Quantifying Entanglement in Quantum Complex Networks

Jincheng Zhang

Quantum Resonance Networks (QRNs) represent a novel approach to algorithmic complexity analysis, leveraging the principles of entanglement and superposition to create dynamic, interconnected networks. This research explores the potential of QRNs as a tool for quantifying complexity, moving beyond traditional approximation methods and offering a fundamentally quantum-based assessment. The core mechanism involves establishing a network where nodes are interconnected through entanglement, generating patterns that are indicative of complex algorithmic behavior. This paper details the framework for QRNs, its underlying principles, and its potential impact on the field of algorithmic complexity.

Qi Han, Shijiao Xing, Xiao Wu +2 more

Logan Edward Hillberry

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…

Taige Wang, Nisarga Paul, Liang Fu

Characterizing the complexity of correlations between subsystems is a fundamental task across information theory, machine learning, and science. In quantum physics, neural networks have found increasing application in learning wavefunctions. Here we introduce the entangling power of an encoder-decoder neural network, which quantifies its ability to generate entanglement between subsystems, dependent on a latent space dimension $K$ and the complexity class of the decoder. We exactly calculate this quantity for polynomial decoders of degree $p$ acting on a $K$-dimensional latent space. Our results establish the exponential entangling power of neural networks with modest resources. More broadly, our work provides a framework for analyzing correlations in machine learning that generalizes the notion of the Schmidt rank in entanglement theory.

H. J. Briegel, S. J. van Enk, J. I. Cirac +9 more

Sami Ul Haq, Farhan Saif