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) $\overline{CCZ}$ 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 $p_c \sim 0.1\%$ in all three codes. We expect that this threshold could be improved by optimisation of the decoder.

中文翻译：

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通过三维表面码在新格子切片中实时解码的数值实现

我们在 B. Brown 最近的工作（Sci. Adv. 6, eaay4929 (2020)）的基础上开发和模拟三个 3D 表面码中的即时解码方案的显式配方，可用于实现三个二维表面代码之间的横向（非 Clifford）$\overline{CCZ}$ 在时间上与代码距离呈线性关系。我们通过 3D 代码展示了一组完整的有界高度晶格切片，这些切片保留了完整 3D 代码的代码距离和测量误差检测属性，并允许从/折叠到支持的 2D 表面代码的维度跳跃过程每个切片的边界。在该过程的每个时间步长上，切片就应该应用 $CCZ$ 的一组公共重叠量子位达成一致。我们使用这些切片来模拟简单 JIT 解码器针对随机 $X$ 和测量误差的性能，并在所有三个代码中找到阈值 $p_c\sim 0.1\%$ 的证据。我们期望可以通过优化解码器来提高这个阈值。