A Classification of Invertible Stabilizer Codes

TL;DR

This paper classifies invertible translation-invariant stabilizer codes using algebraic topology, providing a comprehensive framework for understanding their structure and equivalence in various spatial dimensions.

Contribution

It introduces a new classification scheme for invertible stabilizer codes based on relative L-theory groups, extending the understanding of topological quantum codes.

Findings

Classified invertible stabilizer codes in all spatial dimensions.

Established isomorphism between 3D code classes and 2D topological order Witt group.

Proposed relative L-theory spectrum as a cohomology theory for these codes.

Abstract

We develop a framework for the classification of invertible translation-invariant stabilizer codes modulo condensation and stabilization with simple codes. We introduce generalizations of the Pauli groups of local unitaries for quantum systems of qudits on cubic lattices and analyze stabilizer Hamiltonians whose terms are chosen from these groups. We define invertible stabilizer codes to be ground states of stabilizer Hamiltonians with trivial topological charges and completely classify them in any spatial dimension in terms of relative L-theory groups. In particular, we show that the group of equivalence classes of such codes in three spatial dimensions is isomorphic to the Witt group of abelian topological orders in two spatial dimensions. Additionally, we propose the spectrum of the relative L-theory as a representative of the generalized cohomology theory corresponding to the invertible stabilizer states.