Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature

We introduce a method that ensures efficient computation of one-dimensional quantum systems with long-range interactions across all temperatures. Our algorithm operates within a quasi-polynomial runtime for inverse temperatures up to $\beta={\rm poly}(\ln(n))$. At the core of our approach is the Density Matrix Renormalization Group algorithm, which typically does not guarantee efficiency. We have created a new truncation scheme for the matrix product operator of the quantum Gibbs states, which allows us to control the error analytically. Additionally, our method can be applied to simulate the time evolution of systems with long-range interactions, achieving significantly better precision than that offered by the Lieb-Robinson bound.