Quantum walks from a view point of discrete geometric analysis

2018 Fiscal Year Final Research Report

• Project/Area Number: 25400068
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tohoku University
• Principal Investigator: Tate Tatsuya 東北大学, 理学研究科, 教授 (00317299)
• Project Period (FY): 2013-04-01 – 2019-03-31
• Keywords: 量子ウォーク / 周期的ユニタリ推移作用素 / 無限二面体群 / ベキ乗公式 / 絶対連続スペクトル / 固有値 / 局在化

Outline of Final Research Achievements

The notion of quantum walks is a quantum counterpart of the notion of random walks and it has many applications to computer sciences. For example, the Grover walk, which is one of typical quantum walks, is used to improve Grover's quantum search algorithm as discussed originally by Ambainis et al. In this research program, we focused on rather qualitative aspects of quantum walks because it would be very important to understand its qualitative aspects to find its further applications and develop its theory further in mathematics. One of importances of random walks are its applications to discrete group theory. Also random walks plays important roles in the theory of crystal lattices. Concerning with these facts, we mainly considered discrete geometric analytical problems for quantum walks in this program and we obtained results on such as algebraic structures of 1-dimensional homogeneous quantum walks and setting up of quantum walks on crystal lattices and their spectral behavior.

Free Research Field

幾何解析学

Academic Significance and Societal Importance of the Research Achievements

上述の通り量子ウォークはコンピュータサイエンスなど応用面では活発に研究されているが，数学的な理論構築や定性的な性質の解明は遅れている．数学理論が構築されれば予見も可能となる可能性があり，応用面においてさらなる効果が期待できる．また通常の量子論においては，まずハミルトニアンがあり，それが生成するユニタリ時間発展作用素を考察する．一方量子ウォークはユニタリ作用素の定義する確率分布であり，その解析はユニタリ作用素を直接扱う数学分野として画期的である．今回定義した周期的ユニタリ推移作用素そしてそのスペクトル論的な研究成果はむしろ先駆的な結果であり，今後詳細な挙動を調べる方向の研究が期待される．