Homological stabilizer codes
In this paper we define homological stabilizer codes on qubits which encompass codes such as Kitaev’s toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs...
Gespeichert in:
| Veröffentlicht in: | Annals of physics 2013-03, Vol.330, p.1-22 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| container_end_page | 22 |
|---|---|
| container_issue | |
| container_start_page | 1 |
| container_title | Annals of physics |
| container_volume | 330 |
| creator | Anderson, Jonas T. |
| description | In this paper we define homological stabilizer codes on qubits which encompass codes such as Kitaev’s toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs which are suitable as homological stabilizer codes. We then show that all toric codes are equivalent to homological stabilizer codes on 4-valent graphs. We show that the topological color codes and toric codes correspond to two distinct classes of graphs. We define the notion of label set equivalencies and show that under a small set of constraints the only homological stabilizer codes without local logical operators are equivalent to Kitaev’s toric code or to the topological color codes.
► We show that Kitaev’s toric codes are equivalent to homological stabilizer codes on 4-valent graphs. ► We show that toric codes and color codes correspond to homological stabilizer codes on distinct graphs. ► We find and classify all 2D homological stabilizer codes. ► We find optimal codes among the homological stabilizer codes. |
| doi_str_mv | 10.1016/j.aop.2012.11.007 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_osti_</sourceid><recordid>TN_cdi_osti_scitechconnect_22157076</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0003491612001790</els_id><sourcerecordid>1315680929</sourcerecordid><originalsourceid>FETCH-LOGICAL-c429t-552f5a7cbaf4aafcf6f0f5c9e7f679776d6436356ce2f5a7d43c618f625058b93</originalsourceid><addsrcrecordid>eNp9kEFLAzEQhYMoWKs_QPBQ8OJl15nsZrKLJylqhYIXBW8hzSaast3UZCvor3drRTx5msv3Hm8-xk4RcgSky2WuwzrngDxHzAHkHhsh1JRBIZ732QgAiqyskQ7ZUUpLAMRSVCN2Ngur0IYXb3Q7Sb1e-NZ_2jgxobHpmB043SZ78nPH7On25nE6y-YPd_fT63lmSl73mRDcCS3NQrtSa2ccOXDC1FY6krWU1FBZUCHI2G-wKQtDWDniAkS1qIsxO9_1htR7lYzvrXk1oeus6RXnKCRIGqiLHbWO4W1jU69WPhnbtrqzYZMUFiiogpr_KfxFl2ETu-EHhZwqIgESBwp3lIkhpWidWke_0vFDIaitVbVUg1W1taoQ1WB1yFztMnbw8e5t3M61nbGNj9u1TfD_pL8AQHx8oA</addsrcrecordid><sourcetype>Open Access Repository</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1268665071</pqid></control><display><type>article</type><title>Homological stabilizer codes</title><source>ScienceDirect Journals (5 years ago - present)</source><creator>Anderson, Jonas T.</creator><creatorcontrib>Anderson, Jonas T.</creatorcontrib><description>In this paper we define homological stabilizer codes on qubits which encompass codes such as Kitaev’s toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs which are suitable as homological stabilizer codes. We then show that all toric codes are equivalent to homological stabilizer codes on 4-valent graphs. We show that the topological color codes and toric codes correspond to two distinct classes of graphs. We define the notion of label set equivalencies and show that under a small set of constraints the only homological stabilizer codes without local logical operators are equivalent to Kitaev’s toric code or to the topological color codes.
