Estimating Gibbs partition function with quantum Clifford sampling

The partition function is an essential quantity in statistical mechanics, and its accurate computation is a key component of any statistical analysis of quantum systems and phenomena. However, for interacting many-body quantum systems, its calculation generally involves summing over an exponential number of terms and can thus quickly grow to be intractable. Accurately and efficiently estimating the partition function of its corresponding system Hamiltonian then becomes the key in solving quantum many-body problems. In this paper we develop a hybrid quantum-classical algorithm to estimate the partition function, utilising a novel quantum Clifford sampling technique. Note that previous works on the estimation of partition functions require O(1/epsilon root Delta)-depth quantum circuits (Srinivasan et al 2021 IEEE Int. Conf. on Quantum Computing and Engineering (QCE) pp 112-22; Montanaro 2015 Proc. R. Soc. A 471 20150301), where Delta is the minimum spectral gap of stochastic matrices and epsilon is the multiplicative error. Our algorithm requires only a shallow O(1)-depth quantum circuit, repeated O(n/epsilon(2)) times, to provide a comparable epsilon approximation. Shallow-depth quantum circuits are considered vitally important for currently available noisy intermediate-scale quantum devices.