  Estimating and reducing the overhead of fault tolerance (FT) schemes is a
crucial step toward realizing scalable quantum computers. Of particular
interest are schemes based on two-dimensional (2D) topological codes such as
the surface and color codes which have high thresholds but lack a natural
implementation of a non-Clifford gate. In this work, we directly compare two
leading FT implementations of the T gate in 2D color codes under circuit noise
across a wide range of parameters in regimes of practical interest. We report
that implementing the T gate via code switching to a 3D color code does not
offer substantial savings over state distillation in terms of either space or
space-time overhead. We find a circuit noise threshold of 0.07(1)% for the T
gate via code switching, almost an order of magnitude below that achievable by
state distillation in the same setting. To arrive at these results, we provide
and simulate an optimized code switching procedure, and bound the effect of
various conceivable improvements. Many intermediate results in our analysis may
be of independent interest. For example, we optimize the 2D color code for
circuit noise yielding its largest threshold to date 0.37(1)%, and adapt and
optimize the restriction decoder finding a threshold of 0.80(5)% for the 3D
color code with perfect measurements under Z noise. Our work provides a
much-needed direct comparison of the overhead of state distillation and code
switching, and sheds light on the choice of future FT schemes and hardware
designs.
