  The simulation of quantum systems has been a key aim of quantum technologies
for decades, and the generalisation to open systems is necessary to include
physically realistic systems. We introduce an approach for quantum simulations
of open system dynamics in terms of an environment of minimal size and a
time-dependent Hamiltonian. This enables the implementation of a
continuous-time simulation with a finite environment, whereas state of the art
methods require an infinite environment or only match the simulation at
discrete times. We find necessary and sufficient conditions for this
Hamiltonian to be well behaved and, when these are not met, we show that there
exists an approximate Hamiltonian that is, and look into its applications.
