Additionally, simulating the time-dependent polariton quantum dynamics of the hybrid matter-field systems is often a necessary and essential task, as these polariton photochemical reactions often involve a complex dynamical interplay among the electronic, nuclear, and photonic degrees of freedom (DOFs). However, accurately simulating polaritonic quantum dynamics remains a challenging task and is beyond the paradigm of traditional photochemistry, which does not include quantized photons, and quantum optics which does not have a well-defined theory to include the influence of nuclear degrees of freedom to describe reactivity nor properly account for molecular structures. Over the past years, enormous progress has been made to address this interdisciplinary challenge. We have witnessed how electronic structure theory (Sec. 3.2) and non-adiabatic quantum dynamics (Sec. 4) have actively participated in this progressing this exciting field.