Vanishing correlations in (bi)stochastic controlled circuits

TL;DR

This paper demonstrates that stochastic and bistochastic controlled gates in quantum and classical circuits lead to vanishing multi-point correlations, revealing simple correlation structures despite complex dynamics.

Contribution

It proves that such gates cause correlations to vanish except when operators act on the same site, highlighting a broad class of systems with simple correlation behavior.

Findings

Two-point correlations vanish unless operators act on the same site

Multi-point correlations require rightmost operators to act on the same site

Autocorrelation decays exponentially to an exponentially small value in system size

Abstract

We study the dynamics of circuits composed of stochastic and bistochastic controlled gates. This type of dynamics arises from quantum circuits with random controlled gates, as well as in stochastic circuits and deterministic classical cellular automata. We prove that stochastic and bistochastic controlled gates lead to two-point spatio-temporal correlation functions that vanish everywhere except when the two operators act on the same site. More generally, for multi-point correlations the two rightmost operators must act on the same site. We argue that autocorrelation, while hard to compute, typically decays exponentially towards a value that is exponentially small in the system size. Our results reveal a broad class of quantum systems that exhibit surprisingly simple correlation structures despite their complex microscopic dynamics.