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 this https URL

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)

arXiv:2608.31156 [quant-ph]

(or arXiv:2608.31156v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.31156 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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