  We investigate quantum algorithms derived from tensor networks to simulate
the static and dynamic properties of quantum many-body systems. Using a
sequentially prepared quantum circuit representation of a matrix product state
(MPS) that we call a quantum tensor network (QTN), we demonstrate algorithms to
prepare ground and excited states on a quantum computer and apply them to
molecular nanomagnets (MNMs) as a paradigmatic example. In this setting, we
develop two approaches for extracting the spectral correlation functions
measured in neutron scattering experiments: (a) a generalization of the SWAP
test for computing wavefunction overlaps and, (b) a generalization of the
notion of matrix product operators (MPOs) to the QTN setting which generates a
linear combination of unitaries. The latter method is discussed in detail for
translationally invariant spin-half systems, where it is shown to reduce the
qubit resource requirements compared with the SWAP method, and may be
generalized to other systems. We demonstrate the versatility of our approaches
by simulating spin-1/2 and spin-3/2 MNMs, with the latter being an
experimentally relevant model of a Cr$^{3+}_8$ ring. Our approach has qubit
requirements that are independent of the number of constituents of the
many-body system and scale only logarithmically with the bond dimension of the
MPS representation, making them appealing for implementation on near-term
quantum hardware with mid-circuit measurement and reset.
