dorsal/arxiv
View SchemaNonbinary stabilizer codes over finite fields
| Authors | Avanti Ketkar, Andreas Klappenecker, Santosh Kumar, Pradeep Kiran Sarvepalli |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0508070 |
| URL | https://arxiv.org/abs/quant-ph/0508070 |
Abstract
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over GF(q) in terms of classical codes over GF(q^2) is provided that generalizes the well-known notion of additive codes over GF(4) of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.
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"abstract": "One formidable difficulty in quantum communication and computation is to\nprotect information-carrying quantum states against undesired interactions with\nthe environment. In past years, many good quantum error-correcting codes had\nbeen derived as binary stabilizer codes. Fault-tolerant quantum computation\nprompted the study of nonbinary quantum codes, but the theory of such codes is\nnot as advanced as that of binary quantum codes. This paper describes the basic\ntheory of stabilizer codes over finite fields. The relation between stabilizer\ncodes and general quantum codes is clarified by introducing a Galois theory for\nthese objects. A characterization of nonbinary stabilizer codes over GF(q) in\nterms of classical codes over GF(q^2) is provided that generalizes the\nwell-known notion of additive codes over GF(4) of the binary case. This paper\nderives lower and upper bounds on the minimum distance of stabilizer codes,\ngives several code constructions, and derives numerous families of stabilizer\ncodes, including quantum Hamming codes, quadratic residue codes, quantum Melas\ncodes, quantum BCH codes, and quantum character codes. The puncturing theory by\nRains is generalized to additive codes that are not necessarily pure. Bounds on\nthe maximal length of maximum distance separable stabilizer codes are given. A\ndiscussion of open problems concludes this paper.",
"arxiv_id": "quant-ph/0508070",
"authors": [
"Avanti Ketkar",
"Andreas Klappenecker",
"Santosh Kumar",
"Pradeep Kiran Sarvepalli"
],
"categories": [
"quant-ph",
"cs.IT",
"math.IT"
],
"title": "Nonbinary stabilizer codes over finite fields",
"url": "https://arxiv.org/abs/quant-ph/0508070"
},
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