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International Joint Conference on Neural Networks (IJCNN)

Neural Network Model Reconstructed from Entangled Quantum States

Junwei Zhang, Zhao Li, Jianmao Xiao, Ming Li

Published Jul 18, 2022• 3 citations
Publisher

Abstract

Neural Networks (NNs) have received extensive attention and research due to their ability to extract and combine different features non-manually and to mine the internal relationships between features. In quantum mechanics, the entangled state simultaneously describes the classical and non-classical correlation between subsystems, so it can reveal important quantum phenomena such as non-locality, entanglement, etc., but its essence is the characterization of strong statistical relationships. Considering the superiority of the entangled state, this article attempts to use the entangled state to reconstruct the neurons of NNs to achieve that the network model can characterize the strong statistical relationship between the features, namely, the classical and non-classical correlation. Specifically, based on concurrence, a quantification method of entanglement, we propose a regularizer that can constrain a state vector to an entangled state, and apply it to the optimization process to ensure that the vector passed to the neuron is a legal entangled state. Finally, the entangled state is measured to obtain the output of the neuron. Experimental results show that our model is better than the baseline. Moreover, it performs well compared to the model inspired by quantum entanglement.

DOI
10.1109/ijcnn55064.2022.9892246

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Neural Networks (NNs) have received extensive attention and research due to their ability to extract and combine different features non-manually and to mine the internal relationships between features. In quantum mechanics, the entangled state simultaneously describes the classical and non-classical correlation between subsystems, so it can reveal important quantum phenomena such as non-locality, entanglement, etc., but its essence is the characterization of strong statistical relationships. Considering the superiority of the entangled state, this article attempts to use the entangled state to reconstruct the neurons of NNs to achieve that the network model can characterize the strong statistical relationship between the features, namely, the classical and non-classical correlation. Specifically, based on concurrence, a quantification method of entanglement, we propose a regularizer that can constrain a state vector to an entangled state, and apply it to the optimization process to ensure that the vector passed to the neuron is a legal entangled state. Finally, the entangled state is measured to obtain the output of the neuron. Experimental results show that our model is better than the baseline. Moreover, it performs well compared to the model inspired by quantum entanglement.
Even the most sophisticated artificial neural networks are built by aggregating substantially identical units called neurons. A neuron receives multiple signals, internally combines them, and applies a non-linear function to the resulting weighted sum. Several attempts to generalize neurons to the quantum regime have been proposed, but all proposals collided with the difficulty of implementing non-linear activation functions, which is essential for classical neurons, due to the linear nature of quantum mechanics. Here we propose a solution to this roadblock in the form of a small quantum circuit that naturally simulates neurons with threshold activation. Our quantum circuit defines a building block, the "quantum neuron", that can reproduce a variety of classical neural network constructions while maintaining the ability to process superpositions of inputs and preserve quantum coherence and entanglement. In the construction of feedforward networks of quantum neurons, we provide numerical evidence that the network not only can learn a function when trained with superposition of inputs and the corresponding output, but that this training suffices to learn the function on all individual inputs separately. When arranged to mimic Hopfield networks, quantum neural networks exhibit properties of associative memory. Patterns are encoded using the simple Hebbian rule for the weights and we demonstrate attractor dynamics from corrupted inputs. Finally, the fact that our quantum model closely captures (traditional) neural network dynamics implies that the vast body of literature and results on neural networks becomes directly relevant in the context of quantum machine learning.
In supervised learning, the generative approach is an important one, which obtains the generative model by learning the joint probability between features and categories. In quantum mechanics, Quantum Entanglement (QE) can provide a statistical correlation between subsystems (or attributes) that is stronger than what classical systems are able to produce. It inspires us to use entangled systems (states) to characterize this strong statistical correlation between features and categories, that is, to use the joint probability derived from QE to model the correlation. Based on the separability of the density matrix of entangled systems, this paper formally clarifies the manifestation of the strong statistical correlation revealed by QE, and implements a classification algorithm (called ECA) to verify the feasibility and superiority of the correlation in specific tasks. Since QE arises from the measurement process of entangled systems, the core of ECA is quantum measurement operations. In this paper, we use the GHZ [25] and W [22] states to prepare the entangled system and use a fully connected network layer to learn the measurement operator. It can also be understood as replacing the output layer of the Multi-Layer Perceptron (MLP) with a quantum measurement operation. The experimental results show that ECA is superior to most representative classification algorithms in multiple evaluation metrics.
