Recent advances in strong light-matter interactions have revealed a wealth of new physical phenomena in molecules embedded in optical cavities, including modified chemical reactivity, altered excitation spectra, and novel quantum correlations. To describe these effects from first-principles, the field of ab initio 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.

Advanced Computing Mathematics and Data Division Pacific Northwest National Laboratory Richland Washington 99354 United States

Center for Computational Quantum Physics Flatiron Institute New York New York 10010 United States

Department of Chemistry and Biochemistry Florida State University Tallahassee Florida 32306 4390 United States

Department of Chemistry and Physics New York University New York New York 10003 United States

Department of Chemistry Michigan State University East Lansing Michigan 48824 United States

Department of Chemistry University of North Carolina Charlotte Charlotte North Carolina 28223 United States

Department of Chemistry University of Washington Seattle Washington 98195 United States

Department of Physics City College of New York New York New York 10031 United States

Department of Physics Rutgers University Newark New Jersey 07102 United States

Department of Physics University of Washington Seattle Washington 98195 United States

Faculty of Mathematics and Physics Charles University 12116 Prague 2 Czech Republic

J Heyrovský Institute of Physical Chemistry Academy of Sciences of the Czech Republic v v i Dolejškova 3 18223 Prague 8 Czech Republic

Nanoscience Center and Department of Chemistry University of Jyvaskyla Jyvaskyla 40014 Finland

Physical Sciences Division Pacific Northwest National Laboratory Richland Washington 99352 United States

Simons Center for Computational Physical Chemistry New York University New York New York 10003 United States

Theoretical Division Los Alamos National Laboratory Los Alamos New Mexico 87544 United States

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