Numerical Implementation of Just-In-Time Decoding in Novel Lattice Slices Through the Three-Dimensional Surface Code

TL;DR

This paper develops and simulates a just-in-time decoding scheme for 3D surface codes, enabling transversal non-Clifford gates with a threshold around 0.1%, and introduces a detailed lattice slicing method that preserves code properties.

Contribution

It presents a novel explicit recipe for JIT decoding in 3D surface codes using lattice slices, facilitating transversal CCZ gates and threshold estimation.

Findings

Threshold estimated at ~0.1% for stochastic errors

Lattice slices retain code distance and error detection

Simulation shows potential for decoder optimization

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) $\overline{CCZ}$ 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.