Bosonic codes from compact phase spaces

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.

7 pages, 3 figures + Supplemental Material (30 pages, 4 figures). Code, notebooks, and data at https://doi.org/10.5281/zenodo.20585488