Brownian Gaussian Unitary Ensemble: non-equilibrium dynamics, environment friendly k-design and software in classical shadow tomography
Authors: Haifeng Tang
Summary: We assemble and extensively examine a Brownian generalization of the Gaussian Unitary Ensemble (BGUE). Our evaluation begins with the non-equilibrium dynamics of BGUE, the place we derive express analytical expressions for numerous one-replica and two-replica variables at finite N and t. These variables embrace the spectral kind issue and its fluctuation, the two-point perform and its fluctuation, out-of-time-order correlators (OTOC), the second Rényi entropy, and the body potential for unitary 2-designs. We focus on the implications of those outcomes for hyperfast scrambling, emergence of tomperature, and replica-wormhole-like contributions in BGUE. Subsequent, we examine the low-energy physics of the efficient Hamiltonian for an arbitrarily variety of replicas, deriving long-time outcomes for the body potential. We conclude that the time required for the BGUE ensemble to achieve k-design is linear in okay, per earlier findings in Brownian SYK fashions. Lastly, we apply the BGUE mannequin to the duty of classical shadow tomography, deriving analytical outcomes for the shadow norm and figuring out an optimum time that minimizes the shadow norm, analogous to the optimum circuit depth in shallow-circuit shadow tomography