► We show that Kitaev’s toric codes are equivalent to homological stabilizer codes on 4-valent graphs. ► We show that toric codes and color codes correspond to homological stabilizer codes on distinct graphs. ► We find and classify all 2D homological stabilizer codes. ► We find optimal codes among the homological stabilizer codes.</description><identifier>ISSN: 0003-4916</identifier><identifier>EISSN: 1096-035X</identifier><identifier>DOI: 10.1016/j.aop.2012.11.007</identifier><identifier>CODEN: APNYA6</identifier><language>eng</language><publisher>New York: Elsevier Inc</publisher><subject>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS ; Color ; Color code ; CORRECTIONS ; DIAGRAMS ; Equivalence ; ERRORS ; GRAPH THEORY ; Graphs ; Homology ; LIMITING VALUES ; Operators ; Physics ; QUANTUM CRYPTOGRAPHY ; Quantum error correction ; QUANTUM MECHANICS ; QUANTUM STATES ; QUBITS ; Qubits (quantum computing) ; Topological order ; Topological quantum code ; TOPOLOGY ; Toric code</subject><ispartof>Annals of physics, 2013-03, Vol.330, p.1-22</ispartof><rights>2012 Elsevier Inc.</rights><rights>Copyright © 2013 Elsevier B.V. All rights reserved.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c429t-552f5a7cbaf4aafcf6f0f5c9e7f679776d6436356ce2f5a7d43c618f625058b93</citedby><cites>FETCH-LOGICAL-c429t-552f5a7cbaf4aafcf6f0f5c9e7f679776d6436356ce2f5a7d43c618f625058b93</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.aop.2012.11.007$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>230,314,780,784,885,3548,27923,27924,45994</link.rule.ids><backlink>$$Uhttps://www.osti.gov/biblio/22157076$$D View this record in Osti.gov$$Hfree_for_read</backlink></links><search><creatorcontrib>Anderson, Jonas T.</creatorcontrib><title>Homological stabilizer codes</title><title>Annals of physics</title><description>In this paper we define homological stabilizer codes on qubits which encompass codes such as Kitaev’s toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs which are suitable as homological stabilizer codes. We then show that all toric codes are equivalent to homological stabilizer codes on 4-valent graphs. We show that the topological color codes and toric codes correspond to two distinct classes of graphs. We define the notion of label set equivalencies and show that under a small set of constraints the only homological stabilizer codes without local logical operators are equivalent to Kitaev’s toric code or to the topological color codes.
► We show that Kitaev’s toric codes are equivalent to homological stabilizer codes on 4-valent graphs. ► We show that toric codes and color codes correspond to homological stabilizer codes on distinct graphs. ► We find and classify all 2D homological stabilizer codes. ► We find optimal codes among the homological stabilizer codes.</description><subject>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</subject><subject>Color</subject><subject>Color code</subject><subject>CORRECTIONS</subject><subject>DIAGRAMS</subject><subject>Equivalence</subject><subject>ERRORS</subject><subject>GRAPH THEORY</subject><subject>Graphs</subject><subject>Homology</subject><subject>LIMITING VALUES</subject><subject>Operators</subject><subject>Physics</subject><subject>QUANTUM CRYPTOGRAPHY</subject><subject>Quantum error correction</subject><subject>QUANTUM MECHANICS</subject><subject>QUANTUM STATES</subject><subject>QUBITS</subject><subject>Qubits (quantum computing)</subject><subject>Topological order</subject><subject>Topological quantum code</subject><subject>TOPOLOGY</subject><subject>Toric code</subject><issn>0003-4916</issn><issn>1096-035X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2013</creationdate><recordtype>article</recordtype><recordid>eNp9kEFLAzEQhYMoWKs_QPBQ8OJl15nsZrKLJylqhYIXBW8hzSaast3UZCvor3drRTx5msv3Hm8-xk4RcgSky2WuwzrngDxHzAHkHhsh1JRBIZ732QgAiqyskQ7ZUUpLAMRSVCN2Ngur0IYXb3Q7Sb1e-NZ_2jgxobHpmB043SZ78nPH7On25nE6y-YPd_fT63lmSl73mRDcCS3NQrtSa2ccOXDC1FY6krWU1FBZUCHI2G-wKQtDWDniAkS1qIsxO9_1htR7lYzvrXk1oeus6RXnKCRIGqiLHbWO4W1jU69WPhnbtrqzYZMUFiiogpr_KfxFl2ETu-EHhZwqIgESBwp3lIkhpWidWke_0vFDIaitVbVUg1W1taoQ1WB1yFztMnbw8e5t3M61nbGNj9u1TfD_pL8AQHx8oA</recordid><startdate>20130301</startdate><enddate>20130301</enddate><creator>Anderson, Jonas T.