Brownian Gaussian Unitary Ensemble: non-equilibrium dynamics, efficient $k$-design and application in classical shadow tomography

We construct and extensively study a Brownian generalization of the Gaussian Unitary Ensemble (BGUE). Our analysis begins with the non-equilibrium dynamics of BGUE, where we derive explicit analytical expressions for various one-replica and two-replica variables at finite $N$ and $t$. These variables include the spectral form factor and its fluctuation, the two-point function and its fluctuation, out-of-time-order correlators (OTOC), the second R\'enyi entropy, and the frame potential for unitary 2-designs. We discuss the implications of these results for hyperfast scrambling, emergence of tomperature, and replica-wormhole-like contributions in BGUE. Next, we investigate the low-energy physics of the effective Hamiltonian for an arbitrarily number of replicas, deriving long-time results for the frame potential. We conclude that the time required for the BGUE ensemble to reach $k$-design is linear in $k$, consistent with previous findings in Brownian SYK models. Finally, we apply the BGUE model to the task of classical shadow tomography, deriving analytical results for the shadow norm and identifying an optimal time that minimizes the shadow norm, analogous to the optimal circuit depth in shallow-circuit shadow tomography.