A Classification of Invertible Stabilizer 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.