An algebro-geometric study of two models of quantum computaiton
Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2018
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| Online Access: | http://hdl.handle.net/1721.1/119594 |
| _version_ | 1811092375531945984 |
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| author | Lorgat, Raeez |
| author2 | Peter W. Shor. |
| author_facet | Peter W. Shor. Lorgat, Raeez |
| author_sort | Lorgat, Raeez |
| collection | MIT |
| description | Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017. |
| first_indexed | 2024-09-23T15:17:10Z |
| format | Thesis |
| id | mit-1721.1/119594 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T15:17:10Z |
| publishDate | 2018 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/1195942019-04-10T14:06:56Z An algebro-geometric study of two models of quantum computaiton Lorgat, Raeez Peter W. Shor. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science. Electrical Engineering and Computer Science. Thesis: M. Eng., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2017. Cataloged from PDF version of thesis. Includes bibliographical references (pages 39-40). We investigate geometric aspects of two models of quantum computation: starting with i. the computational complexity of the particle excitations of topological phases of matter in the restricted model of a Topological Quantum Field Theory in the sense of Turaev before leading to ii. where in analogy with ideas of Nielsen et. al., we propose and study a geometric model for grover's search algorithm. i. presents fundamental results in the mathematics and physics literature on conformal field theory as a model for quantum computation, phrased within the algebraic framework of the theory of tensor categories. Our main questions are 1. how does the computational power of these excitations change as a function of the genus of a fixed 2-dimensional space-time? and 2. independent of any particular space-time, what structural properties of a TQFT govern its computational power? When restricted to a space-time with space-like degrees of freedom represented by a smooth surface of genus g, we answer the first question by observing a q⁹-fold degeneracy in the ground state of the TQFT resulting from the presence of abelian anyons with exchange statistics a q-th root of unity. Such a resource is a topologically fault-tolerant quantum memory. The abelian character of the emergent particle statistics leads us to answer the second question via an algebraic realization of non-abelian anyonic excitations in the language of unitary modular tensor categories. Subsequently, ii. studies the quantum mechanical evolution of a particle within the Schröedinger wave-function formalism of quantum mechanics: our primary result is a purely geometric proof of the optimality of Grover's Search Algorithm on n qubits obtained via a study of the geometric structure of a homogenous space for the Unitary group of transformations acting on a single qubit. by Raeez Lorgat. M. Eng. 2018-12-11T21:07:45Z 2018-12-11T21:07:45Z 2017 2017 Thesis http://hdl.handle.net/1721.1/119594 1066742016 eng MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission. http://dspace.mit.edu/handle/1721.1/7582 40 pages application/pdf Massachusetts Institute of Technology |
| spellingShingle | Electrical Engineering and Computer Science. Lorgat, Raeez An algebro-geometric study of two models of quantum computaiton |
| title | An algebro-geometric study of two models of quantum computaiton |
| title_full | An algebro-geometric study of two models of quantum computaiton |
| title_fullStr | An algebro-geometric study of two models of quantum computaiton |
| title_full_unstemmed | An algebro-geometric study of two models of quantum computaiton |
| title_short | An algebro-geometric study of two models of quantum computaiton |
| title_sort | algebro geometric study of two models of quantum computaiton |
| topic | Electrical Engineering and Computer Science. |
| url | http://hdl.handle.net/1721.1/119594 |
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