  Non-commutative polynomial optimization (NPO) problems seek to minimize the
state average of a polynomial of some operator variables, subject to polynomial
constraints, over all states and operators, as well as the Hilbert spaces where
those might be defined. Many of these problems are known to admit a complete
hierarchy of semidefinite programming (SDP) relaxations. NPO theory has found
application in quantum information theory, quantum chemistry and statistical
physics. In this work, we consider a variant of NPO problems where a subset of
the operator variables satisfies a system of ordinary differential equations.
We find that, under mild conditions of operator boundedness, for every such
problem one can construct a standard NPO problem with the same solution. This
allows us to define a complete hierarchy of SDPs to tackle the original
differential problem. We apply this method to extrapolate quantum time series
in a semi-device-independent way and sketch how one can use it to model
Hamiltonian time evolution in many-body quantum systems.
