  Recent advances in numerical methods significantly pushed forward the
understanding of electrons coupled to quantized lattice vibrations. At this
stage, it becomes increasingly important to also account for the effects of
physically inevitable environments. In particular, we study the transport
properties of the Hubbard-Holstein Hamiltonian that models a large class of
materials characterized by strong electron-phonon coupling, in contact with a
dissipative environment. Even in the one-dimensional and isolated case,
simulating the quantum dynamics of such a system with high accuracy is very
challenging due to the infinite dimensionality of the phononic Hilbert spaces.
For this reason, the effects of dissipation on the conductance properties of
such systems have not been investigated systematically so far. We combine the
non-Markovian hierarchy of pure states method and the Markovian quantum jumps
method with the newly introduced projected purified density-matrix
renormalization group, creating powerful tensor-network methods for dissipative
quantum many-body systems. Investigating their numerical properties, we find a
significant speedup up to a factor $\sim 30$ compared to conventional
tensor-network techniques. We apply these methods to study dissipative
quenches, aiming for an in-depth understanding of the formation, stability, and
quasi-particle properties of bipolarons. Surprisingly, our results show that in
the metallic phase dissipation localizes the bipolarons, which is reminiscent
of an indirect quantum Zeno effect. However, the bipolaronic binding energy
remains mainly unaffected, even in the presence of strong dissipation,
exhibiting remarkable bipolaron stability. These findings shed light on the
problem of designing real materials exhibiting phonon-mediated
high-$T_\mathrm{C}$ superconductivity.
