The Quantum Dynamics propagation requires basic operator action of ĤQ |Ψ⟩ operation (see Eq. 17 and Eq. 18) using for example, RK4 method, or Ô |Ψ⟩ (see Eq. 36) in the Chebyshev expansion scheme (Eq. 35). For a K-dimensional system in ĤQ , this would in principle require K 2 operations in matrix multiplications. However, we notice that the matrix of ĤQ in the HTC Hamiltonian is extremely sparse in the single and double excitation subspaces, which are the relevant subspaces for photophysics and spectroscopy measurements. Here, we take advantage of this sparsity to significantly reduce the scaling of the computational costs of quantum dynamics propagation.