Figure 3.

Density estimation and performance test. (a) Points $\overrightarrow{R}$ associated with density criterion for the set of unitary matrices ${\{e^{A_j},e^{D_j}\}}_{j=1}^{50}$ (left column) and ${\{e^{B_j},e^{C_j}\}}_{j=1}^{50}$ (right column). The shaded blue area denotes the Goldilocks region where universality is expected for $N=6$. In turn, the blue heat maps denote the modulus of some particular choices of the unitary $F=e^{i{X}_j}$ matrices, with $X\in \{A,B,C,D\}$. (b) Error norm (log$_{10}^{}$) L in (2) for each unitary matrix under consideration with fifty testing targets per matrix. (c) Mean and standard deviation $N\tilde{\mu }$ and $N\tilde{\sigma }$, respectively, related to the density estimation for each unitary matrix in (a). The horizontal blue and red lines denote the universality threshold for $N\tilde{\mu }$ and $N\tilde{\sigma }$, respectively.