The theoretical technique applies the well-known Chebyshev series expansion approach for quantum dynamics propagation and uses it to simulate the polariton dynamics in the HTC system. The Chebyshev expansion allows us to use a much larger time step for propagation and only requires a few recursive operations of the Polariton Hamiltonian acting on state vectors, which we can further take advantage of the first technique we developed in this work. Due to the block diagonal structure of the HTC-type Hamiltonian, one can decompose Chebyshev expansions for single and double excitation subspaces separately, and a shift in the energy axis of the subspace that significantly reduces the oscillation magnitude of the coefficients in time and allows a much larger propagation time step. These two theoretical approaches are general and can be applied to any trajectory-based non-adiabatic quantum dynamics methods.