We build on recent work by B. Brown (Sci. Adv. 6, eaay4929 (2020)) to develop
and simulate an explicit recipe for a just-in-time decoding scheme in three 3D
surface codes, which can be used to implement a transversal (non-Clifford)
CCZ between three 2D surface codes in time linear in the code
distance. We present a fully detailed set of bounded-height lattice slices
through the 3D codes which retain the code distance and measurement-error
detecting properties of the full 3D code and admit a dimension-jumping process
which expands from/collapses to 2D surface codes supported on the boundaries of
each slice. At each timestep of the procedure the slices agree on a common set
of overlapping qubits on which CCZ should be applied. We use these slices to
simulate the performance of a simple JIT decoder against stochastic X and
measurement errors and find evidence for a threshold pc∼0.1% in all
three codes. We expect that this threshold could be improved by optimisation of
the decoder.Comment: 19 pages, 11 figures. Additional supplementary materials at
https://github.com/tRowans/JIT-supplementary-materials. v2; removed some
claims regarding issues with staircase slices and changed one referenc