This paper presents a novel approach to efficiently preparing thermal states of matter in quantum systems, which is crucial for quantum simulation. The authors prove that a recently introduced dissipative evolution, when implemented efficiently, thermalizes to the Gibbs state in polynomial time with respect to the system size, provided the temperature is high enough and the Hamiltonian satisfies a Lieb-Robinson bound. This result is significant because it establishes the first rigorous proof of efficient preparation of high-temperature Gibbs states and their purifications on a quantum computer. For low temperatures, the authors show that implementing this dissipative evolution for inverse temperatures logarithmic in the system's size is polynomially equivalent to standard quantum computation. The proof techniques involve mapping the generator of the evolution to a Hamiltonian and analyzing its perturbation properties. The results have implications for quantum many-body systems and could potentially mirror the success of classical Monte Carlo methods in the quantum realm. The paper also discusses the adiabatic preparation of purified Gibbs states and establishes a connection between the class of problems solvable by this method and the complexity class BQP, demonstrating that the algorithm can efficiently prepare states with inverse polynomial overlap with the ground states of BQP-hard Hamiltonians.This paper presents a novel approach to efficiently preparing thermal states of matter in quantum systems, which is crucial for quantum simulation. The authors prove that a recently introduced dissipative evolution, when implemented efficiently, thermalizes to the Gibbs state in polynomial time with respect to the system size, provided the temperature is high enough and the Hamiltonian satisfies a Lieb-Robinson bound. This result is significant because it establishes the first rigorous proof of efficient preparation of high-temperature Gibbs states and their purifications on a quantum computer. For low temperatures, the authors show that implementing this dissipative evolution for inverse temperatures logarithmic in the system's size is polynomially equivalent to standard quantum computation. The proof techniques involve mapping the generator of the evolution to a Hamiltonian and analyzing its perturbation properties. The results have implications for quantum many-body systems and could potentially mirror the success of classical Monte Carlo methods in the quantum realm. The paper also discusses the adiabatic preparation of purified Gibbs states and establishes a connection between the class of problems solvable by this method and the complexity class BQP, demonstrating that the algorithm can efficiently prepare states with inverse polynomial overlap with the ground states of BQP-hard Hamiltonians.