  Benchmarking and characterising quantum states and logic gates is essential
in the development of devices for quantum computing. We introduce a Bayesian
approach to self-consistent process tomography, called fast Bayesian tomography
(FBT), and experimentally demonstrate its performance in characterising a
two-qubit gate set on a silicon-based spin qubit device. FBT is built on an
adaptive self-consistent linearisation that is robust to model approximation
errors. Our method offers several advantages over other self-consistent
tomographic methods. Most notably, FBT can leverage prior information from
randomised benchmarking (or other characterisation measurements), and can be
performed in real time, providing continuously updated estimates of full
process matrices while data is acquired.
