Fig. 9: Ensemble average results under restricted Haar ensemble (top) and Haar ensemble (bottom).
From: Dynamical transition in controllable quantum neural networks with large depth
In top panel, we plot a \(\overline{{K}_{\infty }}\) versus O0 with L = 512 fixed, b \(\overline{{\zeta }_{\infty }}\) versus L, \(\overline{{\lambda }_{\infty }}\) versus (c) L and d n with L = 512 at late time in state preparation. We set O0 = 1 for b and d, and O0 = 5 for c. Blue dots in top panels a–c represents numerical results from late-time optimization of n = 5 qubit RPA. Red solid lines represent exact ensemble average with restricted Haar ensemble in Supplementary Equations (256), (313), (279) in Supplementary Note 12. Magenta dashed lines represent asymptotic ensemble average with restricted Haar ensemble in Eq. (35), (36), (37) which overlap with the exact results (red solid). The observable in all cases is \(\left\vert \Phi \right\rangle \left\langle \Phi \right\vert\) with \(\left\vert \Phi \right\rangle\) being a fixed Haar random state. In the inset of b, we fix L = 512. In bottom panel, we plot (e) fluctuation \({{{\rm{SD}}}}[{K}_{0}]/\overline{{K}_{0}}\) versus L, (f) \(\overline{{\zeta }_{0}}\) versus L, \(\overline{{\lambda }_{0}}\) versus (g) L and (h) n with L = 128 under random initialization. Green dots in bottom panel from e–g represent numerical results from random initializations of n = 6 qubit RPA. Brown solid lines represent the exact ensemble average with the Haar ensemble in Supplementary Equations (241), (180), (120) in Supplementary Note 11. Gray dashed lines represent asymptotic ensemble average with a restricted Haar ensemble in Eqs. (45), (39), (40). The observable and target in e–h are XXZ model with J = 2 and \({O}_{0}={O}_{\min }\). Orange solid line in e represents results from ref. 29.
