Microscopic origin of dissipative two-level systems in Al$_2$O$_3$

Resonant absorption of two-level systems (TLS) poses a serious limitation to the performance of superconducting qubit devices for quantum computing. Experiments indicate that the TLS is associated with defects in the dielectric layers of the device. However, the nature of these defects has yet to be established. Using hybrid functional calculations, we investigate possible defects in Al$_2$O$_3$ that can act as sources of resonant absorption. Hydrogen is a ubiquitous impurity, and easily incorporates in the interstitial sites (H$_i$) in Al$_2$O$_3$. In the positive charge state, H$_i$ is bonded to one oxygen atom, but also interacts with a secondary oxygen atom. At particular O-O distances, close to those found in amorphous Al$_2$O$_3$ or near the Al$_2$O$_3$/Al interface, the H atom is effectively in a double well. We calculate the three-dimensional potential energy surface (PES) for the H atom in a so-called ``coincidence configuration,'' which allows for tunneling between two equivalent positions. Using the calculated PES, we solve the Schr\"odinger equation for the tunneling proton and determine the tunneling frequency. We find that H$_i$ gives rise to resonant absorption in the range of 10-100 GHz, in agreement with experimental observations.