Persistent Link:
http://hdl.handle.net/10150/297012
Title:
Hypermap-Homology Quantum Codes
Author:
Leslie, Martin P.
Issue Date:
2013
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular, the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m²,2, m] code as compared to the toric code which is a [2m²,2, m]code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.
Type:
text; Electronic Dissertation
Keywords:
hypermap; hypermap homology; LDPC code; quantum code; toric code; Mathematics; CSS code
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Rychlik, Marek

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleHypermap-Homology Quantum Codesen_US
dc.creatorLeslie, Martin P.en_US
dc.contributor.authorLeslie, Martin P.en_US
dc.date.issued2013-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractWe introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices. We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular, the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m²,2, m] code as compared to the toric code which is a [2m²,2, m]code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjecthypermapen_US
dc.subjecthypermap homologyen_US
dc.subjectLDPC codeen_US
dc.subjectquantum codeen_US
dc.subjecttoric codeen_US
dc.subjectMathematicsen_US
dc.subjectCSS codeen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorRychlik, Mareken_US
dc.contributor.committeememberLux, Klausen_US
dc.contributor.committeememberWehr, Janeken_US
dc.contributor.committeememberTiep, Pham Huuen_US
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