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Quantum Neural Networks with Non-Markovian Dynamics

Jincheng Zhang

Published Aug 29, 2026• 0 citations
Publisher

Abstract

This paper investigates the potential of quantum neural networks (QNNs) enhanced by non-Markovian dynamics. Traditional Markovian QNNs suffer from limitations in capturing long-range dependencies, a crucial aspect in complex data processing. We propose a novel QNN architecture that incorporates quantum memory and entanglement to facilitate non-Markovian information processing. This approach allows the network to retain and utilize the entire past history of the input, fundamentally altering the dynamics of information flow. Through theoretical analysis and mathematical formulations, we demonstrate how this architecture can significantly improve the network's ability to learn and represent complex patterns, particularly those exhibiting long-range correlations. Specifically, we introduce a modified update rule leveraging the concept of quantum memory and entanglement, expressed as: $ \Delta w_{ij} = \alpha \cdot E[\rho(t)|x_1, x_2, ..., x_K] \cdot \langle \psi_i | \rho(t) | \psi_j \rangle - (1 - \alpha) \cdot w_{ij}$ Where: * $ \Delta w_{ij}$ is the change in the synaptic weight between neurons *i* and *j*. * $\alpha$ is the learning rate. * $E[\rho(t)|x_1, x_2, ..., x_K]$ represents the expected state of the quantum memory at time *t*, given the entire input sequence $x_1, x_2, ..., x_K$. * $\rho(t)$ is the density matrix representing the quantum state at time *t*. * $\langle \psi_i | \rho(t) | \psi_j \rangle$ is the inner product between the quantum states $\psi_i$ and $\psi_j$ within the density matrix $\rho(t)$. * $w_{ij}$ is the current synaptic weight between neurons *i* and *j*. The core of our innovation lies in the utilization of quantum memory, modeled as a quantum state $\rho(t)$, which retains information about the entire past sequence. The entanglement between neurons further enables correlated updates, bypassing the limitations of Markovian models that only consider immediate dependencies. We explore the impact of varying the strength of entanglement and the capacity of the quantum memory on the network's performance using mathematical models. The results suggest a significant improvement in accuracy and convergence speed compared to standard Markovian QNNs when dealing with datasets exhibiting long-range dependencies. Further investigation into the scalability and practical implementation of this architecture is warranted.

DOI
10.5281/zenodo.22153939

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Featuring memory of past inputs is a fundamental requirement for machine learning models processing time-dependent data. In quantum reservoir computing, all architectures proposed so far rely on Markovian dynamics, which, as we prove, inherently lead to an exponential decay of past information, thereby limiting long-term memory capabilities. We demonstrate that non-Markovian dynamics can overcome this limitation, enabling extended memory retention. By analytically deriving memory bounds and supporting our findings with numerical simulations, we show that non-Markovian reservoirs can outperform their Markovian counterparts, particularly in tasks that require a coexistence of short- and long-term correlations. We introduce an embedding approach that allows a controlled transition from Markovian to non-Markovian evolution, providing a path for practical implementations. Our results establish quantum non-Markovianity as a key resource for enhancing memory in quantum machine learning architectures, with broad implications in quantum neural networks.
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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.
Quantum Neural Networks (QNNs) represent a novel approach to artificial neural networks, leveraging the unique properties of quantum mechanics to enhance computation capabilities. This paper investigates the design and implementation of a quantum neural network (QNN) that employs adaptive neuron structures, dynamically adjusting their connections based on input data. We explore the potential of quantum entanglement to facilitate this self-adaptive behavior, aiming to achieve improved efficiency and robustness compared to classical neural networks. The core mechanism centers around utilizing quantum entanglement to enable neurons to subtly influence each other's states, promoting a form of self-organization within the network. This research offers a promising avenue for advancing QNNs and opens up new possibilities for complex computational tasks.
Quantum Neural Networks (QNNs) offer a fundamentally different approach to neural network learning compared to classical networks, leveraging the unique properties of quantum mechanics. This paper investigates the potential of QNNs to achieve more efficient and powerful non-linear learning, particularly when handling high-dimensional data and complex patterns. We propose a novel algorithm that utilizes quantum entanglement and superposition to enhance the learning process and feature extraction capabilities. This work explores the potential of these quantum properties to overcome limitations inherent in current neural network architectures, specifically in areas requiring sophisticated pattern recognition. The core of our approach focuses on designing quantum units within the QNN architecture to facilitate enhanced learning and memory. The paper details the proposed algorithm, its theoretical underpinnings, and preliminary experimental results demonstrating its effectiveness in a simulated environment. We conclude with a discussion of future research directions, emphasizing the transformative potential of integrating quantum principles into neural network learning.
This paper proposes a novel architecture for quantum neural networks (QNNs) inspired by biological neural networks, specifically focusing on adaptive learning capabilities. The core idea is to integrate biological mechanisms such as synaptic plasticity and long-range dependencies into the quantum learning process. The traditional approach to QNN training often relies on gradient-based methods, which can be inefficient and susceptible to local minima. This work introduces a bio-inspired approach that leverages the inherent parallelism and quantum superposition of QNNs to achieve more robust and adaptive learning. We formulate the learning process using a modified Hebbian learning rule combined with a quantum measurement process. The key innovation lies in dynamically adjusting the quantum gates based on the network's internal state, mimicking the neural network's ability to adapt to changing environments. The resulting QNN architecture exhibits enhanced learning speed and improved generalization performance compared to conventional QNNs. The theoretical framework is presented, and the underlying mathematical equations are detailed, providing a robust foundation for further research and development in this field.