  The preparation of thermal states of matter is a crucial task in quantum
simulation. In this work, we prove that a recently introduced, efficiently
implementable dissipative evolution thermalizes to the Gibbs state in time
scaling polynomially with system size at high enough temperatures, and 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. To the best of our
knowledge, these are the first results rigorously establishing 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 temperatures logarithmic in the system's size is
polynomially equivalent to standard quantum computation. 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 analysing it as perturbation of the
Hamiltonian corresponding to infinite temperature. For low temperature, we
instead perform a perturbation at zero temperature of the Laplace transform of
the energy observable at fixed runtime, 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.
