Weakly interacting fermions are prototypical quantum systems that are considered solvable in practice yet are beyond exact diagonalizability. They have motivated the development of perturbative methods for quantum many-body systems, including the diagrammatic quantum Monte Carlo method. Despite the empirical success in studying these quantum systems numerically, algorithms with provable runtime guarantees have only recently been discovered. I will present a quantum algorithm based on Lindbladians to compute the finite-temperature properties of weakly interacting fermions, followed by a later classical algorithm based on cluster expansion and belief propagation for computing the log partition function.
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