  Quantum-enhanced computing methods are promising candidates to solve
currently intractable problems. We consider here a variational quantum
eigensolver (VQE), that delegates costly state preparations and measurements to
quantum hardware, while classical optimization techniques guide the quantum
hardware to create a desired target state. In this work, we propose a bosonic
VQE using superconducting microwave cavities, overcoming the typical
restriction of a small Hilbert space when the VQE is qubit based. The
considered platform allows for strong nonlinearities between photon modes,
which are highly customisable and can be tuned in situ, i.e. during running
experiments. Our proposal hence allows for the realization of a wide range of
bosonic ansatz states, and is therefore especially useful when simulating
models involving degrees of freedom that cannot be simply mapped to qubits,
such as gauge theories, that include components which require
infinite-dimensional Hilbert spaces. We thus propose to experimentally apply
this bosonic VQE to the U(1) Higgs model including a topological term, which in
general introduces a sign problem in the model, making it intractable with
conventional Monte Carlo methods.
