  Effective low-energy theories represent powerful theoretical tools to reduce
the complexity in modeling interacting quantum many-particle systems. However,
common theoretical methods rely on perturbation theory, which limits their
applicability to weak interactions. Here we introduce the Variational Adiabatic
Gauge Transformation (VAGT), a non-perturbative hybrid quantum algorithm that
can use nowadays quantum computers to learn the variational parameters of the
unitary circuit that brings the Hamiltonian to either its block-diagonal or
full-diagonal form. If a Hamiltonian can be diagonalized via a shallow quantum
circuit, then VAGT can learn the optimal parameters using a polynomial number
of runs. The accuracy of VAGT is tested trough numerical simulations, as well
as simulations on Rigetti and IonQ quantum computers.
