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Quantum-Classical Hybrid Quantized Neural Network

Chuan Xi Wang, Wen-Xin Li, Hong-Dong Zhu, Qi Gao, Yin Ma, Hai Wei, Kai Wen

Published Jul 7, 2025• 1 citation
Publisher

Abstract

Abstract Here in this work, we present a novel Quadratic Binary Optimization (QBO) model for quantized neural network training, enabling the use of arbitrary activation and loss functions through spline interpolation. We introduce Forward Interval Propagation (FIP), a method designed to tackle the challenges of non-linearity and the multi-layer composite structure in neural networks by discretizing activation functions into linear subintervals. This approach preserves the universal approximation properties of neural networks while allowing complex nonlinear functions to be optimized using quantum computers, thus broadening their applicability in artificial intelligence. We provide theoretical upper bounds on the approximation error and the number of Ising spins required, which is independent of the training sample size. A significant challenge in solving the associated Quadratic Constrained Binary Optimization (QCBO) model on a large scale is the presence of numerous constraints. When employing the penalty method to handle these constraints, tuning a large number of penalty coefficients becomes a critical hyperparameter optimization problem, increasing computational complexity and potentially affecting solution quality. To address this, we employ the Quantum Conditional Gradient Descent (QCGD) algorithm, which leverages quantum computing to directly solve the QCBO problem. We prove the convergence of QCGD under a quantum oracle with randomness and bounded variance in objective value, as well as under limited precision constraints in the coefficient matrix. Additionally, we provide an upper bound on the Time-To-Solution for the QCBO solving process. Experimental results using a coherent Ising machine (CIM) demonstrate a $94.95\%$ accuracy on the Fashion MNIST classification task, with comparisons to classical algorithms highlighting superior efficiency. We further validate the robustness of the spline interpolation method and analyze the impact of varying precision levels on QCGD convergence, underscoring the reliability of our framework.

