Complexity is an intuitively recognized feature of nature, but where does it come from? Is complexity only apparent at the classical level, or can it be found at the underlying quantum level as well? We approach these questions by exploring models consistent with basic quantum theory but which also hold promise for exhibiting complex behaviors – a set of models we call the quantum elementary cellular automata (QECA). Previously, various quantum cellular automata models have been studied for both their quantum information theoretic properties and their ability to simulate key physics equations like the Schrodinger and Dirac equations. We take the unique approach of analyzing QECA as complexity-generating systems. Doing so requires a more precise notion of what we mean by complex. This is done by proposing nine axes of complexity along which the complexity of any system may be quantified. When applied to QECA, we find evidence of complexity using three of these axes, namely diversity, persistent dynamical macrostates, and connectivity. The studies presented are numerical simulations done without approximation using highly optimized exact diagonalization code which supports a Hilbert space of up to 2^27 dimensions. The code is written entirely in the high-level open source programming language Python, making it easily expandable to future projects requiring exact simulation of quantum systems. A careful description of an algorithm critical to our method as well as our use of high performance computing resources on a cluster supercomputer is given. Powerful quantifiers of entanglement and connectivity such as von Neumann entropy and complex network measures computed on quantum mutual information adjacency matrices provide analysis tools for the simulations. Each network measure is defined then tested on well-characterized entangled states from quantum information theory, like the GHZ and W states and singlet state arrays. The network measures known as network density, clustering coefficient, and disparity are specifically considered. We find these network measures offer unique information regarding the structure of two point correlations in the states produced by QECA dynamics, as compared to each other and as quantified by a principal component analysis. Using such measures, we address the complexity of QECA models at three levels of specificity. First, a broad analysis of tens of thousands of simulations gives an overview of the variety of dynamics available to the models. We quantify the diversity of our simulations as the density of simulations which appear, on average, unlike typical entangled quantum states. Second, a more selective analysis identifies QECA by their complexity dynamics, in the frequency domain. A few QECA which exhibit persistent dynamical macrostates in the form of highly structured entanglement are also described. Entanglement dynamics are quantified by the distribution of changes in bond entropy (the von Neumann entropy of all bipartitionings of the QECA system). Finally, we take a detailed…
Complexity is an intuitively recognized feature of nature, but where does it come from? Is complexity only apparent at the classical level, or can it be found at the underlying quantum level as well? We approach these questions by exploring models consistent with basic quantum theory but which also hold promise for exhibiting complex behaviors – a set of models we call the quantum elementary cellular automata (QECA). Previously, various quantum cellular automata models have been studied for both their quantum information theoretic properties and their ability to simulate key physics equations like the Schrodinger and Dirac equations. We take the unique approach of analyzing QECA as complexity-generating systems. Doing so requires a more precise notion of what we mean by complex. This is done by proposing nine axes of complexity along which the complexity of any system may be quantified. When applied to QECA, we find evidence of complexity using three of these axes, namely diversity, persistent dynamical macrostates, and connectivity. The studies presented are numerical simulations done without approximation using highly optimized exact diagonalization code which supports a Hilbert space of up to 2^27 dimensions. The code is written entirely in the high-level open source programming language Python, making it easily expandable to future projects requiring exact simulation of quantum systems. A careful description of an algorithm critical to our method as well as our use of high performance computing resources on a cluster supercomputer is given. Powerful quantifiers of entanglement and connectivity such as von Neumann entropy and complex network measures computed on quantum mutual information adjacency matrices provide analysis tools for the simulations. Each network measure is defined then tested on well-characterized entangled states from quantum information theory, like the GHZ and W states and singlet state arrays. The network measures known as network density, clustering coefficient, and disparity are specifically considered. We find these network measures offer unique information regarding the structure of two point correlations in the states produced by QECA dynamics, as compared to each other and as quantified by a principal component analysis. Using such measures, we address the complexity of QECA models at three levels of specificity. First, a broad analysis of tens of thousands of simulations gives an overview of the variety of dynamics available to the models. We quantify the diversity of our simulations as the density of simulations which appear, on average, unlike typical entangled quantum states. Second, a more selective analysis identifies QECA by their complexity dynamics, in the frequency domain. A few QECA which exhibit persistent dynamical macrostates in the form of highly structured entanglement are also described. Entanglement dynamics are quantified by the distribution of changes in bond entropy (the von Neumann entropy of all bipartitionings of the QECA system). Finally, we take a detailed…
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.
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.
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.
