<p>The preparation of thermal states of matter is a crucial task in quantum simulation. Here we prove that recently introduced, efficiently implementable dissipative evolution thermalizes to the Gibbs state in time scaling polynomially with system size at high-enough temperatures for any Hamiltonian that satisfies a Lieb–Robinson bound, such as local Hamiltonians on a lattice. Furthermore, we show the efficient adiabatic preparation of the associated purifications or ‘thermofield double’ states. These results establish the efficient preparation of high-temperature Gibbs states and their purifications. In the low-temperature regime, we show that implementing this family of dissipative evolutions for inverse temperature polynomial in the system’s size is computationally equivalent to polynomial-time quantum computations. On a technical level, for high temperatures, our proof makes use of the mapping of the generator of the evolution into a Hamiltonian, and then connecting its convergence to that of the infinite temperature limit. For low temperature, we instead perform a perturbation at zero temperature and resort to circuit-to-Hamiltonian mappings akin to the proof of universality of quantum adiabatic computing. Taken together, our results show that a family of quasi-local dissipative evolutions efficiently prepares a large class of quantum many-body states of interest, and has the potential to mirror the success of classical Monte Carlo methods for quantum many-body systems.</p>