A new study introduces a computational technique that drastically reduces the time required to simulate quantum lattice models with long-range interactions. These models are crucial for understanding and designing exotic quantum materials, but their simulation is hampered by the exponential increase of the Hilbert space dimension with system size. The breakthrough enables efficient computation of high-order lattice sums, referred to as graph zeta functions, which were previously only approachable with Monte Carlo methods.

The proposed method factorizes the lattice sum over blocks and applies the cheapest available evaluation strategy depending on each block's treewidth (tw). For basic blocks, analytic forms are obtained in terms of generalized zeta functions. Series-parallel blocks with tw ≤ 2 are computed at linear cost in the number of graph nodes and in the size of the momentum grid, using a semi-analytical algebra based on Epstein zeta functions and rapidly decaying Fourier series. For cases with tw > 2, the technique is combined with tensor-network bucket elimination, yielding polynomial scaling of numerical work and memory with momentum grid size.

This approach reduces the evaluation time for state-of-the-art series expansions from tens of thousands of core-hours to mere minutes. The precision and runtime of the method have been thoroughly analyzed against analytic and numerical benchmarks, and full agreement was achieved when reproducing published Monte Carlo data for the transverse-field Ising model on various 1D, 2D, and 3D lattices. This computational advance is fundamental for research in quantum materials and condensed matter physics.