  We present an exact ansatz for the eigenstate problem of mixed fermion-boson
systems that can be implemented on quantum devices. Based on a generalization
of the electronic contracted Schr\"odinger equation (CSE), our approach guides
a trial wave function to the ground state of any arbitrary mixed Hamiltonian by
directly measuring residuals of the mixed CSE on a quantum device. Unlike
density-functional and coupled-cluster theories applied to electron-phonon or
electron-photon systems, the accuracy of our approach is not limited by the
unknown exchange-correlation functional or the uncontrolled form of the
exponential ansatz. To test the performance of the method, we study the
Tavis-Cummings model, commonly used in polaritonic quantum chemistry. Our
results demonstrate that the CSE is a powerful tool in the development of
quantum algorithms for solving general fermion-boson many-body problems.
