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

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.

19 pages, (v2) removed the need of Clifford QCA of v1 using recent results (v3) some clarification, minor rewording

References in corpus (4)

• Nontrivial Quantum Cellular Automata in Higher Dimensions
• Classification of Quantum Cellular Automata
• On the structure of Clifford quantum cellular automata
• Clifford Quantum Cellular Automata: Trivial group in 2D and Witt group in 3D

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