Nicholas P. Bauman, Leonardo A. Cunha, A. Eugene DePrince, Johannes Flick, Jonathan J. Foley, Niranjan Govind, Gerrit Groenhof, Norah M. Hoffmann, Karol Kowalski, Xiaosong Li, Marcus D. Liebenthal, Neepa T. Maitra, Ruby Manderna, Mikuláš Matoušek, Ilia M. Mazin, Daniel Mejı́a-Rodrı́guez, Ajay Panyala, Bo Peng, Benjamin G. Peyton, Libor Veis, Nam Vu, Jared D. Weidman, Angela K. Wilson, Rhiannon A. Zarotiadis, Yu Zhang
quantum electrodynamics (QED) has emerged as a compelling extension of quantum chemistry that treats electronic and photonic degrees of freedom on equal footing. In this Perspective, we review the growing landscape of many-body QED methods, including Hartree-Fock, density functional theory (QEDFT), time-dependent DFT (QED-TDDFT), configuration interaction (QED-CI), complete active space (QED-CASSCF), coupled cluster (QED-CC), quantum Monte Carlo (QED-QMC), and density matrix renormalization group (QED-DMRG), highlighting recent developments and implementations. We further explore real-time methods, gradient and Hessian formalisms, and the integration of nonadiabatic nuclear dynamics. Applications range from benchmark simulations of polaritonic chemistry to quantum simulations on emerging quantum hardware. We conclude by outlining future directions for theory development and interdisciplinary efforts at the interface of quantum chemistry, condensed matter, and quantum optics.