Quantum dynamics of few-photon pulsed waveguide-QED systems with a single artificial atom have been investigated using two distinct theoretical simulation methods. A frequency-dependent scattering theory approach utilizes input-output scattering matrices to analyze the interaction between photons and the artificial atom, a two-level system or qubit. In contrast, a time-dependent matrix product states (MPS) approach provides an alternative framework for understanding the quantum behavior of these systems. The study directly compares and contrasts these two methods, offering insights into the quantum dynamics of pulsed few-photon scattering1. The artificial atom, a fundamental component of quantum computing architectures, is a crucial element in the development of quantum technologies. As quantum computing continues to advance, understanding the quantum dynamics of such systems is essential for the development of secure cryptographic protocols and robust computing systems. This research matters to practitioners because it sheds light on the underlying quantum mechanics that will inform the next generation of quantum computing and cryptography systems.