Compositional Quantum Surrogates: Circuit-Guided Representations for Quantum Surrogate Learning

Quantum expectation surrogates amortize measurements across related circuit controls, but downstream tasks require different forms of reuse. Existing predictivesurrogate methods learn mean-value functions; how the fitted representation serves ordered predictions, observable reuse, and surrogate-based optimization remains open. We introduce Compositional Quantum Surrogates (CQS) through two circuit-guided architectures: CQS-R, a recurrent model for ordered layers or drive periods, and CQS-P, a masked Fourier product model for observable-dependent interactions. With matched masks and frequency coverage, product composition reduces moment prediction error by 10.5% relative to complete Fourier regression. A query-wise recurrent control reaches similar accuracy on 12-period trajectories but requires 78 updates, compared with 12 shared updates for CQS-R. In VQE, linear moment readouts enable Hamiltonian reuse, and Fourier structure enables analytic continuation; on two-layer grids, continuation mainly improves attainment near a feasible reference. The results favor dependency-aware products for moments, shared recurrence for trajectories, and continuation-stabilized search for VQE pre-training.