Graph Representation of Topological Stabilizer States

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dc.contributor.advisorFeder, David
dc.contributor.advisorSanders, Barry
dc.contributor.authorLiao, Pengcheng
dc.contributor.committeememberBarzanjeh, Shabir
dc.contributor.committeememberSimon, Christoph
dc.contributor.committeememberHøyer, Peter
dc.date2022-02
dc.date.accessioned2022-01-17T16:59:34Z
dc.date.available2022-01-17T16:59:34Z
dc.date.issued2022-01
dc.description.abstractTopological quantum states, especially these in topological stabilizer quantum error correction codes, are currently the focus of intense activity because of their potential for fault-tolerant operations. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. In this thesis, I show that the toric code graph is composed of only two kinds of subgraphs: star graphs and half graphs. The topological order of the toric code is identified with the existence of multiple star graphs, which reveals a nice connection between repetition codes and the toric code. The graph structure readily yields a log-depth and a constant-depth (including ancillae) circuit for state preparation. Next, I derive the necessary and sufficient conditions for a family of graph states to be in TQO-1, a class of quantum error correction code states whose code distance scales macroscopically with the number of physical qubits. Using these criteria, I consider a number of specific graph families, including the star and complete graphs, and the line graphs of complete and completely bipartite graphs, and discuss which are topologically ordered and how to construct the codewords. The formalism is then employed to construct several codes with macroscopic distance, including a three-dimensional topological code generated by local stabilizers that also has a macroscopic number of encoded logical qubits. Last, the connection between the characterization of topological order using graph theory and the hierarchy of topological order is analyzed.en_US
dc.identifier.citationLiao, P. (2022). Graph representation of topological stabilizer states (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca.en_US
dc.identifier.doihttp://dx.doi.org/10.11575/PRISM/39498
dc.identifier.urihttp://hdl.handle.net/1880/114283
dc.language.isoengen_US
dc.publisher.facultyScienceen_US
dc.publisher.institutionUniversity of Calgaryen
dc.rightsUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.en_US
dc.subjecttopological orderen_US
dc.subjectgraph statesen_US
dc.subjectquantum error correctionen_US
dc.subject.classificationCondensed Matteren_US
dc.subject.classificationPhysics--Theoryen_US
dc.titleGraph Representation of Topological Stabilizer Statesen_US
dc.typemaster thesisen_US
thesis.degree.disciplinePhysics & Astronomyen_US
thesis.degree.grantorUniversity of Calgaryen_US
thesis.degree.nameMaster of Science (MSc)en_US
ucalgary.item.requestcopytrueen_US
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