Graph state representation of the toric code

TL;DR

This paper characterizes the graph structure of the toric code, revealing it as composed of star and half graphs, and introduces a new graph-theoretic framework for topological quantum error correction.

Contribution

It identifies the specific graph components of the toric code and develops a log-depth circuit for state preparation, advancing understanding of topological quantum states.

Findings

Toric code graphs are composed of star and half graphs.

Multiple star graphs indicate topological order.

A log-depth circuit for state preparation is proposed.

Abstract

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.