dorsal/arxiv
View SchemaWigner formula of rotation matrices and quantum walks
| Authors | Takahiro Miyazaki, Makoto Katori, Norio Konno |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0611022 |
| URL | https://arxiv.org/abs/quant-ph/0611022 |
| DOI | 10.1103/PhysRevA.76.012332 |
| Journal | Phys. Rev. A76 (2007) 012332/1-14 |
Abstract
Quantization of a random-walk model is performed by giving a qudit (a multi-component wave function) to a walker at site and by introducing a quantum coin, which is a matrix representation of a unitary transformation. In quantum walks, the qudit of walker is mixed according to the quantum coin at each time step, when the walker hops to other sites. As special cases of the quantum walks driven by high-dimensional quantum coins generally studied by Brun, Carteret, and Ambainis, we study the models obtained by choosing rotation as the unitary transformation, whose matrix representations determine quantum coins. We show that Wigner's $(2j+1)$-dimensional unitary representations of rotations with half-integers $j$'s are useful to analyze the probability laws of quantum walks. For any value of half-integer $j$, convergence of all moments of walker's pseudovelocity in the long-time limit is proved. It is generally shown for the present models that, if $(2j+1)$ is even, the probability measure of limit distribution is given by a superposition of $(2j+1)/2$ terms of scaled Konno's density functions, and if $(2j+1)$ is odd, it is a superposition of $j$ terms of scaled Konno's density functions and a Dirac's delta function at the origin. For the two-, three-, and four-component models, the probability densities of limit distributions are explicitly calculated and their dependence on the parameters of quantum coins and on the initial qudit of walker is completely determined. Comparison with computer simulation results is also shown.
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"abstract": "Quantization of a random-walk model is performed by giving a qudit (a\nmulti-component wave function) to a walker at site and by introducing a quantum\ncoin, which is a matrix representation of a unitary transformation. In quantum\nwalks, the qudit of walker is mixed according to the quantum coin at each time\nstep, when the walker hops to other sites. As special cases of the quantum\nwalks driven by high-dimensional quantum coins generally studied by Brun,\nCarteret, and Ambainis, we study the models obtained by choosing rotation as\nthe unitary transformation, whose matrix representations determine quantum\ncoins. We show that Wigner\u0027s $(2j+1)$-dimensional unitary representations of\nrotations with half-integers $j$\u0027s are useful to analyze the probability laws\nof quantum walks. For any value of half-integer $j$, convergence of all moments\nof walker\u0027s pseudovelocity in the long-time limit is proved. It is generally\nshown for the present models that, if $(2j+1)$ is even, the probability measure\nof limit distribution is given by a superposition of $(2j+1)/2$ terms of scaled\nKonno\u0027s density functions, and if $(2j+1)$ is odd, it is a superposition of $j$\nterms of scaled Konno\u0027s density functions and a Dirac\u0027s delta function at the\norigin. For the two-, three-, and four-component models, the probability\ndensities of limit distributions are explicitly calculated and their dependence\non the parameters of quantum coins and on the initial qudit of walker is\ncompletely determined. Comparison with computer simulation results is also\nshown.",
"arxiv_id": "quant-ph/0611022",
"authors": [
"Takahiro Miyazaki",
"Makoto Katori",
"Norio Konno"
],
"categories": [
"quant-ph",
"cond-mat.stat-mech",
"math-ph",
"math.MP",
"math.PR",
"nlin.SI"
],
"doi": "10.1103/PhysRevA.76.012332",
"journal_ref": "Phys. Rev. A76 (2007) 012332/1-14",
"title": "Wigner formula of rotation matrices and quantum walks",
"url": "https://arxiv.org/abs/quant-ph/0611022"
},
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