DOI
10.21203/rs.3.rs-6725857/v1

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Abstract Here in this work, we present a novel Quadratic Binary Optimization (QBO) model for quantized neural network training, enabling the use of arbitrary activation and loss functions through spline interpolation. We introduce Forward Interval Propagation (FIP), a method designed to tackle the challenges of non-linearity and the multi-layer composite structure in neural networks by discretizing activation functions into linear subintervals. This approach preserves the universal approximation properties of neural networks while allowing complex nonlinear functions to be optimized using quantum computers, thus broadening their applicability in artificial intelligence. We provide theoretical upper bounds on the approximation error and the number of Ising spins required, which is independent of the training sample size. A significant challenge in solving the associated Quadratic Constrained Binary Optimization (QCBO) model on a large scale is the presence of numerous constraints. When employing the penalty method to handle these constraints, tuning a large number of penalty coefficients becomes a critical hyperparameter optimization problem, increasing computational complexity and potentially affecting solution quality. To address this, we employ the Quantum Conditional Gradient Descent (QCGD) algorithm, which leverages quantum computing to directly solve the QCBO problem. We prove the convergence of QCGD under a quantum oracle with randomness and bounded variance in objective value, as well as under limited precision constraints in the coefficient matrix. Additionally, we provide an upper bound on the Time-To-Solution for the QCBO solving process. Experimental results using a coherent Ising machine (CIM) demonstrate a 94.95%94.95\%94.95% accuracy on the Fashion MNIST classification task, with comparisons to classical algorithms highlighting superior efficiency. We further validate the robustness of the spline interpolation method and analyze the impact of varying precision levels on QCGD convergence, underscoring the reliability of our framework.
As a dedicated quantum device, Ising machines could solve large-scale binary optimization problems in milliseconds. There is emerging interest in utilizing Ising machines to train feedforward neural networks due to the prosperity of generative artificial intelligence. However, existing methods can only train single-layer feedforward networks because of the complex nonlinear network topology. This paper proposes an Ising learning algorithm to train quantized neural network (QNN), by incorporating two essential techinques, namely binary representation of topological network and order reduction of loss function. As far as we know, this is the first algorithm to train multi-layer feedforward networks on Ising machines, providing an alternative to gradient-based backpropagation. Firstly, training QNN is formulated as a quadratic constrained binary optimization (QCBO) problem by representing neuron connection and activation function as equality constraints. All quantized variables are encoded by binary bits based on binary encoding protocol. Secondly, QCBO is converted to a quadratic unconstrained binary optimization (QUBO) problem, that can be efficiently solved on Ising machines. The conversion leverages both penalty function and Rosenberg order reduction, who together eliminate equality constraints and reduce high-order loss function into a quadratic one. With some assumptions, theoretical analysis shows the space complexity of our algorithm is O(H2L+HLNlog⁡H)\mathcal{O}(H^2L + HLN\log H)O(H2L+HLNlogH), quantifying the required number of Ising spins. Finally, the algorithm effectiveness is validated with a simulated Ising machine on MNIST dataset. After annealing 700 ms, the classification accuracy achieves 98.3%. Among 100 runs, the success probability of finding the optimal solution is 72%. Along with the increasing number of spins on Ising machine, our algorithm has the potential to train deeper neural networks.
I present a novel approach to training Binary Neural Networks (BiNNs) by leveraging Variational Quantum Algorithms (VQAs). My method employs the hybrid quantum-classical framework of VQAs to train simulated quantum circuits as hypernetworks that generate effective BiNN configurations. By developing approximate methods for calculating the VQA objective and gradient, I significantly scale quantum hypernetworks—over two orders of magnitude larger than previous capabilities, overcoming the constraints of exact computations. Quantum circuits are trained to produce effective BiNNs for binary classification tasks on resized and native resolution MNIST datasets. My empirical investigation extends to various VQA hyperparameters, including sample and batch sizes, circuit depth, and Multi-Basis Encoding (MBE). The results reveal that my optimized circuits consistently generate BiNN configurations with competitive cross-entropy losses. This work showcases the practical scalability of quantum hypernetworks and demonstrates significant interdisciplinary contribution in quantum machine learning, inviting further exploration into hybrid quantum-classical computational techniques.
Convolutional Neural Networks (CNNs) are pivotal in computer vision and Big Data analytics but demand significant computational resources when trained on large-scale datasets. Conventional training via back-propagation (BP) with loss functions like Mean Squared Error or Cross-Entropy often requires extensive iterations and may converge sub-optimally. Quantum computing offers a promising alternative by leveraging superposition, tunneling, and entanglement to search complex optimization landscapes more efficiently.In this work, we propose a hybrid optimization method that combines an Unconstrained Binary Quadratic Programming (UBQP) formulation with Stochastic Gradient Descent (SGD) to accelerate CNN training. Evaluated on the MNIST dataset, our approach achieves improvement without compromising the accuracy while maintaining similar execution times. These results illustrate the potential of hybrid quantum-classical techniques in High-Performance Computing (HPC) environments for Big Data and Deep Learning. Fully realizing these benefits, however, requires a careful alignment of algorithmic structures with underlying quantum mechanisms.
Inspired by the principle of superposing states in quantum mechanics,a new kind of feedforward neural network with single hidden layer was designed and named as Quantum Neural Networks.It has mainly been used for data classification.Multilevel activation functions is adopted in Quantum Neural Networks and a different objective function is used for the learning of quantum intervals of the multilevel hidden units;minimizing the class-conditional variances at the outputs of the hidden units.Quantum Neural Networks is endowed with a special ability of identifying uncertainty in data classification by its unique design.However,as different objective functions are used for the learning of synaptic weights and quantum intervals,conflict is observed in the iterative training process which will cause more iteration times for training and degrades the training results.Using the theory of Constrained Optimization,a new training algorithm is proposed in this paper by introducing penalty function into the steepest descent method.In the new training algorithm, the conflict is removed.Experimental results indicate that the new algorithm can increase the training speed and improve the performance of the trained network remarkably.
The training of large-scale machine learning models presents significant computational challenges, primarily due to the inherent complexity of optimization algorithms like stochastic gradient descent (SGD) and its variants. These algorithms often struggle with convergence speed, getting trapped in local minima, and the ability to effectively explore the vast solution space. This paper investigates the potential of quantum-inspired optimization techniques to address these limitations. Specifically, we explore the application of quantum annealing and variational quantum eigensolver (VQE) approaches to accelerate the training and optimization process of large-scale machine learning models. Our core claim is that by mapping the optimization problem to a quantum system, we can leverage quantum computation to overcome the computational bottlenecks associated with classical optimization methods. We propose a framework where the objective function of the machine learning model is translated into a Hamiltonian for a quantum annealer or the cost function for a VQE, enabling a quantum-inspired solution. The results, while theoretical in this initial exploration, demonstrate a potential pathway for significantly reducing training times and improving the overall performance of large-scale machine learning models. Further research is needed to develop efficient hybrid quantum-classical algorithms and explore the practical limitations of this approach.
This research explores the integration of quantum computing with classical machine learning to enhance data classification tasks using Quantum Neural Networks (QNN) and Parameterized Quantum Circuits (PQC). The hybrid approach leverages the advantages of both quantum and classical systems to improve the efficiency and accuracy of data processing. In this model, data is encoded into qubits using amplitude encoding, representing input vectors as amplitudes of quantum states. The QNN is initialized by placing the qubits in superposition using Hadamard gates, followed by data encoding with parameterized rotational gates that map classical data to quantum states using rotation angles. PQC plays a central role by applying layers of parameterized quantum operations to process data in the quantum space. These parameters are optimized during the training process, where a quadratic loss function minimizes the error between the predicted quantum states and the true class labels using gradient descent. Experiments conducted on the MNIST dataset show that the hybrid the hybrid quantum-classical neural network (QCNN) with PQC achieves a classification accuracy of over 95%, highlighting its potential in machine learning applications. The results demonstrate that integrating quantum computing with classical machine learning enhances performance in complex data analysis tasks due to the exponential growth of quantum state space and the parallelism of quantum systems, making hybrid models promising for computer vision and classification tasks.