  In a recently demonstrated quantum-state tomography scheme [Phys. Rev. Lett.
124, 010405 (2020)], a random control field is locally applied to a
multipartite system to reconstruct the full quantum state of the system through
single-observable measurements. Here, we analyze the robustness of such a
tomography scheme against measurement errors. We characterize the sensitivity
to measurement errors using the logarithm of the condition number of a linear
system that fully describes the tomography process. Using results from random
matrix theory we derive the scaling law of the logarithm of this condition
number with respect to the system size when Haar-random evolutions are
considered. While this expression is independent on how Haar randomness is
created, we also perform numerical simulations to investigate the temporal
behavior of the robustness for two specific quantum systems that are driven by
a single random control field. Interestingly, we find that before the mean
value of the logarithm of the condition number as a function of the driving
time asymptotically approaches the value predicted for a Haar-random evolution,
it reaches a plateau whose length increases with the system size.
