{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/106102"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/106102","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Applications of abelian algebraic structures in quantum computation","abstract":"Shor's groundbreaking algorithms for integer factoring and discrete logarithm [58], along with their later generalizations 116, 35, 49, 18], demonstrated a unique ability of quantum computers to solve problems defined on abelian groups. In this thesis, we study ways in which that ability can be leveraged in order to solve problems on more complex structures such as non-abelian groups and hypergroups. This leads to new quantum algorithms for the hidden subgroup problem on nilpotent groups whose order is a product of large primes, the hidden subhypergroup problem on both strongly integral hypergroups and ultragroups, testing equivalence of group extensions, and computing the component parts of the cohomology groups of both group extensions and a generalization of simplicial complexes, amongst other problems. For each of those listed, we also show that no classical algorithm can achieve similar efficiency under standard cryptographic assumptions.","abstract_html":"Shor's groundbreaking algorithms for integer factoring and discrete logarithm [58], along with their later generalizations 116, 35, 49, 18], demonstrated a unique ability of quantum computers to solve problems defined on abelian groups. In this thesis, we study ways in which that ability can be leveraged in order to solve problems on more complex structures such as non-abelian groups and hypergroups. This leads to new quantum algorithms for the hidden subgroup problem on nilpotent groups whose order is a product of large primes, the hidden subhypergroup problem on both strongly integral hypergroups and ultragroups, testing equivalence of group extensions, and computing the component parts of the cohomology groups of both group extensions and a generalization of simplicial complexes, amongst other problems. For each of those listed, we also show that no classical algorithm can achieve similar efficiency under standard cryptographic assumptions.","abstract_has_math":false,"creators":["Zatloukal, Kevin C. (Kevin Chaffee)"],"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":["Aram W. Harrow."],"committee_chairs":[],"committee_members":[],"year":2016,"date_issued":"2016","date_published":"2016","updated_at":"2026-07-22T22:21:55Z","subjects":["Electrical Engineering and Computer Science."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/106102","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Aram W. Harrow."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. 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D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2016.","Cataloged from PDF version of thesis.","Includes bibliographical references (pages 163-168)."]},{"key":"dc:description.abstract","label":"Abstract","values":["Shor's groundbreaking algorithms for integer factoring and discrete logarithm [58], along with their later generalizations 116, 35, 49, 18], demonstrated a unique ability of quantum computers to solve problems defined on abelian groups. In this thesis, we study ways in which that ability can be leveraged in order to solve problems on more complex structures such as non-abelian groups and hypergroups. This leads to new quantum algorithms for the hidden subgroup problem on nilpotent groups whose order is a product of large primes, the hidden subhypergroup problem on both strongly integral hypergroups and ultragroups, testing equivalence of group extensions, and computing the component parts of the cohomology groups of both group extensions and a generalization of simplicial complexes, amongst other problems. For each of those listed, we also show that no classical algorithm can achieve similar efficiency under standard cryptographic assumptions."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Applications of abelian algebraic structures in quantum computation"]}]}],"canonical_facts":{"dc:contributor.advisor":["Aram W. Harrow."],"dc:contributor.department":["Massachusetts Institute of Technology. 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In this thesis, we study ways in which that ability can be leveraged in order to solve problems on more complex structures such as non-abelian groups and hypergroups. This leads to new quantum algorithms for the hidden subgroup problem on nilpotent groups whose order is a product of large primes, the hidden subhypergroup problem on both strongly integral hypergroups and ultragroups, testing equivalence of group extensions, and computing the component parts of the cohomology groups of both group extensions and a generalization of simplicial complexes, amongst other problems. For each of those listed, we also show that no classical algorithm can achieve similar efficiency under standard cryptographic assumptions."],"dc:description.degree":["Ph. D."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/106102"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. 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