An O(n²) Algebraic Decoder for Topological Codes via Super-Tannakian Geometry

We introduce a deterministic, algebraic decoding framework for topological quantum error-correcting codes that replaces combinatorial graph search with structured inversion of the Cartan metric tensor. The resulting Adjugate Transition Bridge (ATB) decoder achieves O (n²) precomputation and O (d³) per-syndrome runtime, providing a sub-cubic alternative to minimum-weight perfect matching (MWPM). The key structural result is a block-tridiagonal decomposition of the Cartan matrix for the rotated surface code under a snake-partition ordering. For all odd code distances d, the Cartan matrix C = HZ HZT decomposes into d+1 blocks of size b = (d-1) /2, with: scalar-identity diagonal blocks (boundary: 2I, bulk: 4I), a universal, position-independent coupling matrix F with | (F) | = 1. This structure enables efficient inversion via the block Thomas algorithm, reducing global decoding complexity to algebraic matrix operations. We prove a Parity Theorem over 𝔽₂: the algebraic supercharge map = HZT C^-1🕼䃒 recovers the logical coset up to a parity offset equal to the error weight wt (e) 2. The proof reduces to the identity (P + I) Z = 1 and relies on a new result: invertibility of the Cartan matrix over 𝔽₂ for all odd distances, established via a block-path determinant argument using the unique perfect matching of an even-length path graph. The decoder operates in three stages: Block-Thomas inversion of the Cartan metric (global algebraic step), Algebraic coset identification via a precomputed vector over 𝔽₂, Local parity acquisition, confined to an O (d) -sized neighborhood due to syndrome locality. This eliminates global search: combinatorial work is reduced to a bounded local check independent of system size. The framework is verified at distances d = 3, 5, 7, with: 100% correction of all single-qubit errors, exact coset identification for all even-weight errors, complete agreement with theoretical predictions across all tested instances. Conceptually, the ATB decoder reframes quantum error correction as algebraic inversion of a geometric object rather than graph optimization. The construction is derived via super-Tannakian reconstruction of code symmetry, positioning decoding within a broader program of algorithmic motives, where computational problems are governed by intrinsic algebraic structures.