  Hamiltonian simulation, i.e., simulating the real time evolution of a target
quantum system, is a natural application of quantum computing. Trotter-Suzuki
splitting methods can generate corresponding quantum circuits; however, a
faithful approximation can lead to relatively deep circuits. Here we start from
the insight that for translation invariant systems, the gates in such circuit
topologies can be further optimized on classical computers to decrease the
circuit depth and/or increase the accuracy. We employ tensor network techniques
and devise a method based on the Riemannian trust-region algorithm on the
unitary matrix manifold for this purpose. For the Ising and Heisenberg models
on a one-dimensional lattice, we achieve orders of magnitude accuracy
improvements compared to fourth-order splitting methods. The optimized circuits
could also be of practical use for the time-evolving block decimation (TEBD)
algorithm.
