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

Andrew Stewart Caldin

Published Jul 28, 2026• 0 citations
Publisher

Abstract

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.com

DOI
10.5281/zenodo.21637270

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Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices

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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.com

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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.com
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 βc=12ln⁡(1+2)≈0.4407 \beta_c = \frac{1}{2} \ln(1+\sqrt{2}) \approx 0.4407 βc​=21​ln(1+2​)≈0.4407. The threshold ratio (error rate per qubit) is found to be pth≈0.109 p_{\text{th}} \approx 0.109 pth​≈0.109 for the hexagonal lattice and pth≈0.112 p_{\text{th}} \approx 0.112 pth​≈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., ν=1 \nu = 1 ν=1, η=1/4 \eta = 1/4 η=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
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FINDING: Surface code error thresholds are governed by the topological order of a 2D lattice, with the XZZX variant achieving optimal thresholds under biased noise via a mapping to a classical disordered magnet (random-bond Ising model). | MATH: Surface code threshold ≈ 10.3% (depolarizing) for standard toric code; XZZX code threshold → 50% under pure Z-biased noise (theoretical limit). The decoding problem maps exactly to the random-bond Ising model (RBIM) on a square lattice with Nishimori condition: βJ=12ln⁡(p1−p) \beta J = \frac{1}{2} \ln\left(\frac{p}{1-p}\right) βJ=21​ln(1−pp​), where ppp is the error rate. The threshold is the Nishimori critical point: pc≈0.1093(2) p_c \approx 0.1093(2) pc​≈0.1093(2) for the standard case, and pc→0.5 p_c \to 0.5 pc​→0.5 for pure Z-bias. The stabilizer group is Z2×Z2 \mathbb{Z}_2 \times \mathbb{Z}_2 Z2​×Z2​ per plaquette/vertex, with anyonic excitations (e, m, ε) obeying fusion rules e×m=ε e \times m = \varepsilon e×m=ε. | 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
FINDING: Topological quantum error correction (TQEC) uses lattice geometry—specifically the toric code and surface codes—to encode logical qubits non-locally, with error thresholds tied to lattice percolation and anyon braiding. MATH: - Toric code: Hamiltonian H=−∑vAv−∑pBp H = -\sum_v A_v - \sum_p B_p H=−∑v​Av​−∑p​Bp​, with star operators Av=∏i∈vXi A_v = \prod_{i \in v} X_i Av​=∏i∈v​Xi​ and plaquette operators Bp=∏i∈pZi B_p = \prod_{i \in p} Z_i Bp​=∏i∈p​Zi​. Ground state degeneracy on genus-ggg surface: 4g 4^g 4g. - Logical operators: non-contractible loops on torus — Xˉ,Zˉ \bar{X}, \bar{Z} Xˉ,Zˉ each have weight L L L (lattice length), minimum weight scales as O(L) O(L) O(L) for L×L L \times L L×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 pc≈0.593 p_c \approx 0.593 pc​≈0.593 (site) and pc≈0.5 p_c \approx 0.5 pc​≈0.5 (bond). - Anyon statistics: e e e (charge) and m m m (flux) excitations obey Z2 \mathbb{Z}_2 Z2​ fusion: e×m=ψ e \times m = \psi e×m=ψ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin-glass systems. The analysis for the corresponding spin-glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be 1-pc=0.1096-8 for the hexagonal lattice and 1-pc=0.1092-3 for the square-octagonal lattice, where 1-p denotes the error probability on each qubit. Hence, both of them are expected to be slightly lower than the probability 1-pc=0.110 028 for the quantum Gilbert-Varshamov bound with a zero encoding rate.
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: PL∝(p/pth)d/2 P_L \propto (p/p_{\text{th}})^{d/2} PL​∝(p/pth​)d/2 for code distance d d d, physical error rate p p 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 Z2 \mathbb{Z}_2 Z2​ 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