  We apply algorithms based on Lieb-Robinson bounds to simulate time-dependent
and thermal quantities in quantum systems. For time-dependent systems, we
modify a previous mapping to quantum circuits to significantly reduce the
computer resources required. This modification is based on a principle of
"observing" the system outside the light-cone. We apply this method to study
spin relaxation in systems started out of equilibrium with initial conditions
that give rise to very rapid entanglement growth. We also show that it is
possible to approximate time evolution under a local Hamiltonian by a quantum
circuit whose light-cone naturally matches the Lieb-Robinson velocity.
Asymptotically, these modified methods allow a doubling of the system size that
one can obtain compared to direct simulation. We then consider a different
problem of thermal properties of disordered spin chains and use quantum belief
propagation to average over different configurations. We test this algorithm on
one dimensional systems with mixed ferromagnetic and anti-ferromagnetic bonds,
where we can compare to quantum Monte Carlo, and then we apply it to the study
of disordered, frustrated spin systems.
