  In the current work we address the problem of quantum process tomography
(QPT) in the case of imperfect preparation and measurement of the states which
are used for QPT. The fuzzy measurements approach which helps us to efficiently
take these imperfections into account is considered. However, to implement such
a procedure one should have a detailed information about the errors. An
approach for obtaining the partial information about them is proposed. It is
based on the tomography of the ideal identity gate. This gate could be
implemented by performing the measurement right after the initial state
preparation. By using the result of the identity gate tomography we were able
to significantly improve further QPT procedures. The proposed approach has been
tested experimentally on the IBM superconducting quantum processor. As a
result, we have obtained an increase in fidelity from 89% to 98% for Hadamard
transformation and from 77% to 95% for CNOT gate.
