LREI: A fast numerical solver for quantum Landau--Lifshitz equations

Abstract

We develop LREI, short for Low-Rank Eigenmode Integration, a memory- and time-efficient numerical scheme for solving quantum Landau–Lifshitz (q-LL) and quantum Lan- dau–Lifshitz–Gilbert (q-LLG) equations, which govern spin dynamics in open quantum systems. Although the system size grows exponentially with the number of spins, our approach benefits from the low-rank structure of the density matrix and the sparsity of system Hamiltonians to avoid costly full matrix computations. By representing density matrices in terms of their low-rank factors and using Krylov subspace techniques for partial eigendecompositions, we reduce the per-step complexity of Runge–Kutta and Adams–Bashforth schemes from O(N 3) to O(r2N ), where N = 2n is the Hilbert space dimension for n spins and r ? N is the effective rank of the density matrix. Likewise, the memory footprint is reduced from O(N 2) to O(rN ), since no full N × N matrices are ever formed. Among several technical improvements we applied, one key idea was to handle the computation of the action of the invariant subspace of the density matrix associated with its zero eigenvalues. This was accomplished by applying Householder reflectors constructed for the dominant eigenspace, thereby enabling the entire solution process to proceed without ever forming any large matrices. As an example, we can now evolve a time step of a twenty-spin system—corresponding to a density matrix size exceeding one million—in just a few seconds on a standard laptop. In addition, both classes of Runge-Kutta and Adams-Bashforth techniques are reformulated to preserve the physical properties of the density matrix throughout the time evolution. The new low-rank algorithm makes it possible to simulate much larger spin systems that were previously computationally infeasible. This, in turn, provides a powerful tool for comparing the q-LL and q-LLG dynamics, assessing the validity of each model, and exploring how quantum features such as correlations and entanglement evolve across different regimes of system size and damping.