In experiments with quantum devices we do not in most cases have access to measurements for reconstruction of the whole wavefunction.
Using probability theory, machine learning or combination of both we can still reconstruct the key parameters for the system dynamics. We implemented these ideas for quantum dots: Phys. Rev. A 96, 052104 (2017) and arXiv:1711.05238 as well as for superconducting qubits: Phys. Rev A 94, 042334 (2016).
Currently I am interested in how to expand these methods to scales that cannot be simulated on classical computers.