</creator><general>Elsevier Inc</general><general>Elsevier BV</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7U5</scope><scope>8FD</scope><scope>L7M</scope><scope>OTOTI</scope></search><sort><creationdate>20130301</creationdate><title>Homological stabilizer codes</title><author>Anderson, Jonas T.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c429t-552f5a7cbaf4aafcf6f0f5c9e7f679776d6436356ce2f5a7d43c618f625058b93</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2013</creationdate><topic>CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS</topic><topic>Color</topic><topic>Color code</topic><topic>CORRECTIONS</topic><topic>DIAGRAMS</topic><topic>Equivalence</topic><topic>ERRORS</topic><topic>GRAPH THEORY</topic><topic>Graphs</topic><topic>Homology</topic><topic>LIMITING VALUES</topic><topic>Operators</topic><topic>Physics</topic><topic>QUANTUM CRYPTOGRAPHY</topic><topic>Quantum error correction</topic><topic>QUANTUM MECHANICS</topic><topic>QUANTUM STATES</topic><topic>QUBITS</topic><topic>Qubits (quantum computing)</topic><topic>Topological order</topic><topic>Topological quantum code</topic><topic>TOPOLOGY</topic><topic>Toric code</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Anderson, Jonas T.</creatorcontrib><collection>CrossRef</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>OSTI.GOV</collection><jtitle>Annals of physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Anderson, Jonas T.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Homological stabilizer codes</atitle><jtitle>Annals of physics</jtitle><date>2013-03-01</date><risdate>2013</risdate><volume>330</volume><spage>1</spage><epage>22</epage><pages>1-22</pages><issn>0003-4916</issn><eissn>1096-035X</eissn><coden>APNYA6</coden><abstract>In this paper we define homological stabilizer codes on qubits which encompass codes such as Kitaev’s toric code and the topological color codes. These codes are defined solely by the graphs they reside on. This feature allows us to use properties of topological graph theory to determine the graphs which are suitable as homological stabilizer codes. We then show that all toric codes are equivalent to homological stabilizer codes on 4-valent graphs. We show that the topological color codes and toric codes correspond to two distinct classes of graphs. We define the notion of label set equivalencies and show that under a small set of constraints the only homological stabilizer codes without local logical operators are equivalent to Kitaev’s toric code or to the topological color codes.
► We show that Kitaev’s toric codes are equivalent to homological stabilizer codes on 4-valent graphs. ► We show that toric codes and color codes correspond to homological stabilizer codes on distinct graphs. ► We find and classify all 2D homological stabilizer codes. ► We find optimal codes among the homological stabilizer codes.</abstract><cop>New York</cop><pub>Elsevier Inc</pub><doi>10.1016/j.aop.2012.11.007</doi><tpages>22</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0003-4916 |
| ispartof | Annals of physics, 2013-03, Vol.330, p.1-22 |
| issn | 0003-4916 1096-035X |
| language | eng |
| recordid | cdi_osti_scitechconnect_22157076 |
| source | ScienceDirect Journals (5 years ago - present) |
| subjects | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS Color Color code CORRECTIONS DIAGRAMS Equivalence ERRORS GRAPH THEORY Graphs Homology LIMITING VALUES Operators Physics QUANTUM CRYPTOGRAPHY Quantum error correction QUANTUM MECHANICS QUANTUM STATES QUBITS Qubits (quantum computing) Topological order Topological quantum code TOPOLOGY Toric code |
| title | Homological stabilizer codes |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-11T11%3A35%3A55IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_osti_&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Homological%20stabilizer%20codes&rft.jtitle=Annals%20of%20physics&rft.au=Anderson,%20Jonas%20T.&rft.date=2013-03-01&rft.volume=330&rft.spage=1&rft.epage=22&rft.pages=1-22&rft.issn=0003-4916&rft.eissn=1096-035X&rft.coden=APNYA6&rft_id=info:doi/10.1016/j.aop.2012.11.007&rft_dat=%3Cproquest_osti_%3E1315680929%3C/proquest_osti_%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=1268665071&rft_id=info:pmid/&rft_els_id=S0003491612001790&rfr_iscdi=true |