To solve TDSE in Eq. 17, one can use many numerical methods, such as the RK4 solution of Eq. 17 or using the symplectic integrator. These approaches can still take advantage of the sparsity properties of the Hamiltonian outlined in Sec. III A. The numerical challenges for these propagation-based methods are the requirement of relatively small ∆t for stable propagation. Due to the large N considered in the HTC Hamiltonian, each propagation step remains computationally expensive.