Graph Representation of Topological Stabilizer States

Topological 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.

Name thesis:degree_name

Master of Science (MSc)

Discipline thesis:degree_discipline

Physics & Astronomy

Grantor dc:publisher.institution

Science

Year dc:date.issued

2022

Author dc:creator

• Liao, Pengcheng

Advisors dc:contributor.advisor

• Feder, David
• Sanders, Barry

Committee members dc:contributor.committeemember

• Barzanjeh, Shabir
• Simon, Christoph
• Høyer, Peter

• topological order
• graph states
• quantum error correction

Statement dc:rights

• University 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.

Language dc:language.iso

eng

DOI

http://dx.doi.org/10.11575/PRISM/39498

OAI identifier oai:identifier

oai:ucalgary.scholaris.ca:1880/114283