  The simulation of out-of-equilibrium dissipative quantum many body systems is
a problem of fundamental interest to a number of fields in physics, ranging
from condensed matter to cosmology. For unitary systems, tensor network methods
have proved successful and extending these to open systems is a natural avenue
for study. In particular, an important question concerns the possibility of
approximating the critical dynamics of non-equilibrium systems with tensor
networks. Here, we investigate this by performing numerical simulations of a
paradigmatic quantum non-equilibrium system with an absorbing state: the
quantum contact process. We consider the application of matrix product states
and the time-evolving block decimation algorithm to simulate the time-evolution
of the quantum contact process at criticality. In the Lindblad formalism, we
find that the Heisenberg picture can be used to improve the accuracy of
simulations over the Schr\"{o}dinger approach, which can be understood by
considering the evolution of operator-space entanglement. Furthermore, we also
consider a quantum trajectories approach, which we find can reproduce the
expected universal behaviour of key observables for a significantly longer time
than direct simulation of the average state. These improved results provide
further evidence that the universality class of the quantum contact process is
not directed percolation, which is the class of the classical contact process.
