  The number of measurements required to reconstruct the states of quantum
systems increases exponentially with the quantum system dimensions, which makes
the state reconstruction of high-qubit quantum systems have a great challenge
in physical quantum computing experiments. Compressive sensing (CS) has been
verified as a effective technique in the reconstruction of quantum state,
however, it is still unknown that if CS can reconstruct quantum states given
the less data measured by nuclear magnetic resonance (NMR). In this paper, we
propose an effective NMR quantum state reconstruction method based on CS.
Different from the conventional CS-based quantum state reconstruction, our
method uses the actual observation data from NMR experiments rather than the
data measured by the Pauli operators. We implement measurements on quantum
states in practical NMR computing experiments and reconstruct states of 2,3,4
qubits using fewer number of measurements, respectively. The proposed method is
easy to implement and performs more efficiently with the increase of the system
dimension size. The performance reveals both efficiency and accuracy, which
provides an alternative for the quantum state reconstruction in practical NMR.
