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arXiv · 2608.31156

Bosonic codes from compact phase spaces

David Roberts·Aaron Slipper·Alireza Parhizkar·Victor V. Albert·Mohammad Hafezi
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Abstract

We present the algebraic structure of bosonic quantum error-correcting codes on genus-two Riemann surfaces. We explicitly construct the code words as automorphic forms and analytically generate the full tower of code spaces at all weights. We prove a fundamental no-go theorem: for any genus greater than one, the stabilizer group is non-amenable, forcing a strictly positive spectral gap in the stabilizer Hamiltonian. Consequently, no normalizable quantum state can satisfy all stabilizer conditions. This sharply contrasts with standard Gottesman-Kitaev-Preskill (GKP) codes, where the amenability of the stabilizer group $\mathbb{Z}^2$ permits approximate code words with arbitrary precision.

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David Roberts, Aaron Slipper, Alireza Parhizkar, Victor V. Albert, Mohammad Hafezi. 2026-08-31. Bosonic codes from compact phase spaces. https://arxiv.org/abs/2608.31156

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