Low-rank propagation for tridiagonalizable open quantum systems: near-linear scaling with system size

Abstract

The quadratic growth of the density matrix with Hilbert-space dimension D is the central obstacle to simulating large open quantum systems. We introduce a deterministic algorithm that eliminates it for Lindblad dynamics whose Hamiltonian consists of a time-dependent diagonal part plus terms that are tridiagonal after reordering the basis. The state is a low-rank ensemble of vectors, propagated by tridiagonal split-operator steps and ensemble rank truncation of short-time Kraus branches, so that memory and cost per step are linear in D at fixed rank. For a driven nitrogen-vacancy-cavity model, rank 16 reproduces full density-matrix observables to relative error below $10^{-5}$, the method is up to two orders of magnitude faster than QuTiP already at $D \approx 500$, and its runtime scales nearly linearly.