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Abstract Interactions between electrons and phonons play a crucial role in quantum materials. Yet, there is no universal method that would simultaneously accurately account for strong electron-phonon interactions and electronic correlations. By combining methods of the variational quantum eigensolver and the variational non-Gaussian solver, we develop a hybrid quantum-classical algorithm suitable for this type of correlated systems. This hybrid method tackles systems with arbitrarily strong electron-phonon coupling without increasing the number of required qubits and quantum gates, as compared to purely electronic models. We benchmark our method by applying it to the paradigmatic Hubbard-Holstein model at half filling, and show that it correctly captures the competition between charge density wave and antiferromagnetic phases, quantitatively consistent with exact diagonalization. 

Variational approaches are among the most powerful techniques toapproximately solve quantum many-body problems. These encompass bothvariational states based on tensor or neural networks, and parameterizedquantum circuits in variational quantum eigensolvers. However,self-consistent evaluation of the quality of variational wavefunctionsis a notoriously hard task. Using a recently developed Hamiltonianreconstruction method, we propose a multi-faceted approach to evaluatingthe quality of neural-network based wavefunctions. Specifically, weconsider convolutional neural network (CNN) and restricted Boltzmannmachine (RBM) states trained on a square latticespin-1/2 $1/2$J_1\!-\!J_2 $J_1-J_2$Heisenberg model. We find that the reconstructed Hamiltonians aretypically less frustrated, and have easy-axis anisotropy near the highfrustration point. In addition, the reconstructed Hamiltonians suppressquantum fluctuations in the largeJ_2 $J_2$limit. Our results highlight the critical importance of thewavefunction’s symmetry. Moreover, the multi-faceted insight from theHamiltonian reconstruction reveals that a variational wave function canfail to capture the true ground state through suppression of quantumfluctuations.