{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/40917"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/40917","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Quantum codes on Hurwitz surfaces","abstract":"Ever since the birth of the first quantum error correcting code, many error correcting techniques and formalism has been constructed so far. Among those, generating a quantum code on a locally planar geometry have lead to some interesting classes of codes. Main idea of this thesis stems from Kitaev's Toric code, which was the first surface code, yet it suffered from having a asymptotically vanishing encoding rate. In this paper, we propose a quantum surface code on a more complicated closed surface which has large genus, namely the Hurwitz surface. This code admits a constant encoding rate in the asymptotic limit that the number of genus goes to infinity. However, we give evidence that t/n, where n is the number of qubits and t is the number of correctible errors, converges to 0 asymptotically. This is based on numerically generating many Hurwitz surfaces and observing the corresponding quantum code in the limit that genus number goes to infinity.","abstract_html":"Ever since the birth of the first quantum error correcting code, many error correcting techniques and formalism has been constructed so far. Among those, generating a quantum code on a locally planar geometry have lead to some interesting classes of codes. Main idea of this thesis stems from Kitaev's Toric code, which was the first surface code, yet it suffered from having a asymptotically vanishing encoding rate. In this paper, we propose a quantum surface code on a more complicated closed surface which has large genus, namely the Hurwitz surface. This code admits a constant encoding rate in the asymptotic limit that the number of genus goes to infinity. However, we give evidence that t/n, where n is the number of qubits and t is the number of correctible errors, converges to 0 asymptotically. This is based on numerically generating many Hurwitz surfaces and observing the corresponding quantum code in the limit that genus number goes to infinity.","abstract_has_math":false,"creators":["Kim, Isaac H. (Isaac Hyun)"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Physics.","school":null,"contributors":[],"advisors":["Peter Shor."],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007","date_published":"2007","updated_at":"2026-07-22T22:21:31Z","subjects":["Physics."],"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/1721.1/40917","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Peter Shor."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Physics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Physics."]},{"key":"dc:creator","label":"Author","values":["Kim, Isaac H. 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Main idea of this thesis stems from Kitaev's Toric code, which was the first surface code, yet it suffered from having a asymptotically vanishing encoding rate. In this paper, we propose a quantum surface code on a more complicated closed surface which has large genus, namely the Hurwitz surface. This code admits a constant encoding rate in the asymptotic limit that the number of genus goes to infinity. However, we give evidence that t/n, where n is the number of qubits and t is the number of correctible errors, converges to 0 asymptotically. This is based on numerically generating many Hurwitz surfaces and observing the corresponding quantum code in the limit that genus number goes to infinity."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["S.B."]},{"key":"dc:title","label":"Title","values":["Quantum codes on Hurwitz surfaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Peter Shor."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Physics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Physics."],"dc:creator":["Kim, Isaac H. (Isaac Hyun)"],"dc:date.accessioned":["2008-03-27T18:23:08Z"],"dc:date.available":["2008-03-27T18:23:08Z"],"dc:date.issued":["2007"],"dc:description":["Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Physics, 2007.","Includes bibliographical references (p. 41-43)."],"dc:description.abstract":["Ever since the birth of the first quantum error correcting code, many error correcting techniques and formalism has been constructed so far. Among those, generating a quantum code on a locally planar geometry have lead to some interesting classes of codes. Main idea of this thesis stems from Kitaev's Toric code, which was the first surface code, yet it suffered from having a asymptotically vanishing encoding rate. In this paper, we propose a quantum surface code on a more complicated closed surface which has large genus, namely the Hurwitz surface. This code admits a constant encoding rate in the asymptotic limit that the number of genus goes to infinity. However, we give evidence that t/n, where n is the number of qubits and t is the number of correctible errors, converges to 0 asymptotically. This is based on numerically generating many Hurwitz surfaces and observing the corresponding quantum code in the limit that genus number goes to infinity."],"dc:description.degree":["S.B."],"dc:identifier.uri":["http://hdl.handle.net/1721.1/40917"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc: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."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Physics."],"dc:title":["Quantum codes on Hurwitz surfaces"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:31Z"}