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

Andrew Stewart Caldin

Published Jul 30, 2026• 0 citations

Abstract

FINDING: Topological color code threshold ratio near unity maps to 2D Ising universality class, revealing a critical constant. | MATH: The accuracy threshold for topological color codes on hexagonal and square-octagonal lattices is determined via mapping to a random-bond Ising model. The critical point of the 2D Ising model is at $ \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

DOI: 10.5281/zenodo.21697714

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FINDING: Surface code logical error rate scales as $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ for distance-$d$ codes, with threshold $p_{th} \approx 0.01$ (circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate $p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$; for odd $d$, exponent $= (d+1)/2$; for even $d$, exponent $= d/2$. Distance $d$ = minimum weight of logical operator (e.g., $d=5$ for 49-qubit rotated code). Qubit count: rotated $= d^2$, unrotated $= 2d^2 - 2d + 1$. Threshold $p_{th} \approx 0.01$ (circuit-level noise), $\approx 0.11$ (code capacity). | CONNECTION: The exponent $\lfloor (d+1)/2 \rfloor$ is the **minimum weight of a logical $Z$ or $X$ operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a **$d \times d$ square patch** of the square lattice — a **$D_4$ root system** projection (checkerboard lattice). The ratio of qubit counts (rot 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

Andrew Stewart Caldin

Published Jul 30, 2026• 0 citations

Publisher

Abstract

FINDING: Topological color code threshold ratio near unity maps to 2D Ising universality class, revealing a critical constant. | MATH: The accuracy threshold for topological color codes on hexagonal and square-octagonal lattices is determined via mapping to a random-bond Ising model. The critical point of the 2D Ising model is at ( \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

DOI: 10.5281/zenodo.21697714

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

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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.comFINDING: 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

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

Andrew Stewart Caldin

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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Andrew Stewart Caldin

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Critical Exponent ν ≈ 2.7 for Metal-Topological Insulator Transition — E8 Intelligence Research

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Andrew Stewart Caldin

FINDING: Quantum error correction (QEC) is demonstrated below the surface code threshold by Google Quantum AI, and a theoretical link is proposed between QEC and the emergence of spacetime in holography. MATH: - Surface code threshold: ~0.57% error rate per physical qubit (logical error rate decreases exponentially below this). - Logical error rate: $P_L \propto (p/p_{\text{th}})^{d/2}$ for code distance $d$ , physical error rate $p$ . - Holographic duality: AdS/CFT correspondence maps bulk gravity to boundary quantum field theory; QEC codes (e.g., tensor networks) realize the holographic map via isometric tensors. - Key constants: No explicit golden ratio or base-60 numbers appear in these results. CONNECTION: - Surface code is a topological lattice (square grid) with stabilizer generators on plaquettes and vertices — a $\mathbb{Z}_2$ gauge theory on a 2D lattice. - Holographic QEC codes (e.g., HaPPY code) use perfect tensors on hyperbolic tilings (e.g., {5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Quantum error correction (QEC) is demonstrated below the surface code threshold by Google Quantum AI, and a theoretical link is proposed between QEC and the emergence of spacetime in holography. MATH: - Surface code threshold: ~0.57% error rate per physical qubit (logical error rate decreases exponentially below this). - Logical error rate: $ P_L \propto (p/p_{\text{th}})^{d/2} $ for code distance $ d $, physical error rate $ p $. - Holographic duality: AdS/CFT correspondence maps bulk gravity to boundary quantum field theory; QEC codes (e.g., tensor networks) realize the holographic map via isometric tensors. - Key constants: No explicit golden ratio or base-60 numbers appear in these results. CONNECTION: - Surface code is a topological lattice (square grid) with stabilizer generators on plaquettes and vertices — a $ \mathbb{Z}_2 $ gauge theory on a 2D lattice. - Holographic QEC codes (e.g., HaPPY code) use perfect tensors on hyperbolic tilings (e.g., {5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Surface Code Error Scaling and Rotated Qubit Efficiency — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Surface code logical error rate scales as $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ for distance- $d$ codes, with threshold $p_{th} \approx 0.01$ (circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate $p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ ; for odd $d$ , exponent $= (d+1)/2$ ; for even $d$ , exponent $= d/2$ . Distance $d$ = minimum weight of logical operator (e.g., $d=5$ for 49-qubit rotated code). Qubit count: rotated $= d^2$ , unrotated $= 2d^2 - 2d + 1$ . Threshold $p_{th} \approx 0.01$ (circuit-level noise), $\approx 0.11$ (code capacity). | CONNECTION: The exponent $\lfloor (d+1)/2 \rfloor$ is the **minimum weight of a logical $Z$ or $X$ operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a ** $d \times d$ square patch** of the square lattice — a ** $D_4$ root system** projection (checkerboard lattice). The ratio of qubit counts (rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Surface code logical error rate scales as $p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$ for distance-$d$ codes, with threshold $p_{th} \approx 0.01$ (circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate $p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}$; for odd $d$, exponent $= (d+1)/2$; for even $d$, exponent $= d/2$. Distance $d$ = minimum weight of logical operator (e.g., $d=5$ for 49-qubit rotated code). Qubit count: rotated $= d^2$, unrotated $= 2d^2 - 2d + 1$. Threshold $p_{th} \approx 0.01$ (circuit-level noise), $\approx 0.11$ (code capacity). | CONNECTION: The exponent $\lfloor (d+1)/2 \rfloor$ is the **minimum weight of a logical $Z$ or $X$ operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a **$d \times d$ square patch** of the square lattice — a **$D_4$ root system** projection (checkerboard lattice). The ratio of qubit counts (rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 3, 20260 citations

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

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 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

P_L \propto (p/p_{\text{th}})^{d/2}

for code distance

d

, physical error rate

p

. - Holographic duality: AdS/CFT correspondence maps bulk gravity to boundary quantum field theory; QEC codes (e.g., tensor networks) realize the holographic map via isometric tensors. - Key constants: No explicit golden ratio or base-60 numbers appear in these results. CONNECTION: - Surface code is a topological lattice (square grid) with stabilizer generators on plaquettes and vertices — a

\mathbb{Z}_2

gauge theory on a 2D lattice. - Holographic QEC codes (e.g., HaPPY code) use perfect tensors on hyperbolic tilings (e.g., {5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Surface code logical error rate scales as

p_L \propto (p/p_{th})^{\lfloor (d+1)/2 \rfloor}

for distance-

d

codes, with threshold

p_{th} \approx 0.01

(circuit-level) and rotated code halving qubit overhead at equal distance. | MATH: Logical error rate

p_L = A (p/p_{th})^{\lfloor (d+1)/2 \rfloor}

; for odd

d

, exponent

= (d+1)/2

; for even

d

, exponent

= d/2

. Distance

d

= minimum weight of logical operator (e.g.,

d=5

for 49-qubit rotated code). Qubit count: rotated

= d^2

, unrotated

= 2d^2 - 2d + 1

. Threshold

p_{th} \approx 0.01

(circuit-level noise),

\approx 0.11

(code capacity). | CONNECTION: The exponent

\lfloor (d+1)/2 \rfloor

is the **minimum weight of a logical

Z

X

operator** — this is the **Manhattan distance** on the square lattice. The rotated surface code is a **

d \times d

square patch** of the square lattice — a **

D_4

root system** projection (checkerboard lattice). The ratio of qubit counts (rot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com