  Estimating the dimension of an Hilbert space is an important component of
quantum system identification. In quantum technologies, the dimension of a
quantum system (or its corresponding accessible Hilbert space) is an important
resource, as larger dimensions determine e.g. the performance of quantum
computation protocols or the sensitivity of quantum sensors. Despite being a
critical task in quantum system identification, estimating the Hilbert space
dimension is experimentally challenging. While there have been proposals for
various dimension witnesses capable of putting a lower bound on the dimension
from measuring collective observables that encode correlations, in many
practical scenarios, especially for multiqubit systems, the experimental
control might not be able to engineer the required initialization, dynamics and
observables.
  Here we propose a more practical strategy, that relies not on directly
measuring an unknown multiqubit target system, but on the indirect interaction
with a local quantum probe under the experimenter's control. Assuming only that
the interaction model is given and the evolution correlates all the qubits with
the probe, we combine a graph-theoretical approach and realization theory to
demonstrate that the dimension of the Hilbert space can be exactly estimated
from the model order of the system. We further analyze the robustness in the
presence of background noise of the proposed estimation method based on
realization theory, finding that despite stringent constrains on the allowed
noise level, exact dimension estimation can still be achieved.
