Symplectic Barnes-Wall GKP Codes: Deterministic $O(N \log^2 N)$ Decoding and Logarithmic Rate Scaling

• URL: http://arxiv.org/abs/2608.00601v1
• Date: Sat, 01 Aug 2026 11:36:03 GMT
• Title: Symplectic Barnes-Wall GKP Codes: Deterministic $O(N \log^2 N)$ Decoding and Logarithmic Rate Scaling
• Abstract summary: We construct a family of multimode Gottesman-Kitaev-Preskill codes with encoding rate $R = frac12log N$ and a deterministic $O(Nlog2 N)$ bounded-distance decoder.<n>This construction provides a deterministic, space-efficient paradigm for GKP error correction in platforms supporting non-local modular connectivity.
• Score: 12.999017083466264
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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