Simpler (Classical) and Faster (Quantum) Algorithms for Gibbs Partition Functions

We give classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Specifically, we modify the classical algorithm of Štefankovič, Vempala and Vigoda (J. ACM, 56(3), 2009) to improve its sample complexity; and we quantize this new algorithm, improving upon the previously best quantum algorithm for computing Gibbs partition functions due to Harrow and Wei (SODA 2020).The conventional approach to estimate partition functions requires approximating the mean of Gibbs distributions at nearby inverse temperatures that satisfy certain properties; this set of temperatures is called a cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Štefankovič, Vempala and Vigoda with the paired-product estimator of Huber (Ann. Appl. Probab., 25(2), 2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Štefankovič, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.