Replication package: Bosonic codes from compact phase spaces

Abstract

Replication package — code, Jupyter notebooks, and precomputed data — for "Bosonic codes from compact phase spaces" (D. Roberts, A. Slipper, A. Parhizkar, V. V. Albert, and M. Hafezi). The paper develops the correspondence between Gottesman–Kitaev–Preskill (GKP) codes and elliptic curves — established concurrently by Conrad, Burchards & Flammia [1] and the present authors [2] — and extends it, via geometric quantization, to bosonic quantum codes associated with algebraic curves of arbitrary genus. Following geometric quantization, a quantized phase space is a family of coherent states holomorphically parametrized by a complex manifold; this work realizes the phase spaces parametrized by a compact Riemann surface. Code states are then holomorphic sections of a line bundle over that surface (automorphic forms), and the code dimension is fixed by Riemann–Roch — recovering GKP codes when the surface is an elliptic curve (genus 1) and yielding a new class of hyperbolic CV quantum codes at genus greater than or equal to 2. The package also numerically illustrates the central no-go result: an amenability dichotomy under which arbitrarily accurate approximate code words exist at genus one but not at any higher genus. The genus-two Bolza surface — the most symmetric genus-two curve — serves as the explicit worked example throughout, and is what the numerics compute. For that example the package: constructs the k=2 code space, verifies the Veronese factorization Y^2 = XZ that generates the full tower of code spaces, and renders the Husimi Q-functions of the code basis (main-text Fig. 2); builds the Poincaré theta series spanning the code space and studies their convergence; verifies the weight-2 canonical gate representation — V_R, V_U in the main text, ρ(R), ρ(U) in the notebooks — for the automorphism group Aut(C) = GL(2, F_3), symbolically and numerically; derives the circuit-QED conditional two-mode-squeezing Hamiltonian via a lab-frame dispersive Schrieffer–Wolff reduction; simulates the conditional-kick stabilization dynamics (main-text Fig. 3) and diagonalizes the stabilizer Hamiltonian to exhibit a spectral gap, a genus-two instance of the amenability dichotomy. Every computational figure in the main text and Supplemental Material is regenerated; the two non-computational schematics (main-text Fig. 1 and the SM circuit diagram) are illustrations, not code output. See README.md for the full file-to-figure map. [1] https://arxiv.org/abs/2407.03270 (2024)[2] https://github.com/droberts01/higher-genus-gkp (2024)