The ability to engineer non-Gaussian quantum resources underlies quantum technologies from communication and metrology to universal computation. However, while a number of canonical works have set no-go limits for attaining such resources from Gaussian operations, it is widely assumed that such resources can be tuned freely by non-Gaussian Hamiltonian dynamics. Here we prove a general no-go theorem for such resource shaping: no smooth Hamiltonian dynamics can modify higher-order statistical moments of a continuous-variable state without simultaneously changing its mean and covariance. This analytic constraint implies a rigidity theorem for Hamiltonian quantum control: only quadratic (symplectic) generators preserve the Gaussian moment hierarchy, while every nonquadratic term necessarily couples the Gaussian and non-Gaussian sectors. The theorem identifies the symplectic algebra as the unique invariant subalgebra whose differential representations terminate at finite (second) order within the otherwise infinite Hamiltonian algebra. It thereby defines the analytic boundary between classically simulable Gaussian dynamics and the fully universal non-Gaussian regime, the continuous-variable analog of the Gottesman-Knill frontier.

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• Quantum formalism
• Quantum foundations
• Quantum information theory