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We introduce a class of 3D color codes, which we call stacked codes, togetherwith a fault-tolerant transformation that will map logical qubits encoded intwo-dimensional (2D) color codes into stacked codes and back. The stacked codeallows for the transversal implementation of a non-Clifford $\pi/8$ logicalgate, which when combined with the logical Clifford gates that are transversalin the 2D color code give a gate set which is both fault-tolerant and universalwithout requiring nonstabilizer magic states. We then show that the layersforming the stacked code can be unfolded and arranged in a 2D layout. As onlyClifford gates can be implemented transversally for 2D topological stabilizercodes, a non-local operation must be incorporated in order to allow for thistransversal application of a non-Clifford gate. Our code achieves thisoperation through the transformation from a 2D color code to the unfoldedstacked code induced by measuring only geometrically local stabilizers andgauge operators within the bulk of 2D color codes together with a nonlocaloperator 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 arealistic one for 2D stabilizer layouts and would be beneficial in avoiding thelarge overheads caused by magic state distillation.

Stacked codes: universal fault-tolerant quantum computation in a two-dimensional layout