  Starting from a simple mapping of a generator of local stochastic dynamics to
a quantum Hamiltonian, we derive a condition, which allows us to use the
quasi-adiabatic evolution and so relate gapped quantum phases with
non-equilibrium's. This leads us to a study of invertible matrix product
operators. Finally, we present an ansatz for constructing local stochastic
dynamics for which the Perron-Frobenius vector has a Matrix Product
Representation. Additionally, we get for free that the dynamics satisfy a
generalized form of detailed balance.
