Gauge and Homological Structures in Quantum Error Correction

Gauge theory, quantum error correction, and homology theory share a common mathematical backbone that, when made explicit, becomes a practical toolkit for fault-tolerant quantum computation. A CSS code is naturally a length-2 chain complex in which the X-stabilizers act as Gauss-law generators and the code space is the gauge-invariant subspace, the toric code being the prototypical realization of a Z_2 lattice gauge theory. Building on this correspondence, I present two results. First, I introduce a gauge-field formalism in which logical gates are written as exponentials of polynomials of operator-valued cochains—the lattice gauge fields—on the underlying chain complex. Requiring no special structure on the code, the construction applies to general CSS codes and yields explicit physical-gate decompositions of logical S, H, CZ, and T gates whose action depends only on the cohomology class of the logical qubits. Second, I show that the transversal implementability of logical Pauli-Z rotations has a purely homological origin: their logical action is classified by a Z_{2^m}-module extending logical Pauli operators to higher levels of the Clifford hierarchy, and transversality is governed by compatibility and lifting obstructions on homology classes beyond the usual Z_2 coefficient. From a high-energy-physics viewpoint, a level-m transversal gate is a gauge-invariant "2^{m-1}-th root of a Wilson loop." Together these results offer a unifying language for designing logical gates and point toward fault-tolerance from lattice gauge theory and algebraic topology. This talk is based on arXiv:2511.15224 and arXiv:2602.14499.