Scaling Laws and Quasi-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. MATH: For a surface code with distance \(d\), logical error rate \(p_L \approx C (p/p_{th})^{(d+1)/2}\), where \(p\) is physical error rate and \(p_{th}\) is the threshold (~1% for standard surface codes under circuit-level noise). Quasi-threshold \(p_q(d)\) satisfies \(p_L(p_q) = p_q\), giving \(p_q \approx p_{th} (1 - \text{const}/d)\). Stabilizer formalism: code space is joint \(+1\) eigenspace of \(n-k\) independent Pauli operators \(S_i\), with \(S_i^2 = I\), \([S_i,S_j]=0\). Logical operators \(L\) satisfy \([L,S_i]=0\), \(L \notin \langle S_i \rangle\). Decoding via minimum-weight perfect matching (MWPM) on the syndrome graph — a planar lattice with \(O(d^2)\) vertices. CONNECTION: The surface code is a topological code on a square lattice — a \(Z_2\) g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com