  To obtain a complete description of a quantum system, one usually employs
standard quantum state tomography, which however requires exponential number of
measurements to perform and hence is impractical when the system's size grows
large. In this work, we introduce a self-learning tomographic scheme based on
the variational hybrid quantum-classical method. The key part of the scheme is
a learning procedure, in which we learn a control sequence capable of driving
the unknown target state coherently to a simple fiducial state, so that the
target state can be directly reconstructed by applying the control sequence
reversely. In this manner, the state tomography problem is converted to a
state-to-state transfer problem. To solve the latter problem, we use the
closed-loop learning control approach. Our scheme is further experimentally
tested using techniques of a 4-qubit nuclear magnetic resonance. {Experimental
results indicate that the proposed tomographic scheme can handle a broad class
of states including entangled states in quantum information, as well as
dynamical states of quantum many-body systems common to condensed matter
physics.
