Quantum belief propagation: An algorithm for thermal quantum systems

We present an accurate numerical algorithm, called quantum belief propagation, for simulation of one-dimensional quantum systems at nonzero temperature. The algorithm exploits the fact that quantum effects are short-range in these systems at nonzero temperature, decaying on a length scale inversely proportional to the temperature. We compare to exact results on a spin-$1∕2$ Heisenberg chain. Even a very modest calculation, requiring diagonalizing only ten-by-ten matrices, reproduces the peak susceptibility with a relative error of less than ${10}^{-5}$, while more elaborate calculations further reduce the error.