  Simulating time evolution of quantum systems is one of the most promising
applications of quantum computing and also appears as a subroutine in many
applications such as Green's function methods. In the current era of NISQ
machines we assess the state of algorithms for simulating time dynamics with
limited resources. We propose the Jaynes-Cummings model and extensions to it as
useful toy models to investigate time evolution algorithms on near-term quantum
computers. Using these simple models, direct Trotterisation of the time
evolution operator produces deep circuits, requiring coherence times out of
reach on current NISQ hardware. Therefore we test two alternative responses to
this problem: variational compilation of the time evolution operator, and
variational quantum simulation of the wavefunction ansatz. We demonstrate
numerically to what extent these methods are successful in time evolving this
system. The costs in terms of circuit depth and number of measurements are
compared quantitatively, along with other drawbacks and advantages of each
method. We find that computational requirements for both methods make them
suitable for performing time evolution simulations of our models on NISQ
hardware. Our results also indicate that variational quantum compilation
produces more accurate results than variational quantum simulation, at the cost
of a larger number of measurements.
