Toric Code Topology: Homological Ground States and p4m Symmetry — E8 Intelligence Research

Abstract

FINDING: The toric code is a topological quantum error-correcting code whose logical operators are non-contractible loops on a torus, with the p4m wallpaper group (square lattice) providing the underlying spatial symmetry; its ground state degeneracy and logical operator algebra are governed by the first homology group of the torus, \( H_1(T^2) = \mathbb{Z}^2 \). | MATH: Hamiltonian \( H = -\sum_v A_v - \sum_p B_p \), where \( A_v = \prod_{i \in v} X_i \), \( B_p = \prod_{i \in p} Z_i \). Logical operators: \( \bar{X}_1, \bar{Z}_1 \) along one non-contractible cycle, \( \bar{X}_2, \bar{Z}_2 \) along the other; they satisfy \( \bar{X}_i \bar{Z}_j = (-1)^{\delta_{ij}} \bar{Z}_j \bar{X}_i \). Ground state degeneracy = \( |H_1(T^2)| = 4 \) (for \( \mathbb{Z}_2 \) coefficients). The p4m group (order 8, generated by 90° rotation and reflections) acts on the lattice, and the code is invariant under this wallpaper group. | CONNECTION: The square lattice is the \( A_1 \times A_1 \) root lattice Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com