ShadowFluid: Task-Aligned Reduced Dynamics for Quantum Fluid Simulation

Quantum fluid simulation typically evolves the full quantum state, even when downstream tasks require only a small set of coarse-grained observables. We present ShadowFluid, a task-aligned, operator-first framework that constructs and evolves a reduced representation sufficient for target observables such as low-frequency density and energy statistics. Given a Hamiltonian and target observables, ShadowFluid builds a multi-reference dictionary of Fourier-mode coherence operators whose expectations form a reduced density-matrix block governed by a low-dimensional shadow generator. The reduced dynamics are exact when the dictionary is invariant under commutation with the Hamiltonian; otherwise, we derive a truncated evolution with a state-independent commutator-leakage indicator computable before simulation. We further establish a Frobenius-norm error bound with linear-in-time growth. Experiments on two-dimensional Schr\"odinger-flow benchmarks show exact recovery of the low-pass dynamics in the closed regime and sub-1% density error under nontrivial mode coupling at moderate cutoffs, with substantially smaller errors on target observables. These results demonstrate that task-relevant fluid dynamics can be extracted through compact, error-controlled representations without evolving and reconstructing full quantum state.