  Realisation of experiments even on small and medium-scale quantum computers
requires an optimisation of several parameters to achieve high-fidelity
operations. As the size of the quantum register increases, the characterisation
of quantum states becomes more difficult since the number of parameters to be
measured grows as well and finding efficient observables in order to estimate
the parameters of the model becomes a crucial task. Here we propose a method
relying on application of Bayesian inference that can be used to determine
systematic, unknown phase shifts of multi-qubit states. This method offers
important advantages as compared to Ramsey-type protocols. First, application
of Bayesian inference allows the selection of an adaptive basis for the
measurements which yields the optimal amount of information about the phase
shifts of the state. Secondly, this method can process the outcomes of
different observables at the same time. This leads to a substantial decrease in
the resources needed for the estimation of phases, speeding up the state
characterisation and optimisation in experimental implementations. The proposed
Bayesian inference method can be applied in various physical platforms that are
currently used as quantum processors.
