  The advancement of quantum simulators motivates the development of a
theoretical framework to assist with efficient state preparation in quantum
many-body systems. Generally, preparing a target entangled state via unitary
evolution with time-dependent couplings is a challenging task and very little
is known about the existence of solutions and their properties. In this work we
develop a constructive approach for preparing matrix product states (MPS) via
continuous unitary evolution. We provide an explicit construction of the
operator which exactly implements the evolution of a given MPS along a
specified direction in its tangent space. This operator can be written as a sum
of local terms of finite range, yet it is in general non-Hermitian. Relying on
the explicit construction of the non-Hermitian generator of the dynamics, we
demonstrate the existence of a Hermitian sequence of operators that implements
the desired MPS evolution with the error which decreases exponentially with the
operator range. The construction is benchmarked on an explicit periodic
trajectory in a translationally invariant MPS manifold. We demonstrate that the
Floquet unitary generating the dynamics over one period of the trajectory
features an approximate MPS-like eigenstate embedded among a sea of
thermalizing eigenstates. These results show that our construction is useful
not only for state preparation and control of many-body systems, but also
provides a generic route towards Floquet scars -- periodically driven models
with quasi-local generators of dynamics that have exact MPS eigenstates in
their spectrum.
