dorsal/arxiv
View SchemaHomological Error Correction: Classical and Quantum Codes
| Authors | H. Bombin, M. A. Martin-Delgado |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605094 |
| URL | https://arxiv.org/abs/quant-ph/0605094 |
| DOI | 10.1063/1.2731356 |
| Journal | J.Math.Phys.48:052105,2007 |
Abstract
We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for non-orientable surfaces it is impossible to construct homological codes based on qudits of dimension $D>2$, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension $D$. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor's 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.
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"abstract": "We prove several theorems characterizing the existence of homological error\ncorrection codes both classically and quantumly. Not every classical code is\nhomological, but we find a family of classical homological codes saturating the\nHamming bound. In the quantum case, we show that for non-orientable surfaces it\nis impossible to construct homological codes based on qudits of dimension\n$D\u003e2$, while for orientable surfaces with boundaries it is possible to\nconstruct them for arbitrary dimension $D$. We give a method to obtain planar\nhomological codes based on the construction of quantum codes on compact\nsurfaces without boundaries. We show how the original Shor\u0027s 9-qubit code can\nbe visualized as a homological quantum code. We study the problem of\nconstructing quantum codes with optimal encoding rate. In the particular case\nof toric codes we construct an optimal family and give an explicit proof of its\noptimality. For homological quantum codes on surfaces of arbitrary genus we\nalso construct a family of codes asymptotically attaining the maximum possible\nencoding rate. We provide the tools of homology group theory for graphs\nembedded on surfaces in a self-contained manner.",
"arxiv_id": "quant-ph/0605094",
"authors": [
"H. Bombin",
"M. A. Martin-Delgado"
],
"categories": [
"quant-ph",
"cond-mat.other",
"hep-th"
],
"doi": "10.1063/1.2731356",
"journal_ref": "J.Math.Phys.48:052105,2007",
"title": "Homological Error Correction: Classical and Quantum Codes",
"url": "https://arxiv.org/abs/quant-ph/0605094"
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