  We study the non-equilibrium dynamics of the dissipative quantum East model
via numerical tensor networks. We use matrix product states to represent
evolution under quantum-jump unravellings for sizes beyond those accessible to
exact diagonalisation. This allows us to demonstrate that dynamical
heterogeneity accompanies slow relaxation, in analogy with what is seen in
classical glassy systems. Furthermore, using variational matrix product
operators we: (i) compute the spectral gap of the Lindbladian, and show that
glassiness is enhanced in the presence of weak quantum fluctuations compared to
the pure classical case, and (ii) obtain the dynamical large deviations by
calculating the leading eigenvector of the tilted Lindbladian, and find clear
evidence for a first-order active-inactive dynamical phase transition. We also
show how to directly sample the rare quantum trajectories associated to the
large deviations.
