Andreas Bauer

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.

We propose a family of explicit geometrically local circuits realizing any abelian non-chiral topological phase as an actively error-corrected fault-tolerant memory. These circuits are constructed from measuring 1-form symmetries in discrete fixed-point path integrals, which we express through cellular cohomology and higher-order cup products. The specific path integral we use is the abelian Dijkgraaf-Witten state sum on a 3-dimensional cellulation, which is a spacetime representation of the twisted quantum double model. The resulting circuits are based on a syndrome extraction circuit of the (qudit) stabilizer toric code, into which we insert non-Clifford phase gates that implement the ``twist''. The overhead compared to the toric code is moderate, in contrast to known constructions for twisted abelian phases. The simplest non-trivial example is a fault-tolerant circuit for the double-semion phase, defined on the same set of qubits as the stabilizer toric code, with 12 controlled-S gates in addition to the 8 controlled-X gates and 2 single-qubit measurements of the toric code per spacetime unit cell. We also show that other architectures for the (qudit) toric code phase, like measurement-based topological quantum computation or Floquet codes, can be enriched with phase gates to implement twisted quantum doubles instead of their untwisted versions. As a further result, we prove fault tolerance under arbitrary local (including non-Pauli) noise for a very general class of topological circuits that we call 1-form symmetric fixed-point circuits. This notion unifies the circuits in this paper as well as the stabilizer toric code, subsystem toric code, measurement-based topological quantum computation, or the (CSS) honeycomb Floquet code. We also demonstrate how our method can be adapted to construct fault-tolerant circuits for specific non-Abelian phases.