2018 Fiscal Year Final Research Report
A study of limit theorems for quantum walks with gaps in distribution
| Project/Area Number |
16K17648
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Nihon University |
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Principal Investigator |
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| Research Collaborator |
Grünbaum F. Alberto
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | 量子ウォーク / 極限定理 |
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| Outline of Final Research Achievements |
I published one book, one book chapter, and four research papers from international journals. Also, I gave one talk at an international conference and eight talks at domestic meetings. The results I have got are as follows; 2-state quantum walk on the line which is involved in a Weyl equation and whose probability distribution holds a gap, 3-state quantum walks on the line whose probability distribution holds both a gap and localization, 4-state quantum walk on the line motivated by a quantum game, 2-state quantum walk on the half-line. I succeeded in analyzing those quantum walks by Fourier analysis which resulted in long-time limit distributions. The limit distributions which I derived, reproduced the probability distributions of the quantum walkers and told us how the quantum walkers acted after they iterated their dynamics a lot of times.
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| Free Research Field |
量子ウォーク
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| Academic Significance and Societal Importance of the Research Achievements |
量子ウォークの長時間極限定理は、ウォーカーが十分多くの時間発展を繰り返した後のウォーカーの振舞いを漸近的に記述する．得られたおもな研究成果は、数学的手法で計算された極限定理である．解析したそれぞれの量子ウォークモデルの確率分布は、得られた研究成果である長時間極限定理から構成される近似関数によって、よく再現されており、結果としてそれぞれの確率分布の特徴を数学的に明らかにすることができた．また、量子物理学のある方程式に関係した量子ウォークモデルが、その確率分布にギャップ構造をもつことを発見できたことは新しく、学術的に意義があった研究成果といえる．
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