Preparing ground states and thermal states is of key importance to many applications of a quantum computer. Simulating open system dynamics is a natural candidate for preparing such states. We introduce improved quantum algorithmic techniques for efficiently simulating Nature-inspired quantum Master Equations (Lindbladians) utilizing novel primitives such as the operator Fourier transform. We also develop improved general purpose Lindbaldian simulation algorithms utilizing a weak measurement scheme, whose effciency can be boosted to achieve essentially optimal complexity. Our techniques also lead to an efficient algorithm for preparing certain purified Gibbs states (called thermal field double states in high-energy physics) of rapidly thermalizing systems benefitting from a Szegedy-type quadratic improvement with respect to the spectral gap. Our algorithms' cost has a favorable dependence on temperature, accuracy, and the mixing time (or spectral gap) of the relevant Lindbladians.
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