# Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices

## TL;DR

This paper proves that all translation invariant topological Pauli stabilizer codes on 2D prime-dimensional qudit lattices are equivalent to multiple copies of the toric code via local Clifford circuits, with no additional assumptions.

## Contribution

It establishes a complete classification of such codes without assuming nonchirality or CSS structure, showing they decompose into toric codes under local Clifford circuits.

## Key findings

- All codes decompose into toric codes via constant-depth local Clifford circuits.
- Number of toric code copies is a complete topological invariant.
- No assumptions of nonchirality or CSS structure are needed.

## Abstract

We prove that on any two-dimensional lattice of qudits of a prime dimension, every translation invariant Pauli stabilizer group with local generators and with code distance being the linear system size, is decomposed by a local Clifford circuit of constant depth into a finite number of copies of the toric code (abelian discrete gauge theory) stabilizer group. This means that under local Clifford circuits the number of toric code copies is the complete invariant of topological Pauli stabilizer codes. Previously, the same conclusion was obtained under the assumption of nonchirality for qubit codes or the Calderbank-Shor-Steane structure for prime qudit codes; we do not assume any of these.