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

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