  We discuss methods of quantum state tomography for solid-state systems with a
large nuclear spin $I=3/2$ in nanometer-scale semiconductors devices based on a
quantum well. Due to quadrupolar interactions, the Zeeman levels of these
nuclear-spin devices become nonequidistant, forming a controllable four-level
quantum system (known as quartit or ququart). The occupation of these levels
can be selectively and coherently manipulated by multiphoton transitions using
the techniques of nuclear magnetic resonance (NMR) [Yusa et al., Nature
(London) 434, 101 (2005)]. These methods are based on an unconventional
approach to NMR, where the longitudinal magnetization $M_z$ is directly
measured. This is in contrast to the standard NMR experiments and tomographic
methods, where the transverse magnetization $M_{xy}$ is detected. The
robustness against errors in the measured data is analyzed by using the
condition number based on the spectral norm. We propose several methods with
optimized sets of rotations yielding the highest robustness against errors, as
described by the condition number equal to 1, assuming an ideal experimental
detection. This robustness is only slightly deteriorated, as given by the
condition number equal to 1.05, for a more realistic "noisy" $M_z$ detection
based on the standard cyclically-ordered phase sequence (CYCLOPS) method.
