Algorithm A1 Monte Carlo simulation of SYK capacity saturation

1: Initialization:   2: ${\chi }_i\left(0\right)=0$   3: ${\rho }_i=|0\rangle \langle 0|$   4: $D=0.25$   5: ${\chi }_{\mathrm{crit}}={\displaystyle \frac{1}{4{\mathcal{\ell }}_P^2}}$   6: $J_{ijkl}\sim \mathcal{N}\left(0,{\displaystyle \frac{J}{\sqrt{\left(\genfrac{}{}{0pt}{}{N}{4}\right)}}}\right)$   7: $t_{\mathrm{hop}}=0.1$   8:for realization $=1$ to ${10}^5$ do                                  ▹ Monte Carlo   9:    for $t=1$ to $T_{max}$ do 10:         SYK Hamiltonian: $$H_{\mathrm{SYK}}^{\left(i\right)}=\sum _{ijkl}{J}_{ijkl}{\psi }_{i,k}{\psi }_{i,l}{\psi }_{i,m}{\psi }_{i,n}$$ (A47) $$H_{\mathrm{hop}}^{(i,i+1)}=t_{\mathrm{hop}}{\psi }_i^{\top }{\psi }_{i+1}$$ (A48) $$H_{\mathrm{tot}}=H_{\mathrm{SYK}}+H_{\mathrm{hop}}$$ (A49) 11:         Time evolution: $$U\left(t\right)=e^{-i{H}_{\mathrm{tot}}\Delta t}$$ (A50) $${\rho }_i\leftarrow U\left(t\right){\rho }_i{U}^{†}\left(t\right)$$ (A51) 12:         Bulk drive + diffusion: $${\chi }_i(t+\Delta t)={\chi }_i\left(t\right)+D{\nabla }^2{\chi }_i\left(t\right)+J_{\mathrm{bulk}}(i,t)$$ (A52) $${\xi }_i\left(t\right)=\sqrt{2D\Delta t}{\eta }_i\left(t\right),{\eta }_i\sim \mathcal{N}(0,1)$$ (A53) 13:        Saturation condition: 14:        if ${\chi }_i(t+\Delta t)>{\chi }_{\mathrm{crit}}$ then 15:           ${\rho }_i\leftarrow {\displaystyle \frac{\mathbb{I}}{dim{\mathcal{H}}_i}}$ 16:           ${\chi }_i(t+\Delta t)\leftarrow {\chi }_{\mathrm{crit}}$ 17:        end if 18:         Von Neumann entropy: $$S_i\left(t\right)=-\mathrm{Tr}\left({\rho }_{i}log{\rho }_i\right)$$ (A54) 19:    end for 20:    ${\tau }_{\mathrm{sat}}^{\left(r\right)}=$ first passage time s.t. ${\chi }_i>{\chi }_{\mathrm{crit}}$ 21: end for 22: $D_{\mathrm{SYK}}=\langle D\left[S\right]\rangle$ 23: $\langle {\tau }_{\mathrm{sat}}\rangle =\langle {\tau }_{\mathrm{sat}}^{\left(r\right)}\rangle$