Classical algorithms, correlation decay, and complex zeros of partition functions of quantum many-body systems

Basic statistical properties of quantum many-body systems in thermal equilibrium can be obtained from their partition function. In this talk, I will present a quasi-polynomial time classical algorithm that estimates the partition function of quantum many-body systems at temperatures above the thermal phase transition point. It is known that in the worst case, the same problem is NP-hard below this temperature. This shows that the transition in the phase of a quantum system is also accompanied by a transition in the computational hardness of estimating its statistical properties. The key to this result is a characterization of the phase transition and the critical behavior of the system in terms of the complex zeros of the partition function. I will also discuss the relation between these complex zeros and another signature of the thermal phase transition, namely, the exponential decay of correlations. I will show that in a system of n particles above the phase transition point, where the complex zeros are far from the real axis, the correlation between two observables whose distance is at least log(n) decays exponentially. This is based on joint work with Aram Harrow and Saeed Mehraban.