Spin Hamiltonian as Matrix-Free Linear Map
We present an algorithm that computes the action of a generic spin Hamiltonian on a state vector on the fly, entirely avoiding explicit matrix assembly. This is achieved through mixed-radix indexing of the full tensor-product basis, which translates local spin operations into simple integer offsets. The result is an explicit framework for evaluating single- and two-site terms across arbitrary spin lattices, including mixed-spin systems. Our construction bridges the basis-indexing logic familiar from exact diagonalization with the matrix-free state-update philosophy of address-based frameworks. By writing the indexing logic in closed form, a single uniform loop applies to every site regardless of its local Hilbert-space dimension. The method is parallelizable and memory-conserving, and can be extended to restricted basis or truncated bosonic levels.