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We present an algorithm that prepares thermal Gibbs states of one dimensional quantum systems on a quantum computer without any memory overhead, and in a time significantly shorter than other known alternatives. Specifically, the time complexity is dominated by the quantity {NJ/T}, where N is the size of the system, J is a bound on the operator norm of the local terms of the Hamiltonian (coupling energy), and T is the temperature. Given other results on the complexity of thermalization, this overall scaling is likely optimal. For higher dimensions, our algorithm lowers the known scaling of the time complexity with the dimension of the system by one.