Estimating the properties of interacting quantum systems is a central problem in physics and quantum computing. In this talk, I will present a quasi-polynomial time approximation algorithm for estimating quantum partition functions using the locus of their complex zeros. I will also explore the connection between these complex zeros and the decay of correlations in the quantum Gibbs state. These results extend techniques developed for classical counting problems, including a seminal work of Dobrushin and Shlosman and a recent approach due to Barvinok, to quantum settings. This is based on joint work with Aram Harrow and Saeed Mehraban.