A graph-based formalism for surface codes and twists

Understand

Twist defects in surface codes can be used to encode more logical qubits, improve the code rate, and implement logical gates.

• In this work we provide a rigorous formalism for constructing surface codes with twists generalizing the well-defined homological formalism introduced by Kitaev for describing CSS surface codes.
• In particular, we associate a surface code to any graph $G$ embedded on any 2D-manifold, in such a way that (1) qubits are associated to the vertices of the graph, (2) stabilizers are associated to faces, (3) twist defects are associated to odd-degree vertices.
• In this way, we are able to reproduce the variety of surface codes, with and without twists, in the literature and produce some new examples.