Tianci Zhou: Compressing the Influence Matrices for Local Quantum Dynamics

The influence matrices provide an alternative route to the classical simulation of quantum dynamics. These objects describe the effective bath seen by a finite subsystem and, since they retain information only on the local dynamics, they are expected to be easier to simulate than the full wavefunction. Recent work, however, has shown that the influence matrices carry strong temporal correlations even in maximally chaotic systems, which rules out their efficient representation. In this work, we demonstrate that one can nevertheless efficiently store the reduced transition matrix, the combination of influence matrices that directly determines local expectation values. We will first show that the truncation errors are controlled by the singular spectrum of this object, which naturally motivates a low-rank approximation. We then prove that, for chaotic dual-unitary circuits, the associated entropy grows at most logarithmically in time. Our conclusions follow from exact results for random dual-unitary circuits and are supported by numerical results for fixed instances of both dual-unitary circuits and generic circuits.