Algebraic Phase Theory III: Structural Quantum Codes over Frobenius Rings

TL;DR

This paper develops a purely algebraic framework for quantum stabiliser codes over finite Frobenius rings, deriving quantum phase, Weyl operators, and stabiliser codes from algebraic duality without relying on traditional Hilbert space structures.

Contribution

It introduces an algebraic phase theory over Frobenius rings, providing a new foundation for quantum codes independent of Hilbert spaces and analytic inner products.

Findings

Quantum stabiliser codes are canonically identified with self-orthogonal submodules.

CSS constructions are a special case; general Frobenius rings admit non-CSS stabilisers.

Quantum layers protected by ring torsion are algebraically invisible to Weyl errors.

Abstract

We develop the quantum component of Algebraic Phase Theory by showing that quantum phase, Weyl noncommutativity, and stabiliser codes arise as unavoidable algebraic consequences of Frobenius duality. Working over finite commutative Frobenius rings, we extract nondegenerate phase pairings, Weyl operator algebras, and quantum stabiliser codes directly from admissible phase data, without assuming Hilbert spaces, analytic inner products, or an externally imposed symplectic structure. Within this framework, quantum state spaces appear as minimal carriers of faithful phase action, and stabiliser codes are identified canonically with self-orthogonal submodules under the Frobenius phase pairing. CSS-type constructions arise only as a special splitting case, while general Frobenius rings admit intrinsically non-CSS stabilisers. Nilpotent and torsion structure in the base ring give rise to algebraically protected quantum layers that are invisible to admissible Weyl-type errors. These results place quantum stabiliser theory within Algebraic Phase Theory: quantisation emerges as algebraic phase induction rather than analytic completion, and quantum structure is information-complete at the level of algebraic phase relations alone. Throughout, we work over finite Frobenius rings, which are precisely the base rings for which admissible phase data become strongly admissible, and in this regime the full quantum formalism is forced by Frobenius duality.