Anyonic quantum computation via F-matrices and duality in quantum group theory
Abstract/Contents
- Abstract
- Quantum computers promise to solve problems far beyond our current capabilities. Topological quantum computers promise a scalable approach to quantum computation using intrinsically error-protected bits encoded in two-dimensional materials supporting anyons. This thesis studies the mathematical foundations of Topological Quantum Computing. This means we answer questions about ribbon fusion categories (RFCs) and establish results in the representation theory of quantum groups under the guise of understanding the computational power of theorized topological quantum computers. The main objective to understand the braid group representations arising from anyon systems, since they describe the sets of logic gates available anyonic computers for information processing. In the first part we explain the process of anyonic quantum computing and simulate it by constructing braid group representations arising from RFCs. We express each braid generator explicitly using polynomials in the category's F- and R-matrices. The main novelty is that we designed and implemented an F-matrix solver for SageMath that can explicitly compute unitary braid group representations arising from fusion rings associated to simple complex Lie algebras. In the second part we focus on metaplectic anyons, modeled by the spinor object in the fusion rings associated to orthogonal Drinfeld-Jimbo quantum groups. We obtain the associated braid representations using a detailed description of the centralizer algebras over tensor products of the spinor module These algebras elude the usual methods because the braiding operators have too many eigenvalues. We carve a new path. In the quantum group world, the spinor object can be viewed as a certain exterior algebra, so understanding the desired centralizers is tantamount to proving a quantized skew Howe duality theorem. We develop a unified approach to skew Howe duality via Clifford algebras. First we construct commuting actions of quantum general linear groups that factor through a quantized Clifford algebra and then we extend to orthogonal types through a seesaw. The actions de-quantize appropriately, so our approach unifies various quantum and classical results in the literature
Description
| Type of resource | text |
|---|---|
| Form | electronic resource; remote; computer; online resource |
| Extent | 1 online resource |
| Place | California |
| Place | [Stanford, California] |
| Publisher | [Stanford University] |
| Copyright date | 2022; ©2022 |
| Publication date | 2022; 2022 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Author | Aboumrad Sidaoui, Guillermo Antonio |
|---|---|
| Degree supervisor | Bump, Daniel, 1952- |
| Thesis advisor | Bump, Daniel, 1952- |
| Thesis advisor | Conrad, Brian, 1970- |
| Thesis advisor | Rowell, Eric |
| Thesis advisor | Ying, Lexing |
| Degree committee member | Conrad, Brian, 1970- |
| Degree committee member | Rowell, Eric |
| Degree committee member | Ying, Lexing |
| Associated with | Stanford University, Institute for Computational and Mathematical Engineering |
Subjects
| Genre | Theses |
|---|---|
| Genre | Text |
Bibliographic information
| Statement of responsibility | Guillermo (Willie) Aboumrad |
|---|---|
| Note | Submitted to the Institute for Computational and Mathematical Engineering |
| Thesis | Thesis Ph.D. Stanford University 2022 |
| Location | https://purl.stanford.edu/fx598gh2531 |
Access conditions
- Copyright
- © 2022 by Guillermo Antonio Aboumrad Sidaoui
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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