Learning and simulating bosonic systems via finite-energy locality

Bosonic devices promise applications in simulation, sensing and quantum error correction, but infinite-dimensional local Hilbert spaces obstruct the locality tools that make qubit dynamics efficiently learnable and simulable. We establish a finite-energy locality principle for geometrically local bosonic open systems satisfying photon-number moment propagation. It compares unbounded generators with Galerkin cutoffs, transferring finite-dimensional Lieb--Robinson, product-formula and circuit techniques to bosons with explicit errors. For this moment-controlled class, we obtain, to our knowledge, the first model-independent weak Lieb--Robinson bounds beyond Bose--Hubbard-type dynamics, together with quantitative Trotter and simulation guarantees for polynomial bosonic GKSL generators. As a central application, coherent-state preparation and local heterodyne detection suffice to learn coefficients of a known bounded-degree polynomial Hamiltonian ansatz local on a bounded-growth interaction graph to accuracy $\varepsilon$ and failure probability $\delta$, with sample complexity and total evolution time both $\widetilde{\mathcal{O}}(\varepsilon^{-2}\log(m/\delta))$, where $m$ counts on-site and interaction terms. This matches the best known finite-dimensional locality-assisted scaling in accuracy and system size, up to polylogarithmic factors. The assumptions hold for Bose--Hubbard and quadratic dynamics without added dissipation; for more general local polynomial Hamiltonians they can be supplied natively or engineered by known multi-photon loss in stabilized bosonic architectures.