Surface Codes as B2 Root Systems with Percolation-Like Error Thresholds — E8 Intelligence Research

FINDING: Surface codes implement topological quantum error correction via a square lattice whose stabilizer structure maps directly onto the B2 root system, with error thresholds tied to percolation-like critical ratios. MATH: - Surface code: qubits on edges of a square lattice; stabilizers = vertex operators \(A_v = \prod_{e \ni v} X_e\) and plaquette operators \(B_p = \prod_{e \in \partial p} Z_e\), with \(A_v^2 = B_p^2 = I\), \([A_v, B_p] = 0\). - Logical operators: non-contractible loops — \(X\)-type along a horizontal cycle, \(Z\)-type along a vertical cycle; logical Pauli algebra \(\bar{X}, \bar{Z}\) with \(\bar{X}\bar{Z} = -\bar{Z}\bar{X}\). - Distance \(d\) = minimum weight of logical operator = lattice dimension \(L\) (for \(L \times L\) patch); code rate \(k/n = 1/(2L^2 - 2L + 1)\). - Error threshold: \(p_{\text{th}} \approx 0.103\) (for depolarizing noise) — a critical probability, not a simple ratio but related to percolation threshold on square lattice \(p_c \app Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com