Quantum Error Correction: Exponential Suppression of Logical Errors via Code Distance Scaling and Physical Qubit Encoding — E8 Intelligence Research

FINDING: Quantum error correction (QEC) achieves fault‑tolerant logical qubits by encoding a single logical qubit into multiple physical qubits, enabling error rates to be suppressed exponentially with code distance. | MATH: Logical error rate \(p_L \approx \alpha (p/p_{\text{th}})^{d/2}\) where \(p\) is physical error probability, \(p_{\text{th}}\) the threshold, \(d\) the code distance, and \(\alpha\) a constant; Shor code uses 9 qubits, Steane code uses 7, surface code uses a 2‑D lattice with distance \(d\). | CONNECTION: The surface code lattice exhibits a 4‑fold rotational symmetry and a checkerboard pattern akin to crystallographic \(p4mm\) symmetry; the logical operators correspond to non‑contractible loops on a torus, reflecting topological order. No direct appearance of golden ratio or base‑60 numbers. | DEPTH: 8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com