  The simulation of many-body open quantum systems is key to solving numerous
outstanding problems in physics, chemistry, material science, and in the
development of quantum technologies. Near-term quantum computers may bring
considerable advantage for the efficient simulation of their static and
dynamical properties, thanks to hybrid quantum-classical variational algorithms
to approximate the dynamics of the density matrix describing the quantum state
in terms of an ensemble average. Here, a variational quantum algorithm is
developed to simulate the real-time evolution of the density matrix governed by
the Lindblad master equation, under the assumption that the quantum state has a
bounded entropy along the dynamics, entailing a low-rank representation of its
density matrix. The algorithm encodes each pure state of the statistical
mixture as a parametrized quantum circuit, and the associated probabilities as
additional variational parameters stored classically, thereby requiring a
significantly lower number of qubits than algorithms where the full density
matrix is encoded in the quantum memory. Two variational Ans\"atze are
proposed, and their effectiveness is assessed in the simulation of the dynamics
of a 2D dissipative transverse field Ising model. The results underscore the
algorithm's efficiency in simulating the dynamics of open quantum systems in
the low-rank regime with limited quantum resources on a near-term quantum
device.
