Scaling Laws and Thresholds in Surface Code Error Correction — E8 Intelligence Research

FINDING: Surface code logical error rate scales exponentially with code distance, with a quasi-threshold defining the crossover where error correction becomes beneficial; stabilizer formalism provides the algebraic framework for this scaling. MATH: For a surface code with distance \(d\), logical error rate \(p_L ≈ C (p/pₜₕ)d/2\) (or \(p_L ∝ (p/pₜₕ)(d+1)/2\) depending on boundary conditions), where \(p\) is physical error rate and \(pₜₕ\) is the threshold (~1% for standard surface codes under circuit-level noise). The quasi-threshold \(p^*\) satisfies \(p_L(p^*) = p^*\), i.e., the fixed point of the error correction map. Stabilizer formalism: code space is the simultaneous +1 eigenspace of generators \(S_i\) in the Pauli group, with \(S_i^2 = I\), \([S_i,S_j]=0\). Logical operators \(X_L, Z_L\) commute with all \(S_i\) but are not in the stabilizer group. The number of encoded qubits \(k = n - r\) (n physical, r independent stabilizer generat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com