Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code
Thomas R. Scruby, Dan E. Browne, Paul Webster, Michael Vasmer
Abstract
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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mrow><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:mrow><mml:mo accent="false">&#x00AF;</mml:mo></mml:mover></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi><mml:mi>C</mml:mi><mml:mi>Z</mml:mi></mml:math> should be applied. We use these slices to simulate the performance of a simple JIT decoder against stochastic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>X</mml:mi></mml:math> and measurement errors and find evidence for a threshold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>&#x223C;</mml:mo><mml:mn>0.1</mml:mn><mml:mi mathvariant="normal">&#x0025;</mml:mi></mml:math> in all three codes. We expect that this threshold could be improved by optimisation of the decoder.