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.
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机译：我们介绍了一类3D颜色代码（称为堆叠代码）以及容错转换，该转换将将二维（2D）颜色代码编码的逻辑量子位映射到堆叠代码中并返回。堆叠的代码允许非Clifford $ \ pi / 8 $逻辑门的横向实现，将其与在2D颜色代码中横向遍历的逻辑Clifford门相结合，即可提供既容错又通用的门集，而无需使用不稳定的魔术状态。然后，我们显示形成堆叠代码的各层可以展开并以2D布局排列。由于只能针对2D拓扑稳定器代码横向实现Clifford门，因此必须合并非本地操作，以允许非Clifford门的这种横向应用。我们的代码通过从2D颜色代码转换为通过仅测量大量2D颜色代码中的几何局部稳定器和量规运算符以及在此类2D代码之间具有一维边界支持的非局部运算符而引起的展开的堆叠代码来实现此操作。我们认为，该提出的实现非局部操作的方法是二维稳定器布局的一种方法，将有助于避免由魔术状态蒸馏引起的大开销。
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