Graph state representation of the toric code
| Autor: | Pengcheng Liao, David L. Feder |
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| Rok vydání: | 2021 |
| Předmět: |
Physics
Discrete mathematics Quantum Physics Toric code FOS: Physical sciences Star (graph theory) Graph state 01 natural sciences 010305 fluids & plasmas Quantum circuit Quantum state Quantum error correction 0103 physical sciences Topological order 010306 general physics Error detection and correction Quantum Physics (quant-ph) |
| DOI: | 10.48550/arxiv.2103.12268 |
| Popis: | Given their potential for fault-tolerant operations, topological quantum states are currently the focus of intense activity. Of particular interest are topological quantum error correction codes, such as the surface and planar stabilizer codes that are equivalent to the celebrated toric code. While every stabilizer state maps to a graph state under local Clifford operations, the graphs associated with topological stabilizer codes remain unknown. We show that the toric code graph is composed of only two kinds of subgraphs: star graphs (which encode Greenberger-Horne-Zeilinger states) and half graphs. The topological order is identified with the existence of multiple star graphs, which reveals a connection between the repetition and toric codes. The graph structure readily yields a log-depth quantum circuit for state preparation, assuming geometrically non-local gates, which can be reduced to a constant depth including ancillae and measurements at the cost of increasing the circuit width. The results provide a new graph-theoretic framework for the investigation of topological order and the development of novel topological error correction codes. Comment: 18 pages, 8 figures. One section about encoding arbitrary state into toric code is added |
| Databáze: | OpenAIRE |
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