dorsal/arxiv
View SchemaJarlskog's Parametrization of Unitary Matrices and Qudit Theory
| Authors | Kazuyuki Fujii, Kunio Funahashi, Takayuki Kobayashi |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0508006 |
| URL | https://arxiv.org/abs/quant-ph/0508006 |
| Journal | Int.J.Geom.Meth.Mod.Phys. 3(2006) 269-283 |
Abstract
In the paper (math-ph/0504049) Jarlskog gave an interesting simple parametrization to unitary matrices, which was essentially the canonical coordinate of the second kind in the Lie group theory (math-ph/0505047). In this paper we apply the method to a quantum computation based on multi-level system (qudit theory). Namely, by considering that the parametrization gives a complete set of modules in qudit theory, we construct the generalized Pauli matrices which play a central role in the theory and also make a comment on the exchange gate of two-qudit systems. Moreover we give an explicit construction to the generalized Walsh-Hadamard matrix in the case of n=3, 4 and 5. For the case of n=5 its calculation is relatively complicated. In general, a calculation to construct it tends to become more and more complicated as n becomes large. To perform a quantum computation the generalized Walsh-Hadamard matrix must be constructed in a quick and clean manner. From our construction it may be possible to say that a qudit theory with $n\geq 5$ is not realistic. This paper is an introduction towards Quantum Engineering.
{
"annotation_id": "3bb5dd59-935d-4f64-bdb6-0a6c84532e6e",
"date_created": "2026-03-02T18:02:19.830000Z",
"date_modified": "2026-03-02T18:02:19.830000Z",
"file_hash": "1dc3cbd8fa6fa0f1dda059e3944f148f0d3c885fc6d6397743eca7bfd06d0e52",
"private": false,
"record": {
"abstract": "In the paper (math-ph/0504049) Jarlskog gave an interesting simple\nparametrization to unitary matrices, which was essentially the canonical\ncoordinate of the second kind in the Lie group theory (math-ph/0505047).\n In this paper we apply the method to a quantum computation based on\nmulti-level system (qudit theory). Namely, by considering that the\nparametrization gives a complete set of modules in qudit theory, we construct\nthe generalized Pauli matrices which play a central role in the theory and also\nmake a comment on the exchange gate of two-qudit systems.\n Moreover we give an explicit construction to the generalized Walsh-Hadamard\nmatrix in the case of n=3, 4 and 5. For the case of n=5 its calculation is\nrelatively complicated. In general, a calculation to construct it tends to\nbecome more and more complicated as n becomes large.\n To perform a quantum computation the generalized Walsh-Hadamard matrix must\nbe constructed in a quick and clean manner. From our construction it may be\npossible to say that a qudit theory with $n\\geq 5$ is not realistic.\n This paper is an introduction towards Quantum Engineering.",
"arxiv_id": "quant-ph/0508006",
"authors": [
"Kazuyuki Fujii",
"Kunio Funahashi",
"Takayuki Kobayashi"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"journal_ref": "Int.J.Geom.Meth.Mod.Phys. 3(2006) 269-283",
"title": "Jarlskog\u0027s Parametrization of Unitary Matrices and Qudit Theory",
"url": "https://arxiv.org/abs/quant-ph/0508006"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "f93c1e44-ee66-468b-ba76-83fca9a1d9da",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}