The Classification of Pauli Stabilizer Codes: A Lattice and Continuum Treatise

TL;DR

This paper classifies Pauli stabilizer codes using algebraic L-theory, establishing a connection with framed TQFTs and revealing a bulk-boundary correspondence via Clifford QCAs.

Contribution

It introduces a novel algebraic framework for classifying stabilizer codes and relates lattice models to continuum TQFTs through categorical and algebraic tools.

Findings

Classification of stabilizer codes via algebraic L-theory.

Establishment of a bulk-boundary correspondence with Clifford QCAs.

Identification of structural differences between lattice codes and TQFTs.

Abstract

We classify mobile Pauli stabilizer codes up to gapped interfaces and coarse-graining using the framework of algebraic $\mathrm{L}$-theory. We compare this classification with that of framed TQFTs, theories that arise naturally in the continuum, highlighting a close structural relationship between the two. Our approach is formulated in the category of perfect chain complexes equipped with quadratic functor over the Laurent polynomial ring $R = \mathbb{Z}/p[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$, within which the collection of topological operators of Pauli stabilizer codes arise naturally as objects. In particular, we establish a bulk-boundary correspondence for lattice theories: the equivalence class of a Pauli stabilizer code up to gapped interface is described by a Clifford QCA in one dimension higher. This is done using the universal target category for stabilizer codes, which is the categorical spectrum whose existence and universal properties are introduced in this work. We conclude by highlighting subtle differences between the classification of Pauli stabilizer codes and TQFTs, leading to qualitative distinctions between lattice and continuum theories.