Homological invariants of Pauli stabilizer codes

TL;DR

This paper introduces homological invariants called charge modules for translationally invariant Pauli stabilizer codes with arbitrary qudit dimensions, revealing their properties and physical implications, especially for codes with mobile excitations.

Contribution

It develops a homological framework to define and analyze charge modules in stabilizer codes, extending previous results to non-uniform qudit dimensions and higher dimensions.

Findings

Charge modules have Krull dimension zero for certain codes, indicating excitation mobility.

All excitations in these codes are mobile, allowing construction of higher-dimensional excitations and symmetries.

A braiding pairing between charge modules in complementary degrees is established.

Abstract

We study translationally invariant Pauli stabilizer codes with qudits of arbitrary, not necessarily uniform, dimensions. Using homological methods, we define a series of invariants called charge modules. We describe their properties and physical meaning. The most complete results are obtained for codes whose charge modules have Krull dimension zero. This condition is interpreted as mobility of excitations. We show that it is always satisfied for translation invariant 2D codes with unique ground state in infinite volume, which was previously known only in the case of uniform, prime qudit dimension. For codes all of whose excitations are mobile we construct a $p$-dimensional excitation and a $(D-p-1)$-form symmetry for every element of the $p$-th charge module. Moreover, we define a braiding pairing between charge modules in complementary degrees. We discuss examples which illustrate how charge modules and braiding can be computed in practice.