dorsal/arxiv
View SchemaOn Self-Dual Quantum Codes, Graphs, and Boolean Functions
| Authors | Lars Eirik Danielsen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0503236 |
| URL | https://arxiv.org/abs/quant-ph/0503236 |
Abstract
A short introduction to quantum error correction is given, and it is shown that zero-dimensional quantum codes can be represented as self-dual additive codes over GF(4) and also as graphs. We show that graphs representing several such codes with high minimum distance can be described as nested regular graphs having minimum regular vertex degree and containing long cycles. Two graphs correspond to equivalent quantum codes if they are related by a sequence of local complementations. We use this operation to generate orbits of graphs, and thus classify all inequivalent self-dual additive codes over GF(4) of length up to 12, where previously only all codes of length up to 9 were known. We show that these codes can be interpreted as quadratic Boolean functions, and we define non-quadratic quantum codes, corresponding to Boolean functions of higher degree. We look at various cryptographic properties of Boolean functions, in particular the propagation criteria. The new aperiodic propagation criterion (APC) and the APC distance are then defined. We show that the distance of a zero-dimensional quantum code is equal to the APC distance of the corresponding Boolean function. Orbits of Boolean functions with respect to the {I,H,N}^n transform set are generated. We also study the peak-to-average power ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove that PAR_IHN of a quadratic Boolean function is related to the size of the maximum independent set over the corresponding orbit of graphs. A construction technique for non-quadratic Boolean functions with low PAR_IHN is proposed. It is finally shown that both PAR_IHN and APC distance can be interpreted as partial entanglement measures.
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"abstract": "A short introduction to quantum error correction is given, and it is shown\nthat zero-dimensional quantum codes can be represented as self-dual additive\ncodes over GF(4) and also as graphs. We show that graphs representing several\nsuch codes with high minimum distance can be described as nested regular graphs\nhaving minimum regular vertex degree and containing long cycles. Two graphs\ncorrespond to equivalent quantum codes if they are related by a sequence of\nlocal complementations. We use this operation to generate orbits of graphs, and\nthus classify all inequivalent self-dual additive codes over GF(4) of length up\nto 12, where previously only all codes of length up to 9 were known. We show\nthat these codes can be interpreted as quadratic Boolean functions, and we\ndefine non-quadratic quantum codes, corresponding to Boolean functions of\nhigher degree. We look at various cryptographic properties of Boolean\nfunctions, in particular the propagation criteria. The new aperiodic\npropagation criterion (APC) and the APC distance are then defined. We show that\nthe distance of a zero-dimensional quantum code is equal to the APC distance of\nthe corresponding Boolean function. Orbits of Boolean functions with respect to\nthe {I,H,N}^n transform set are generated. We also study the peak-to-average\npower ratio with respect to the {I,H,N}^n transform set (PAR_IHN), and prove\nthat PAR_IHN of a quadratic Boolean function is related to the size of the\nmaximum independent set over the corresponding orbit of graphs. A construction\ntechnique for non-quadratic Boolean functions with low PAR_IHN is proposed. It\nis finally shown that both PAR_IHN and APC distance can be interpreted as\npartial entanglement measures.",
"arxiv_id": "quant-ph/0503236",
"authors": [
"Lars Eirik Danielsen"
],
"categories": [
"quant-ph",
"cs.IT",
"math.IT"
],
"title": "On Self-Dual Quantum Codes, Graphs, and Boolean Functions",
"url": "https://arxiv.org/abs/quant-ph/0503236"
},
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