Figure 3

Density estimation and performance test. (a) Points \(\vec {R}\) 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 \(F=e^{iX_{j}}\) 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 \(N\widetilde{\mu }\) and \(N\widetilde{\sigma }\), respectively, related to the density estimation for each unitary matrix in (a). The horizontal blue and red lines denote the universality threshold for \(N\widetilde{\mu }\) and \(N\widetilde{\sigma }\), respectively.