| 研究課題/領域番号 |
19K14548
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| 研究種目 |
若手研究
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| 配分区分 | 基金 |
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| 審査区分 |
小区分12010:基礎解析学関連
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| 研究機関 | 東北大学 |
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研究代表者 |
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| 研究期間 (年度) |
2019-04-01 – 2024-03-31
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| 研究課題ステータス |
交付 (2022年度)
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| 配分額 *注記 |
3,640千円 (直接経費: 2,800千円、間接経費: 840千円)
2022年度: 780千円 (直接経費: 600千円、間接経費: 180千円)
2021年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
2020年度: 1,040千円 (直接経費: 800千円、間接経費: 240千円)
2019年度: 910千円 (直接経費: 700千円、間接経費: 210千円)
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| キーワード | Index theory / Quantum walk / Topological phases / Operator Algebras / Gapped ground state / SPT phase / operator algebras / gapped ground state / ground state / spectral flow / K-theory |
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| 研究開始時の研究の概要 |
We will use tools from operator algebras to prove precise statements that characterise gapped ground states of quantum spin chains. Cohomology theories can be incorporated into this framework which will clarify the manifestly topological nature of such ground states.
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| 研究実績の概要 |
The project has been quite successful in adapting techniques from operator algebras, index theory and noncommutative geometry to certain classes of quantum mechanical Hamiltonians and gapped ground states.
Building from our established framework, we began a systematic study of topological and index theoretic properties of quantum walks, which can be regarded as quantum (discrete) analogues of random walks. In particular, quantum walks are expected to be useful in the implementation of algorithms in quantum computing. Determining robust properties of such quantum walks via a concrete link to topological invariants is therefore of significant value.
We were able to adapt our mathematical techniques in the study of Hamiltonians and gapped ground states to quantum walk systems and define several topologically robust indices. This work has been submitted for publication and is currently under review.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The key expected outcomes and goals of the submitted proposal have largely been accomplished. Robust topological indices have been defined for a large class of Hamiltonians that support a unique gapped ground state. However, due to the COVID pandemic, most of the proposed travel plans have had to be delayed or cancelled. As such, we have extended the project for one more year to further expand upon the project's original goals as well as reschedule delayed travel plans.
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| 今後の研究の推進方策 |
Our new results on topological phases of quantum walks provide a strong starting point to further explore new and novel connections between index theory and operator algebras with quantum information theory.
As a first step in this direction, our aim is to characterise quantum walks with additional anti-linear symmetries such as charge-conjugation symmetry. We also aim to study quantum walks that arise from periodically driven systems and their connection with secondary invariants.
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