AI 中文总结
该研究构造了具有对数编码效率的辛巴恩斯-沃尔GKP码，实现了确定性O(N log²N)译码，为支持非局域模块化连接的平台提供了空间高效的GKP纠错方案。
AI 中文摘要
我们构造了巴恩斯-沃尔格的显式辛实现，得到一组多模Gottesman-Kitaev-Preskill（GKP）码，其编码速率R=1/2 log₂N，且具有确定性O(N log²N)的有界距离译码器。递归生成元Gₘ₊₁=(Gₘ 0; Gₘ Rₘ Gₘ)（其中Rₘ=I+Ω）同时保证有效量子稳定子的辛整性，并通过一系列等距同构保留精确的巴恩斯-沃尔译码结构。码距在2π单位下恒为Δ²=1，构成显式的距离-速率权衡，以非缩放保护为代价实现对数编码效率。该构造为支持非局域模块化连接的平台提供了一种确定性、空间高效的GKP纠错范式。
英文摘要
We construct an explicit symplectic realization of the Barnes-Wall lattice that yields a family of multimode Gottesman-Kitaev-Preskill (GKP) codes with encoding rate $R = \frac{1}{2}\log_2 N$ and a deterministic $O(N\log^2 N)$ bounded-distance decoder. The recursive generator $G_{m+1} = \bigl(\begin{smallmatrix} G_m & 0 \\ G_m & R_m G_m \end{smallmatrix}\bigr)$ with $R_m = I + Ω$ simultaneously guarantees symplectic integrality for valid quantum stabilizers and preserves the exact Barnes-Wall decoding structure through a chain of isometric isomorphisms. The code distance is constant at $Δ^2 = 1$ (in units of $2π$), representing an explicit distance--rate tradeoff in which logarithmic encoding efficiency is achieved at the cost of non-scaling protection. This construction provides a deterministic, space-efficient paradigm for GKP error correction in platforms supporting non-local modular connectivity.
