Abstract: We present a framework for closure of spatiotemporal dynamics governed by partial differential equations using quantum mechanics. This approach builds quantum mechanical systems that are embedded in the discretization mesh of the coarse model, and act as data-driven surrogate models for the unresolved degrees of freedom of the original dynamics. The coupled classical-quantum system evolves through a prediction--correction cycle, whereby the quantum mechanical models provide estimates of the subgrid fluxes through expectation values of quantum observables, and the classical coarse model conditions the quantum mechanical states via a quantum mechanical Bayes rule. Moreover, by combining methods from operator-valued kernels and delay embedding, the quantum mechanical systems capture a compressed representation of the dynamics that is positivity-preserving and invariant under spatial symmetries of the original dynamics. We illustrate this scheme with applications to closure of the shallow-water equations in a periodic domain.