  We investigate quantum state tomography (QST) for pure states and quantum
process tomography (QPT) for unitary channels via $adaptive$ measurements. For
a quantum system with a $d$-dimensional Hilbert space, we first propose an
adaptive protocol where only $2d-1$ measurement outcomes are used to accomplish
the QST for $all$ pure states. This idea is then extended to study QPT for
unitary channels, where an adaptive unitary process tomography (AUPT) protocol
of $d^2+d-1$ measurement outcomes is constructed for any unitary channel. We
experimentally implement the AUPT protocol in a 2-qubit nuclear magnetic
resonance system. We examine the performance of the AUPT protocol when applied
to Hadamard gate, $T$ gate ($\pi/8$ phase gate), and controlled-NOT gate,
respectively, as these gates form the universal gate set for quantum
information processing purpose. As a comparison, standard QPT is also
implemented for each gate. Our experimental results show that the AUPT protocol
that reconstructing unitary channels via adaptive measurements significantly
reduce the number of experiments required by standard QPT without considerable
loss of fidelity.
