  Given the wide applicability of topological states for fault-tolerant quantum
computation, an important outstanding question is how to generate these states
within the quantum circuit model, and if it is possible to do so efficiently.
In this work, toric code states on the torus are mapped to an equivalent graph
state under local Clifford operations. The graph is composed of only two
subgraphs, both of which can be generated in log depth, assuming geometrically
non-local gates. Including ancillae and measurements, the circuit depth can be
reduced to a constant at the cost of increasing the circuit width. Among many
insights, the toric code graph structure reveals a close connection between
repetition and topological codes. Due to their local equivalence to the toric
code, all two-dimensional translationally invariant topological stabilizer
codes are therefore efficiently preparable on a quantum circuit. The results
provide a new graph-theoretic framework for the development of novel
topological error correction codes, beyond geometrical considerations.
