Entanglement Dynamics of Integrable and Chaotic Quantum Cellular Automata: Towards a Quantum Many-body Kolmogorov-Arnold-Moser Theory

Abstract: 

One-dimensional Quantum cellular automata (QCA) in quantum circuits provide an experimentally realizable quantum computing testbed for quantum entangled dynamics, spanning both integrable and quantum many-body chaotic extremes.  In this work, we establish a quantum many-body Kolmogorov-Arnold-Moser (KAM) framework in 1D QCA, characterizing the breakdown of integrability through a state-dependent hierarchy of conservation laws. Starting from the integrable limit of Goldilocks rules that map exactly onto free-fermion dynamics, we introduce controlled, locality-preserving perturbations via symmetric Strang splitting. We investigate the breakdown of integrability by tracking the deformation of the first 13 local conserved charges directly within the native discrete-time circuit dynamics.

Our central finding in the circuit picture is the emergence of a stability hierarchy of charges  determined by the algebraic structure of the perturbation generator, classified into three distinct tiers: (i) robust invariants which remain exactly conserved independent of perturbation strength; (ii) resonant actions which drift immediately at first order; and (iii) KAM-like candidates, in particular . We identify  as weakly non-resonant: it exhibits anomalous super-delayed deformation under general initial conditions but remains conserved when initialized in an eigenstate of a specific Abelian charge subset.

Complementing this study of quantum circuits in discrete time, we demonstrate rigorously that the associated continuous-time QCA Hamiltonian --- constructed via projector embeddings --- defines a fundamentally distinct dynamical system, conserving only an Abelian subclass of the 13 first charges from Goldilocks QCA.  Within this QCA-like Hamiltonian model, we characterize the broader phenomenology of the integrability-to-chaos crossover. We observe a universal transition from Poisson to Wigner-Dyson spectral statistics and analyze the power-law growth of out-of-time-ordered correlators. Furthermore, using Hamiltonian-based charge autocorrelators, we map the stability of  to a regime of “confined chaos,” where algebraic symmetries shield specific Hilbert space sectors from rapid thermalization, providing a continuous-time counter-part of the KAM stability observed in the discrete circuit.

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