Researchers have developed a new formulation for simulating quantum systems with algebraically decaying interactions using infinite matrix product states (iMPS). Traditionally, to apply standard matrix product operator (MPO) algorithms to Hamiltonians with long-range interactions, approximations were used, replacing the original Hamiltonian with a finite-pole sum-of-exponentials surrogate. This approximation introduced a residual error in the Hamiltonian representation, which could bias simulation results, especially at critical points.

The new approach avoids the need for this surrogate, allowing direct calculation of the variational energy for an iMPS with a fixed bond dimension D. The key lies in directly summing the algebraic tail of the interactions through the connected transfer matrix. This is achieved by representing the interaction term $e^{\mathrm{i} Qr}/r^\alpha$ using a matrix function $F_{\alpha,Q}(\widetilde{T}_A)$, where $F_{\alpha,Q}(z)=\operatorname{Li}_\alpha(e^{\mathrm{i} Q}\,z)/z$. The action of this matrix function is evaluated using a Krylov method, and stable gradients are obtained by combining a Fréchet adjoint with implicit fixed-point differentiation.

The validity of this transfer-matrix-function formulation has been demonstrated through benchmarks on long-range free fermions and the inverse-square Heisenberg family, including the Haldane-Shastry point. A calculation on a long-range Ising chain illustrates a practical advantage: while finite-pole surrogate Hamiltonians can bias a critical diagnostic away from criticality at a known critical field, the new matrix-function calculation retains the expected critical signatures of the target algebraic Hamiltonian. This opens the door to more precise simulations of complex quantum systems with long-range interactions.