Low Rank Structure of the Reduced Transition Matrix

TL;DR

This paper demonstrates that the reduced transition matrix in quantum dynamics can be efficiently approximated using low-rank methods, especially in chaotic systems, supported by theoretical and numerical evidence.

Contribution

It introduces a low-rank approximation approach for the reduced transition matrix, showing controlled truncation error and logarithmic entropy growth in chaotic dual-unitary circuits.

Findings

Truncation error is controlled by the singular-value spectrum.

Entropy grows at most logarithmically in time for chaotic dual-unitary circuits.

Numerical results support the theoretical findings.

Abstract

The influence-matrix formalism provides an alternative route to the classical simulation of quantum dynamics. Because influence matrices retain information only about the effective bath seen by local observables, they are expected to be easier to simulate than the full wavefunction. Recent work, however, has shown that they carry strong temporal correlations even in maximally chaotic systems, making them difficult to represent efficiently. Here we show that the reduced transition matrix, a suitable combination of influence matrices that directly determines local expectation values, can nevertheless be efficiently approximated. We first show that the truncation error is controlled by its singular-value spectrum, 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 further supported by numerical results for fixed instances of both dual-unitary and random circuits.