A study of duplications between quantum walks

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K03625
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12040:Applied mathematics and statistics-related
• Research Institution: Nihon University
• Principal Investigator: MACHIDA Takuya 日本大学, 生産工学部, 准教授 (20637144)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: 量子ウォーク / 極限定理 / 複写可能性

Outline of Final Research Achievements

I published one book chapter and three research papers from international journals. I was invited for domestic meetings and gave two talks there. The results I have got are as follows; a limit theorem for an open quantum walk on the line, a limit distribution of a 2-period time-dependent quantum walk on the half line, and a limit distribution of a quantum walk on the line driven by a five-diagonal unitary matrix. All the results were long-time limit distributions and they were proved by Fourier analysis. I found a duplication between two quantum walks in the study of the 2-period time-dependent quantum walk. As a result, I succeeded in getting a long-time limit distribution of the quantum walk. I visited Math department of University of California, Berkeley in August 2019 and March 2023 for discussion with researchers at the university.

Free Research Field

量子ウォーク

Academic Significance and Societal Importance of the Research Achievements

量子ウォークの研究において、時間発展を繰り返した後のウォーカーの振舞いを、物理システムを用いた実験で統計的に解析することは難しく、理論計算による解析が必要となる。本研究で得られた研究成果（長時間極限定理）は数学的な手法を用いて発見され、時間発展後のウォーカーの振舞いを漸近的に記述する。その結果、いくつかの量子ウォークの特徴を明らかにすることができた。特に、フーリエ解析で直接的に計算が難しい量子ウォークモデルを、これまでのフーリエ解析の方法が適用できるような量子ウォークモデルに複写することで長時間極限定理を導出できることを示したことは新しく、学術的に意義があった研究成果といえる。