Entanglement dynamics in quantum cellular automata

M. Andrecut, Mazhar Ali

Wout Merbis, Calvin C. Bakker

Cellular automata (CA) exemplify systems where simple local interaction rules can lead to intricate and complex emergent phenomena at large scales. The various types of dynamical behavior of CA are usually categorized empirically into Wolfram's complexity classes. Here, we propose a quantitative measure, rooted in quantum information theory, to categorize the complexity of classical deterministic cellular automata. Specifically, we construct a Matrix Product Operator (MPO) of the transition matrix on the space of all possible CA configurations. We find that the growth of entropy of the singular value spectrum of the MPO reveals the complexity of the CA and can be used to characterize its dynamical behavior. This measure defines the concept of operator entanglement entropy for CA, demonstrating that quantum information measures can be meaningfully applied to classical deterministic systems.

Susumu Adachi, Yoshimasa Isawa

Jincheng Zhang

Quantum entanglement, a phenomenon where two or more particles become linked and share the same fate, has captivated physicists and philosophers alike. While established in quantum mechanics, its application to complex systems, particularly cellular automata, presents a fascinating challenge and opportunity. This paper explores the potential of extending the concept of quantum entanglement beyond individual particles to propose a dynamic, multi-level entanglement structure within cellular automata simulations. We hypothesize that this approach, driven by chaotic behavior observed in cellular automata, could yield emergent phenomena, including "quantum foam" behavior, offering novel insights into the fundamental nature of reality. The core mechanism involves introducing a novel evolution rule based on chaotic dynamics, incorporating feedback loops and probabilistic branching to generate complex entanglement patterns. This work aims to advance our understanding of entanglement's role in complex systems and potentially lead to new computational models.

Gerhard Grössing, Anton Zeilinger

Logan Edward Hillberry, Matthew Timothy Jones, David L. Vargas +6 more

Cellular automata are interacting classical bits that display diverse emergent behaviors, from fractals to random-number generators to Turing-complete computation. We discover that quantum cellular automata (QCA) can exhibit complexity in the sense of the complexity science that describes biology, sociology, and economics. QCA exhibit complexity when evolving under 'Goldilocks rules' that we define by balancing activity and stasis. Our Goldilocks rules generate robust dynamical features (entangled breathers), network structure and dynamics consistent with complexity, and persistent entropy fluctuations. Present-day experimental platforms—Rydberg arrays, trapped ions, and superconducting qubits—can implement our Goldilocks protocols, making testable the link between complexity science and quantum computation exposed by our QCA.