Topological Error Suppression: Exponential Scaling in Surface Code Quantum Correction — E8 Intelligence Research

Abstract

FINDING: Surface code logical error rate scales exponentially with code distance (lattice size), enabling practical quantum error correction via topological order. | MATH: Logical error rate \(P_L \propto (\Lambda)^{-d}\), where \(d\) = code distance (linear lattice dimension), \(\Lambda = P_{th}/P_{phys}\) (error suppression factor per round). Threshold \(P_{th} \approx 0.01\) (1% per qubit per cycle). Physical-to-logical overhead scales as \(O(d^2)\) qubits per logical qubit. | CONNECTION: The surface code is a **Z₂ toric code** — a lattice gauge theory on a square lattice with **crystallographic symmetry (p4m plane group)**. The logical operators are **non-contractible loops** on the torus (genus-1 surface), whose weight scales as \(d\) — directly tied to the **topological winding number** and the **fundamental group \(\pi_1(T^2) = \mathbb{Z} \times \mathbb{Z}\)**. The threshold ~1% is not a golden ratio, but the **error suppression per distance** \(\Lambda\) often approaches values Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com