  This work discusses the solution of partial differential equations (PDEs)
using matrix product states (MPS). The study focuses on the search for the
lowest eigenstates of a Hamiltonian equation, for which five algorithms are
introduced: imaginary-time evolution, steepest gradient descent, an improved
gradient descent, an implicitly restarted Arnoldi method, and density matrix
renormalization group (DMRG) optimization. The first four methods are
engineered using a framework of limited-precision linear algebra, where
operations between MPS and matrix product operators (MPOs) are implemented with
finite resources. All methods are benchmarked using the PDE for a quantum
harmonic oscillator in up to two dimensions, over a regular grid with up to
$2^{28}$ points. Our study reveals that all MPS-based techniques outperform
exact diagonalization techniques based on vectors, with respect to memory
usage. Imaginary-time algorithms are shown to underperform any type of gradient
descent, both in terms of calibration needs and costs. Finally, Arnoldi like
methods and DMRG asymptotically outperform all other methods, including exact
diagonalization, as problem size increases, with an exponential advantage in
memory and time usage.
