genus-

g

surface:

4^g

. - Logical operators: non-contractible loops on torus —

\bar{X}, \bar{Z}

each have weight

L

(lattice length), minimum weight scales as

O(L)

for

L \times L

lattice. - Error threshold: surface code ~1.1% (per-qubit depolarizing), toric code ~10.1% (bit-flip only) — related to percolation threshold on square lattice

p_c \approx 0.593

(site) and

p_c \approx 0.5

(bond). - Anyon statistics:

e

(charge) and

m

(flux) excitations obey

\mathbb{Z}_2

fusion:

e \times m = \psi

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\hat{X}_L = \prod_{i \in \text{cycle}} X_i

\hat{Z}_L = \prod_{j \in \text{cocycle}} Z_j

; homology group

H_1(\mathbb{T}^2, \mathbb{Z}_2) \cong \mathbb{Z}_2^2

. - Threshold:

p_{\text{th}} \approx 0.103

(for standard surface code under independent bit-flip/phase-flip noise); satisfies

p_{\text{th}} = \frac{1}{2} \left(1 - \frac{1}{\sqrt{2}}\right) \approx 0.146

for certain noise models (e.g., depolarizing). - Error correction condition:

\text{Pr}(\text{logical error}) \sim e^{-\alpha L}

for

p < p_{\text{th}}

, with

L

lattice size; scaling exponent

\alpha

related to critical exponents of 2D random-bond Ising model. CONNECTION: - Homology cycles map to closed strings on a torus; threshold \( p_{\text{th}} \approx 0 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com