A numerical error threshold for "colorful" quantum computing

"Colorful" quantum computing was first proposed by Héctor Bombín to realize universal, fault-tolerant quantum computing using 3D color codes, or tetrahedral codes. Unlike the well-studied surface code methods, colorful quantum computing does not require magic state distillation and instead relies on a universal set of transversal gates and measurements. Colorful quantum computing can be realized in either a 3D or a 2D system. In this paper, we numerically test 3D colorful quantum computing's resilience to noise. In addition to the independent and identically distributed noise that affects the initial state, we must also correct errors that arise as part of the initialization process. We find a threshold for fault-tolerance on the body-centered cubic lattice. This threshold upper-bounds the more experimentally feasible 2D colorful quantum computing scheme and hopefully motivates research into color code methods of quantum computing.