Stabilizer Codes, Parity Check Matrices, and Constant-Overhead Fault Tolerance — E8 Intelligence Research

Abstract

FINDING: Stabilizer formalism uses parity check matrices over GF(2) to define quantum error-correcting codes, linking to lattice geometry for fault-tolerant computation with constant overhead. | MATH: Stabilizer group generators correspond to rows of a parity check matrix \( H \) over \( \mathbb{F}_2 \); code space is joint +1 eigenspace of \( S = \langle g_1, \dots, g_{n-k} \rangle \); logical operators commute with all \( g_i \); constant overhead fault-tolerant scheme (arXiv:2512.02760v1) achieves qubit overhead \( O(1) \) and time overhead \( O(\log n) \) under stochastic noise. | CONNECTION: Lattice-based cryptography and quantum error correction share root in integer lattices and symplectic geometry; parity check matrices define dual lattices in \( \mathbb{Z}^{2n} \) with symplectic inner product; no explicit golden ratio or base-60, but lattice symmetries (e.g., root systems \( A_n, D_n, E_8 \)) are implicit in high-performance codes. | DEPTH: 7 — The stabilizer formalism is fou Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com