dorsal/arxiv
View SchemaQudit surface codes and gauge theory with finite cyclic groups
| Authors | Stephen S. Bullock, Gavin K. Brennen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0609070 |
| URL | https://arxiv.org/abs/quant-ph/0609070 |
| DOI | 10.1088/1751-8113/40/13/013 |
| Journal | J.Phys.A40:3481-3505,2007; J.Phys.A40:3481-3506,2007 |
Abstract
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 $d^\ell$ for $d$ prime and $\ell \geq 0$, and therefore under tensor decomposition, to arbitrary finite levels. The Hamiltonian describes an exact $\mathbb{Z}_d\cong\mathbb{Z}/d\mathbb{Z}$ 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.
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"abstract": "Surface codes describe quantum memory stored as a global property of\ninteracting spins on a surface. The state space is fixed by a complete set of\nquasi-local stabilizer operators and the code dimension depends on the first\nhomology group of the surface complex. These code states can be actively\nstabilized by measurements or, alternatively, can be prepared by cooling to the\nground subspace of a quasi-local spin Hamiltonian. In the case of spin-1/2\n(qubit) lattices, such ground states have been proposed as topologically\nprotected memory for qubits. We extend these constructions to lattices or more\ngenerally cell complexes with qudits, either of prime level or of level\n$d^\\ell$ for $d$ prime and $\\ell \\geq 0$, and therefore under tensor\ndecomposition, to arbitrary finite levels. The Hamiltonian describes an exact\n$\\mathbb{Z}_d\\cong\\mathbb{Z}/d\\mathbb{Z}$ gauge theory whose excitations\ncorrespond to abelian anyons. We provide protocols for qudit storage and\nretrieval and propose an interferometric verification of topological order by\nmeasuring quasi-particle statistics.",
"arxiv_id": "quant-ph/0609070",
"authors": [
"Stephen S. Bullock",
"Gavin K. Brennen"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/1751-8113/40/13/013",
"journal_ref": "J.Phys.A40:3481-3505,2007; J.Phys.A40:3481-3506,2007",
"title": "Qudit surface codes and gauge theory with finite cyclic groups",
"url": "https://arxiv.org/abs/quant-ph/0609070"
},
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