{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/119594"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/119594","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"An algebro-geometric study of two models of quantum computaiton","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Lorgat, Raeez"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.","school":null,"contributors":[],"advisors":["Peter W. Shor."],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017","date_published":"2017","updated_at":"2026-07-22T22:22:11Z","subjects":["Electrical Engineering and Computer Science."],"languages":["eng"],"rights":["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."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/119594","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Peter W. Shor."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/119594"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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)."]},{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["M. Eng."]},{"key":"dc:title","label":"Title","values":["An algebro-geometric study of two models of quantum computaiton"]}]}],"canonical_facts":{"dc:contributor.advisor":["Peter W. Shor."],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science."],"dc:contributor.other":["Massachusetts Institute of Technology. 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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)."],"dc:description.abstract":["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."],"dc:description.degree":["M. Eng."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/119594"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["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."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Electrical Engineering and Computer Science."],"dc:title":["An algebro-geometric study of two models of quantum computaiton"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:22:11Z"}