Every process of our universe is based on the fundamental elements of nature: information and computation.The basic motivation behind the study of quantum cellular automata (QCA) is the wish to analyze the processes of nature.QCA provide a natural framework within which to describe many classically undescribable and uncomputable physical phenomena, such as the properties of quantum physical systems and the complex background of quantum dynamics.Quantum cellular automata models are based on the working mechanism of classical cellular automata models and use the power of reversible quantum computation.Cellular automata can be used in many fields of science, such as parallel computation, artificial intelligence, image processing, biological systems, simulation of physics systems, hardware design, algorithm theory, and many more.In the first part of this chapter, we give a brief overview of the basic properties of quantum information processing and analyze the quantum versions of classical cellular automata models.We present all materials in a clear, perspicuous, and comprehensible manner, without using a complex mathematical background.After reviewing physical QCA implementations, we sketch future directions, and then conclude the first part.In the second part of the chapter, we examine one possible application of QCA, which uses quantum computing to realize real-life based, truly random network organization.This abstract machine is called a Quantum Cellular Machine (QCM), and we design it for controlling a truly random biologically-inspired network, and to integrate quantum learning algorithms and quantum searching into a controlled, self-organizing system.The selforganizing processes in classical systems cannot be truly random.Using our quantum probabilistic QCM unit, we can add truly random behavior to the self-organizing processes of biological networks.A quantum mechanical-based quantum cellular machine (QCM) controls the self-organizing processes of the network and uses a closed, non-classical quantum mechanical-based language inside the QCM.The proposed QCM solution has deep relevance in the evolution of truly random quantum probabilistic self-organizing network structures.The QCM controls the evolution of the system, changes its environment and creates plans without any human interaction, using truly random quantum probabilistic decisions.In a classical system, the classical circuits can only exhibit deterministic behavior.In a quantum probabilistic control system, the quantum circuits can follow both deterministic and quantum probabilistic quantum cellular machine control behaviors.The QCM has classical and quantum communication layers, it uses the classical layer to detect www.intechopen.comCellular Automata -Innovative Modelling for Science and Engineering 114 the network environment.The lower layer of the quantum cellular machine contains the non-deterministic quantum probabilistic decisions and interacts with the classical level.The quantum cellular machine model with the power of quantum computing can be used for the development of a real-life based network organism.We show, that a real, biologically inspired, non-deterministic, truly random network model can be achieved by the discussed QCM model.In the third part of this chapter, we show that a very efficient quantum searching algorithm can be integrated into the QCM, to find the best solution to a given network input command.We present a quantum searching based method specially designed for quantum probabilistic self-organizing networks, to reduce the complexity of the classical traditional search in the network.The proposed QCM can process both quantum and classical information, and accomplish both deterministic and quantum probabilistic tasks.The information unit is a quantum bit, which can lie in a coherent superposition state of logical states zero and one, and can thus simultaneously store zero and one.Using quantum bits, we can speed up the solutions of classical problems, and even solve some hard problems that classical computers can't solve.The key aspect behind the optimal decisions of a QCM is to design a high-efficiency searching algorithm.A QCM updates the probability amplitudes of its quantum register, according to a given reward value, derived from the network environment.The QCM repeatedly applies a unitary transformation to the quantum states, thus it can enhance, for example, the probability amplitude of the optimal path in the self-organizing network environment, while suppressing the amplitude of all other solutions.The QCM's quantum searching algorithm can help resolve many hard tasks, for example it could be applied to find an optimal logical path, using the effects of quantum mechanics.In the numerical analysis we will show that the quantum communication layer could improve the performance of classical systems.
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.
Information theoretic measures have helped to sharpen our understanding of many-body quantum states. As perhaps the most well-known example, the entanglement entropy (or more generally, the bipartite mutual information) has become a powerful tool for characterizing the dynamical growth of quantum correlations. By contrast, although computable, the bipartite mutual information (MI) is almost never explored in classical many particle systems; this owes in part to the fact that computing the MI requires keeping track of the evolution of the full probability distribution, a feat which is rarely done (or thought to be needed) in classical many-body simulations. Here, we utilize the MI to analyze the spreading of information in 1D elementary cellular automata (CA). Broadly speaking, we find that the behavior of the MI in these dynamical systems exhibits a few different types of scaling that roughly correspond to known CA universality classes. Of particular note is that we observe a set of automata for which the MI converges parametrically slowly to its thermodynamic value. We develop a microscopic understanding of this behavior by analyzing a two-species model of annihilating particles moving in opposite directions. Our work suggests the possibility that information theoretic tools such as the MI might enable a more fine-grained characterization of classical many-body states and dynamics.