Interferometry is a powerful method for studying many fundamental phenomena ranging from gravitational waves to anyon excitations. Besides loss of coherence in the interferometer paths, quantum mechanics sets an incompressible bound for linear interferometry arising from the quantum noise of the light beam, called the shot noise limit. Here, we implement a non-linear quantum interferometer that uses a microwave cavity and a magnetic field resilient anharmonic superconducting quantum circuit made of granular aluminum. The circuit enables interferometry beyond the shot noise limit. A direct application of our findings is the detection of dark matter (axions or dark photons), high-frequency gravitational waves or astronomical masers.