Witt Groups and Bulk-Boundary Correspondence for Stabilizer States
Błażej Ruba, Bowen Yang
TL;DR
This work develops a rigorous bulk–boundary framework for translation‑invariant stabilizer codes using modules over Laurent polynomial rings. It introduces the boundary operator module $P_{\partial}$ and the broader theory of quasi‑symplectic modules, organized by a Witt group, to classify boundary data and relate it to bulk excitations via a boundary‑to‑bulk map. In two dimensions, the boundary description yields a rigorous link between stabilizer states and abelian anyon models with gappable boundaries, resolving a conjecture in the field, while the formalism generalizes to fracton systems and higher dimensions through Ext/Tor techniques and algebraic $L$‑theory. The results provide a unifying algebraic framework that bridges stabilizer code classification, boundary phenomena, and topological order, with concrete reductions to prime power cases and clear connections to Clifford QCAs in the split case.
Abstract
We establish a bulk--boundary correspondence for translation-invariant stabilizer states in arbitrary spatial dimension, formulated in the framework of modules over Laurent polynomial rings. To each stabilizer state restricted to half-space geometry we associate a boundary operator module. Boundary operator modules provide examples of quasi-symplectic modules, which are objects of independent mathematical interest. In their study, we use ideas from algebraic L-theory in a setting involving non-projective modules and non-unimodular forms. Our results about quasi-symplectic modules in one spatial dimension allow us to resolve the conjecture that every stabilizer state in two dimensions is characterized by a corresponding abelian anyon model with gappable boundary. Our techniques are also applicable beyond two dimensions, such as in the study of fractons.