  The driven-dissipative many-body problem remains one of the most challenging
unsolved problems in quantum mechanics. The advent of quantum computers may
provide a unique platform for efficiently simulating such driven-dissipative
systems. But there are many choices for how one can engineer the reservoir. One
can simply employ ancilla qubits to act as a reservoir and then digitally
simulate them via algorithmic cooling. A more attractive approach, which allows
one to simulate an infinite reservoir, is to integrate out the bath degrees of
freedom and describe the driven-dissipative system via a master equation, that
can also be simulated on a quantum computer. In this work, we consider the
particular case of non-interacting electrons on a lattice driven by an electric
field and coupled to a fermionic thermostat. Then, we provide two different
quantum circuits: the first one reconstructs the full dynamics of the system
using Trotter steps, while the second one dissipatively prepares the final
non-equilibrium steady state in a single step. We run both circuits on the IBM
quantum experience. For circuit (i), we achieved up to 5 Trotter steps. When
partial resets become available on quantum computers, we expect that the
maximum simulation time can be significantly increased. The methods developed
here suggest generalizations that can be applied to simulating interacting
driven-dissipative systems.
