Numerical investigation of a MST conjecture for random Haar matrices

The state-to-state control time of finite-dimensional conservative bilinear quantum systems is bounded by the weight of minimum spanning trees (MST) derived from the drift matrices. For a spectrum scaling as k^3/2 , determining whether the control time vanishes as the dimension d tends to infinity remains an open problem due to the global correlations imposed by the Haar measure. We formulate two conjectures regarding the asymptotic behavior of the MST weight\,: a general decay to zero, and a scaling in 1/sqrt(ln d) derived from an independent Gaussian surrogate model. Numerical investigations up to dimension d = 18101 validate the general decay, confirm that the exact Haar model and the independent surrogate are statistically indistinguishable in mean at all tested dimensions, and extract the thermodynamic limit $α$ = 10/3, which both models approach from below once the scaling variable 1/ ln d is adopted.