This paper investigates the potential of quantum entanglement to significantly enhance the accuracy of neural simulations, particularly within complex neural networks. The core claim posits that leveraging quantum entanglement can provide a novel approach to modeling neuronal behavior, moving beyond traditional simulation methods. The proposed mechanism involves mapping synaptic connections within neurons to quantum bits (Qubits) and utilizing entanglement to perform parallel computations across multiple synaptic connections. Quantum measurements are then employed to simulate the probabilistic nature of neuronal firing. This research explores a fundamental shift – directly applying quantum mechanical principles to neural simulation rather than treating them as mere heuristic optimization tools. The theoretical framework developed presents a pathway for exploring the impact of quantum effects on neural processing and potentially leading to more accurate and efficient models of biological neural networks. This work lays the groundwork for future investigations into the quantum computational properties of the brain.
This paper proposes a novel approach to deep learning leveraging the phenomenon of quantum entanglement for enhanced training efficiency and generalization capabilities. The core idea is to construct a neural network architecture where neurons are interconnected via entangled quantum states, enabling non-local data dependency learning. We introduce a framework where input data is encoded into quantum states, subjected to quantum gate transformations, and utilizes entanglement for information propagation. This fundamentally shifts the paradigm from classical, locally-connected networks to a system exploiting inherent quantum parallelism. The theoretical underpinning of this approach suggests potential advantages in terms of computational speed and model performance, offering a new direction for deep learning research. We explore the mathematical formalization of this process, emphasizing the key components and their relationships. The primary goal is to demonstrate the feasibility and potential benefits of integrating quantum mechanics into the architecture and training of deep neural networks. The exploration focuses on the design principles and initial mathematical framework for this entanglement-based learning system. ---
This paper proposes a novel approach to computation by leveraging quantum entanglement within a simulated neural network. The core idea is to utilize the inherent parallelism of quantum mechanics, specifically quantum entanglement, to achieve computational speeds beyond those attainable by classical neural networks. We introduce a model where artificial neurons are interconnected to mimic quantum entanglement, employing quantum algorithms such as Grover's algorithm for distributed computation. The model incorporates neural network synaptic connections to simulate quantum state superposition and interference. The potential for significant computational advantages through this hybrid approach is explored. The key equations demonstrating the core model are presented below: Let sis_isi​ represent the state of neuron iii, where i=1,2,...,Ki = 1, 2, ..., Ki=1,2,...,K. si∈{0,1}s_i \in \{0, 1\}si​∈{0,1} The entanglement between neurons iii and jjj is represented as: Eij=12cos⁡(θij)E_{ij} = \sqrt{\frac{1}{2}} \cos(\theta_{ij})Eij​=21​​cos(θij​) where θij\theta_{ij}θij​ is the angle of entanglement between neurons iii and jjj. The update rule for neuron iii is given by: si(t+1)=∑j=1KGijsj(t)s_i^{(t+1)} = \sum_{j=1}^{K} G_{ij} s_j^{(t)}si(t+1)​=∑j=1K​Gij​sj(t)​ where GijG_{ij}Gij​ is the interaction coefficient between neurons iii and jjj. Grover's Algorithm can be represented as: O=∑i=1Nai∣i⟩∣i⟩⟨i∣O = \sum_{i=1}^{N} a_i \frac{|i\rangle}{\sqrt{|i\rangle\langle i|}}O=∑i=1N​ai​∣i⟩⟨i∣​∣i⟩​ where NNN is the number of qubits. The oracle operator OOO flips the amplitude of the marked state. The Grover iteration is: U=I+2O+λσxU = \mathcal{I} + 2O + \lambda \sigma_xU=I+2O+λσx​ where I\mathcal{I}I is the identity operator, σx\sigma_xσx​ is the Pauli-X operator, and λ\lambdaλ is the Grover iteration number.
This paper proposes a novel neural network architecture, termed Entanglement-Based Neural Networks (EBNNs), inspired by the principles of quantum entanglement. Conventional classical neural networks often face challenges in effectively capturing and representing intricate correlations within data, particularly when dealing with high-dimensional or complex datasets. The core idea behind EBNNs is to construct networks where nodes are interconnected in a manner that mirrors the correlated states observed in quantum entanglement, even across spatial separation. This is achieved through a specifically designed network topology that introduces dependencies between nodes, allowing for a more efficient representation of complex relationships. The network's structure is defined by a set of interconnected nodes, each possessing a state (s_i) represented by a real number. The connections between nodes are characterized by a correlation function (C_{ij}), which dictates the degree to which the state of node *j* is influenced by the state of node *i*. The architecture utilizes a layered approach, where layers are composed of interconnected nodes. The propagation of information through the network is governed by a modified version of the Hebbian learning rule, adapted to incorporate the entanglement-like correlation functions. This approach aims to improve the network's ability to learn and represent complex patterns by leveraging inherent correlations rather than relying solely on independent node activations. The key contributions of this work are the conceptual design of EBNNs and the formulation of the core mathematical framework for their implementation. We present a detailed description of the network architecture, the correlation function definition, and the learning rule, offering a foundation for further research and development in this area. Finally, we demonstrate the potential advantages of EBNNs in terms of improved learning efficiency and representation capacity.