{"id":{"repo_id":"maryland","oai_identifier":"oai:drum.lib.umd.edu:1903/34101"},"canonical_url":"https://search.dev.ndltd.org/etd/maryland/oai:drum.lib.umd.edu:1903/34101","repository":{"repo_id":"maryland","name":"University of Maryland","base_url":"https://api.drum.lib.umd.edu/server/oai/request"},"display":{"title":"Quantum Codes, Transversal Gates, and Representation Theory","abstract":"Recently an algorithm has been constructed that shows the binary icosahedral group2I together with a T-like gate forms the most efficient single-qubit universal gate set [54]. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. We fill this void by constructing a family of distance d = 3 codes that all implement 2I transversally [39, 42]. To do this, we introduce twisted unitary t-groups, a generalization of unitary t-groups under a twisting by an irreducible representation. We then apply representation theoretic methods to the Knill-Laflamme error correction conditions to show that twisted unitary t-groups automatically correspond to quantum codes with distance d = t + 1. Moreover, these methods produce many other quantum codes with interesting transver- sal gates. In particular, we use our methods to construct families of d ≥ 2 quantum codes realizing nearly all the possible transversal gate groups that are unitary 2-designs or better [41]. We also classify certain groups of two qubit gates that may occur as the transversal gate group of a quantum code with two logical qubits, describing the groups by their entanglement structure [40]. Finally, inspired by unitary 2-design groups, we consider the problem of finding Lie primitive subgroups of a simple Lie group.","abstract_html":"Recently an algorithm has been constructed that shows the binary icosahedral group2I together with a T-like gate forms the most efficient single-qubit universal gate set [54]. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. We fill this void by constructing a family of distance d = 3 codes that all implement 2I transversally [39, 42]. To do this, we introduce twisted unitary t-groups, a generalization of unitary t-groups under a twisting by an irreducible representation. We then apply representation theoretic methods to the Knill-Laflamme error correction conditions to show that twisted unitary t-groups automatically correspond to quantum codes with distance d = t + 1. Moreover, these methods produce many other quantum codes with interesting transver- sal gates. In particular, we use our methods to construct families of d ≥ 2 quantum codes realizing nearly all the possible transversal gate groups that are unitary 2-designs or better [41]. We also classify certain groups of two qubit gates that may occur as the transversal gate group of a quantum code with two logical qubits, describing the groups by their entanglement structure [40]. Finally, inspired by unitary 2-design groups, we consider the problem of finding Lie primitive subgroups of a simple Lie group.","abstract_has_math":false,"creators":["Teixeira, Ian Gershon"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Rosenberg, Jonathan M"],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025","date_published":"2025","updated_at":"2026-07-24T03:02:25Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://doi.org/10.13016/2ded-uurs"],"render_values":[{"text":"https://doi.org/10.13016/2ded-uurs","href":"https://doi.org/10.13016/2ded-uurs","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1903/34101","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Rosenberg, Jonathan M"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Teixeira, Ian Gershon"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-08-08T11:45:25Z"]},{"key":"dc:date.issued","label":"Date","values":["2025"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://doi.org/10.13016/2ded-uurs"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1903/34101"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Recently an algorithm has been constructed that shows the binary icosahedral group2I together with a T-like gate forms the most efficient single-qubit universal gate set [54]. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. We fill this void by constructing a family of distance d = 3 codes that all implement 2I transversally [39, 42]. To do this, we introduce twisted unitary t-groups, a generalization of unitary t-groups under a twisting by an irreducible representation. We then apply representation theoretic methods to the Knill-Laflamme error correction conditions to show that twisted unitary t-groups automatically correspond to quantum codes with distance d = t + 1. Moreover, these methods produce many other quantum codes with interesting transver- sal gates. In particular, we use our methods to construct families of d ≥ 2 quantum codes realizing nearly all the possible transversal gate groups that are unitary 2-designs or better [41]. We also classify certain groups of two qubit gates that may occur as the transversal gate group of a quantum code with two logical qubits, describing the groups by their entanglement structure [40]. Finally, inspired by unitary 2-design groups, we consider the problem of finding Lie primitive subgroups of a simple Lie group."]},{"key":"dc:title","label":"Title","values":["Quantum Codes, Transversal Gates, and Representation Theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Rosenberg, Jonathan M"],"dc:contributor.department":["Mathematics"],"dc:creator":["Teixeira, Ian Gershon"],"dc:date.accessioned":["2025-08-08T11:45:25Z"],"dc:date.issued":["2025"],"dc:description.abstract":["Recently an algorithm has been constructed that shows the binary icosahedral group2I together with a T-like gate forms the most efficient single-qubit universal gate set [54]. To carry out the algorithm fault tolerantly requires a code that implements 2I transversally. We fill this void by constructing a family of distance d = 3 codes that all implement 2I transversally [39, 42]. To do this, we introduce twisted unitary t-groups, a generalization of unitary t-groups under a twisting by an irreducible representation. We then apply representation theoretic methods to the Knill-Laflamme error correction conditions to show that twisted unitary t-groups automatically correspond to quantum codes with distance d = t + 1. Moreover, these methods produce many other quantum codes with interesting transver- sal gates. In particular, we use our methods to construct families of d ≥ 2 quantum codes realizing nearly all the possible transversal gate groups that are unitary 2-designs or better [41]. We also classify certain groups of two qubit gates that may occur as the transversal gate group of a quantum code with two logical qubits, describing the groups by their entanglement structure [40]. Finally, inspired by unitary 2-design groups, we consider the problem of finding Lie primitive subgroups of a simple Lie group."],"dc:identifier":["https://doi.org/10.13016/2ded-uurs"],"dc:identifier.uri":["http://hdl.handle.net/1903/34101"],"dc:language.iso":["en"],"dc:title":["Quantum Codes, Transversal Gates, and Representation Theory"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T03:02:25Z"}