Fused and intertwined subsystem surface codes on a tetrahedron for low-overhead non-Clifford gates

Code switching using a three dimensional subsystem code has been proposed as a alternative to magic state distillation. Three dimensional codes offer a number of attractive features such as a geometric locality in three dimensions, single-shot decoding, and access to a universal set of logical gates implementable in constant time, independent of code distance. The most studied example of such codes is Bombin's gauge color code. Comparative studies between code switching and magic state distillation have found that code switching with the gauge color code is only competitive in the very low noise regime which can be attributed, at least in part, to the high-weight of the check operators that one must measure. In this work, leveraging recent constructions of subsystem surface codes, we construct three-dimensional subsystem and stabilizer codes analogous to Bombin's color and gauge color codes but with lower weight checks: weight 1, 3, and 4 compared to weight 4 and 6 for the gauge color code. In various gauge fixings, the subsystem code has access to a universal set of logical gates implemented via low-depth circuits.