Bosonic codes from compact phase spaces

• URL: http://arxiv.org/abs/2608.31156v1
• Date: Mon, 31 Aug 2026 17:55:49 GMT
• Title: Bosonic codes from compact phase spaces
• Abstract summary: We analytically generate the full tower of code spaces at all weights.<n>For any genus greater than one, the stabilizer group is non-amenable.<n>No normalizable quantum state can satisfy all stabilizer conditions.
• Score: 0.15534429177540246
• License: http://creativecommons.org/licenses/by/4.0/
• 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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