Equivalence relations of quantum walks

• Project/Area Number: 17K05274
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Shinshu University
• Principal Investigator: Ohno Hiromichi 信州大学, 学術研究院工学系, 准教授 (90554585)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 量子ウォーク / ユニタリ同値類 / ユニタリ同値性

Outline of Final Research Achievements

A quantum walk is a mathematical representation of the motion of a quantum, such as a photon or an electron, and has many applications, including search algorithms, which are one of the quantum algorithms. Quantum walks have an equivalence relation called unitary equivalence, and quantum walks that are unitary equivalence can be regarded as the same quantum walk. In this study, we clarify the necessary and sufficient conditions for unitary equivalence for several quantum walks, including one-dimensional, two-dimensional, and quantum walks on a circle.

Academic Significance and Societal Importance of the Research Achievements

ユニタリ同値である量子ウォークは、確率分布やスペクトルといった重要な性質のほとんどが一致する。そのため、量子ウォークがユニタリ同値である必要十分条件がわかると、解析すべき量子ウォークの数を減らすことができるようになる。例えば、スプリットステップ量子ウォークでは、もともと４個のパラメータが必要だったのに対し、ユニタリ同値の研究後は、２個のパラメータのみ必要であることがわかった。これにより、量子ウォークの研究をより平易に行うことができるようになった。

Research Products

• [Journal Article] Unitary equivalence classes of split-step quantum walks (2021)
  • Author(s): Narimatsu Akihiro、Ohno Hiromichi、Wada Kazuyuki
  • Journal Title: Quantum Information Processing Volume: 20 Issue: 11 Pages: 1-23
  • DOI: 10.1007/s11128-021-03323-6
  • Peer Reviewed
• [Journal Article] Parameterization of quantum walks on cycles (2021)
  • Author(s): Kuriki Shuji、Nirjhor Md Sams Afif、Ohno Hiromichi
  • Journal Title: Quantum Information Processing Volume: 20 Issue: 1
  • DOI: 10.1007/s11128-020-02950-9
  • Peer Reviewed
• [Journal Article] Parameterization of Translation-Invariant Two-Dimensional Two-State Quantum Walks (2018)
  • Author(s): Ohno Hiromichi
  • Journal Title: Acta Mathematica Vietnamica Volume: 43 Issue: 4 Pages: 737-747
  • DOI: 10.1007/s40306-018-0278-8
  • NAID: 120007100054
  • Peer Reviewed
• [Journal Article] Unitary equivalence classes of one-dimensional quantum walks II (2017)
  • Author(s): Ohno Hiromichi
  • Journal Title: Quantum Information Processing Volume: 16 Issue: 12
  • DOI: 10.1007/s11128-017-1741-5
  • NAID: 120007100055
  • Peer Reviewed / Open Access
• [Journal Article] Non-commutative hypergroup of order five (2017)
  • Author(s): Yasumichi Matsuzawa, Hiromichi Ohno, Akito Suzuki, Tatsuya Tsurii, Satoe Yamanaka
  • Journal Title: Journal of Algebra and Applications Volume: 印刷中 Issue: 07 Pages: 1750127-1750127
  • DOI: 10.1142/s0219498817501274
  • NAID: 120007100052
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] 円上の量子ウォークとスプリットステップ量子ウォークのユニタリ同値類 (2022)
  • Author(s): 大野博道、成松明廣、M.S.A.Nirjhor、和田和幸
  • Organizer: 日本数学会2022年度年会
• [Presentation] Unitary equivalence classes of quantum walks (2020)
  • Author(s): 大野博道
  • Organizer: Workshop on “Non-commutative Probability and Related Fields”
• [Presentation] 2次元2状態量子ウォークのユニタリ同値類 (2019)
  • Author(s): 大野博道
  • Organizer: 2019年度秋季総合分科会
• [Presentation] 量子ウォークのユニタリ同値類 (2019)
  • Author(s): 大野博道
  • Organizer: 信州数理物理サマースクール”量子ウォークにおける対称性とトポロジー”
• [Presentation] 量子ウォークと量子マルコフ連鎖 (2019)
  • Author(s): 大野博道
  • Organizer: 第8回信州関数解析シンポジウム
• [Presentation] Unitary equivalence classes of one-dimensional quantum walks (2018)
  • Author(s): Hiromichi Ohno
  • Organizer: 2018 International Conference on Matrix Inequalities and Matrix Equations
  • Int'l Joint Research
• [Presentation] Unitary equivalence classes of some types of quantum walks (2018)
  • Author(s): Hiromichi Ohno
  • Organizer: The 7th International Conference on Matrix Analysis and Applications
  • Int'l Joint Research
• [Presentation] 1次元量子ウォークのユニタリ同値類 (2018)
  • Author(s): 大野 博道
  • Organizer: 日本数学会2018年度年会
• [Presentation] Unitary equivalent classes of one-dimensional quantum walks (2017)
  • Author(s): Hiromichi Ohno
  • Organizer: 6th International Conference on Matrix Analysis and Applications
  • Int'l Joint Research
• [Book] 量子ウォークの新展開［数理構造の深化と応用］ (2019)
  • Author(s): 今野紀雄、井出勇介編著、大野博道（分担執筆）他
  • Total Pages: 321
  • Publisher: 培風館
  • ISBN: 9784563011628
• [Remarks] 信州大学学術情報オンラインシステムSOAR
  • URL: http://soar-rd.shinshu-u.ac.jp/profile/ja.yFTmbpkh.html
• [Remarks] SOAR 研究者総覧
  • URL: http://soar-rd.shinshu-u.ac.jp/profile/ja.yFTmbpkh.html
• [Remarks] SOAR 研究者総覧
  • URL: http://soar-rd.shinshu-u.ac.jp/profile/ja.yFTmbpkh.html
• [Funded Workshop] The 7th International Conference on Matrix Analysis and Applications (2018)