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

Understand

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.