p4m Symmetry as the Natural Framework for Quantum Stabilizer Codes — E8 Intelligence Research

FINDING: The p4m wallpaper group (crystallographic space group, square lattice with 4-fold rotations and mirror/glide reflections) is the natural symmetry framework for quantum stabilizer codes on a square lattice, with Clifford theory providing the representation-theoretic bridge between the wallpaper group's normal subgroup structure and code subspaces. MATH: - p4m = semidirect product \( \mathbb{Z}^2 \rtimes D_4 \), where \( D_4 \) is the dihedral group of order 8 (generators: 90° rotation \( r \), reflection \( s \); relations \( r^4 = s^2 = (rs)^2 = 1 \)). - Stabilizer code: \( \mathcal{C} = \{ g \in G : g|\psi\rangle = |\psi\rangle \} \) for \( G \) a finite subgroup of the Pauli group on \( n \) qubits. For square-lattice codes, \( G \) contains translations \( T_x, T_y \) and the point group \( D_4 \). - Clifford theory (from arXiv:1603.02493v4): For a chain \( 1 \triangleleft G_1 \triangleleft \cdots \triangleleft G_d = G \), the irreducible representations of \( G \) d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com