Impact of Non-Local Electrodynamics on Flux Noise and Inductance of Superconducting Qubits*

We present explicit numerical calculations of current density, inductance, and impurity-induced flux noise of simple superconducting devices in the non-local Pippard regime, which occurs when the Pippard coherence length is larger than the London penetration depth. In this regime the current density displays a peak away from the surface of the superconductor, signalling a breakdown of the usual approximation of local London electrodynamics with a renormalized penetration depth. Our calculations show that the internal inductance and the bulk flux noise power increases with increasing non-locality. In contrast, the kinetic inductance is reduced and the surface flux noise remains the same. As a result, spins in the bulk may dominate the flux noise in superconducting qubits in the Pippard regime, such as the ones using aluminum superconductors with large electron mean free path.