Entangled Dynamics of Quantum Cellular Automata: Towards a KAM Theory for Quantum Many-Body Systems

Understanding the robustness of integrable systems in quantum many-body physics under perturbations remains a fundamental challenge. Drawing parallels to the classical Kolmogorov-Arnold-Moser (KAM) theorem—which describes the persistence of special quasi-periodic tori under small perturbations—we explore a quantum analogue within Quantum Cellular Automata (QCA), where integrability is governed by local unitary operations and the local rules dictating their application. We consider the Goldilocks' rule (type T6), known for preserving clustering—an upper limit on quantum correlations based on mutual information—well above the Random State ansatz. By varying the local unitary transformations from the free-fermionic form that renders the system integrable, we observe a decay in average clustering and clustering fluctuations, indicating a breakdown of integrability and suggesting near-integrable behavior under the Goldilocks' rule. Conversely, fixing the local unitary and varying the local rules from T6 to T1, we observe a sharp decay of clustering towards the random State ansatz. This reveals a second mechanism for breaking integrability, leading to fully non-integrable dynamics. We discuss the implications of these mechanisms and prospects for developing a KAM-like framework applicable to quantum many-body dynamics.