The cost of universality: A comparative study of the overhead of state distillation and code switching with color codes

Optimizing and estimating the overhead of existing fault tolerance schemes is a crucial step toward realizing scalable quantum computers. Many of the most promising schemes are based upon two-dimensional (2D) topological codes such as the surface and color codes, with a universal gate set consisting of readily implementable Cliffords along with the much more costly T -gate. In our work, we compare the cost of fault-tolerantly implementing the T -gate in color codes using two leading approaches: state distillation and code switching. We report that despite being treated less favorably in our analysis, state distillation is more resource-efficient than code switching, in terms of both qubit overhead and space time overhead. In particular, we find a T-gate threshold via code switching of 0.07% under circuit noise, almost an order of magnitude below that for distillation. To arrive at this result, we implement an end-to-end code switching simulation and accomplish many intermediate goals. For instance: (i) we optimize the 2D color code for circuit noise yielding it’s largest threshold to date 0.037(1)% and (ii) we adapt and optimize the restriction decoder and find a threshold of 0.8% for the 3D color code with perfect measurements under Z noise. We not only find numerical estimates of the overhead of the explicit code switching protocol, but also lower bound the overhead with various conceivable improvements.