Julio C. Magdalena de la Fuente

We show how to perform scalable fault-tolerant non-Clifford gates in two dimensions by introducing domain walls between the surface code and a non-Abelian topological code whose codespace is stabilized by Clifford operators. We formulate a path integral framework which provides both a macroscopic picture for different logical gates as well as a way to derive the associated microscopic circuits. We present explicit protocols and planar non-Clifford circuits that implement non-Clifford logic gates on both surface codes as well as color codes on different geometries. The logical action of the protocol is determined by the spacetime geometry, using the same bulk circuit, composed of simple 2D local circuits of similar complexity to commonly used stabilizer-readout circuits. We present fault-tolerant schemes for logical Clifford measurements as well as diagonal unitary gates in the third level of the Clifford hierarchy such as T, CS and CCZ gate. We also show an equivalence between our approach and prior proposals where a 2D array of qubits reproduces the action of a transversal gate in a 3D stabilizer code over time, thus, establishing a new connection between 3D codes and 2D non-Abelian topological phases. We prove a threshold theorem for our protocols under local stochastic circuit noise using a just-in-time decoder to correct the non-Abelian code.