  We employ the compressed sensing (CS) algorithm and a heavily reduced data
set to experimentally perform true quantum process tomography (QPT) on an NMR
quantum processor. We obtain the estimate of the process matrix $\chi$
corresponding to various two- and three-qubit quantum gates with a high
fidelity. The CS algorithm is implemented using two different operator bases,
namely, the standard Pauli basis and the Pauli-error basis. We experimentally
demonstrate that the performance of the CS algorithm is significantly better in
the Pauli-error basis, where the constructed $\chi$ matrix is maximally sparse.
We compare the standard least square (LS) optimization QPT method with the
CS-QPT method and observe that, provided an appropriate basis is chosen, the
CS-QPT method performs significantly better as compared to the LS-QPT method.
In all the cases considered, we obtained experimental fidelities greater than
0.9 from a reduced data set, which was approximately five to six times smaller
in size than a full data set. We also experimentally characterized the reduced
dynamics of a two-qubit subsystem embedded in a three-qubit system, and used
the CS-QPT method to characterize processes corresponding to the evolution of
two-qubit states under various $J$-coupling interactions.
