Stacked codes: Universal fault tolerant quantum computation in a two dimensional layout
We introduce a class of 3D color codes, which we call stacked codes, together with a fault tolerant transformation that will map logical qubits encoded in two dimensional (2D) color codes into stacked codes and back. The stacked code allows for the transversal implementation of a non Clifford \pi/8$ logical gate, which when combined with the logical Clifford gates that are transversal in the 2D color code give a gate set which is both fault tolerant and universal without requiring nonstabilizer magic states. We then show that the layers forming the stacked code can be unfolded and arranged in a 2D layout. As only Clifford gates can be implemented transversally for 2D topological stabilizer codes, a non local operation must be incorporated in order to allow for this transversal application of a non Clifford gate. Our code achieves this operation through the transformation from a 2D color code to the unfolded stacked code induced by measuring only geometrically local stabilizers and gauge operators within the bulk of 2D color codes together with a nonlocal operator that has support on a one dimensional boundary between such 2D codes. We believe that this proposed method to implement the non local operation is a realistic one for 2D stabilizer layouts and would be beneficial in avoiding the large overheads caused by magic state distillation.