2021/03/31 by Pengcheng Liao, David L. Feder · 8 citations
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #Algorithm #Block code #Combinatorics #Computer science #Decoding methods #Discrete mathematics #Graph #Graphene research and applications #Linear code #Mathematics #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum and electron transport phenomena #Quantum convolutional code #Quantum mechanics #Topological graph #Topological order #Topology (electrical circuits) #Toric code #quant-ph
paper · pdf · doi:10.1103/physreva.104.012432
published in Physical Review A 104(1) (American Physical Society) · 18 pages, 8 figures. One section about encoding arbitrary state into toric code is added
arxiv created 2021/06/25 · openalex publication_date 2021/07/30 · arxiv updated 2021/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Given their potential for fault-tolerant operations, topological quantum states are currently a 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 nonlocal 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 graph-theoretic framework for the investigation of topological order and the development of topological error correction codes.