Exponential Error Suppression in Surface Codes: Scaling with Code Distance — E8 Intelligence Research

FINDING: Surface code logical qubit error suppression scales with code distance \(d\), with logical error rate \(P_L \propto (\Lambda)^{-(d+1)/2}\) where \(\Lambda\) is the error-limit parameter; Google's 2022 experiment demonstrated exponential suppression across \(d=3,5,7\). MATH: - Logical error rate: \(P_L \propto \Lambda^{-(d+1)/2}\) (exponential in code distance) - Threshold condition: physical error rate \(p < p_{th} \approx 0.01\) (for standard surface code with depolarizing noise) - Stabilizer weight: \(w = 4\) (X and Z plaquette operators on square lattice) - Logical operator weight: \(w_L = d\) (minimum weight path across lattice) - Lattice: square grid, \(d \times d\) qubit array, \(d^2\) data qubits + \((d^2-1)\) ancilla qubits - Key ratio: \(P_L / P_{phys} \sim (\Lambda)^{-(d+1)/2}\) where \(\Lambda \approx p_{th}/p\) CONNECTION: - Square lattice = \(Z^2\) root system (crystallographic, order 4 symmetry) — stabilizer generators are 4-body plaquette opera Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com