Qudit surface codes and gauge theory with finite cyclic groups

Stephen S. Bullock, Gavin K. Brennen

Research output: Contribution to journalArticlepeer-review

45 Citations (Scopus)


Surface codes describe quantum memory stored as a global property of interacting spins on a surface. The state space is fixed by a complete set of quasi-local stabilizer operators and the code dimension depends on the first homology group of the surface complex. These code states can be actively stabilized by measurements or, alternatively, can be prepared by cooling to the ground subspace of a quasi-local spin Hamiltonian. In the case of spin- 1/2 (qubit) lattices, such ground states have been proposed as topologically protected memory for qubits. We extend these constructions to lattices or more generally cell complexes with qudits, either of prime level or of level d for d prime and ℓ ≥ 0, and therefore under tensor decomposition, to arbitrary finite levels. The Hamiltonian describes an exact ℤ/dℤ gauge theory whose excitations correspond to Abelian anyons. We provide protocols for qudit storage and retrieval and propose an interferometric verification of topological order by measuring quasi-particle statistics.

Original languageEnglish
Pages (from-to)3481-3505
Number of pages25
JournalJournal of Physics A: Mathematical and Theoretical
Issue number13
Publication statusPublished - 30 Mar 2007
Externally publishedYes


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