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, by deriving the sample complexity of the empirical risk minimization problem, from an optimization perspective. 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 only 1.1-bit precision.
Abstract (translated)
在这项工作中，我们提出了一种新颖的二次二进制优化（QBO）模型用于量化神经网络训练，并通过样条插值实现任意激活和损失函数的应用。我们还引入了前向区间传播（FIP），这是一种旨在解决非线性和多层复合结构带来的挑战的方法，它将激活函数离散化为线性子区间。这种方法保留了神经网络的通用近似性质，同时允许使用量子计算机对复杂的非线性功能进行优化，从而扩大其在人工智能领域的适用范围。 我们从优化的角度推导出经验风险最小化问题的样本复杂度，并提供理论上的误差上限和所需的伊辛自旋数量，以此来衡量近似的精度。解决与QBO模型相关的二次约束二进制优化（QCBO）问题的一个重大挑战在于存在大量的约束条件。在使用惩罚方法处理这些限制时，调整大量惩罚系数成为一个关键的超参数优化问题，并可能增加计算复杂性和影响解决方案质量。 为了解决这个问题，我们采用了量子条件梯度下降（QCGD）算法，该算法利用量子计算机直接解决QCBO问题。我们在随机量子Oracle和目标值有界方差以及系数矩阵精度有限的情况下证明了QCGD的收敛性，并提供了QCBO求解过程的时间复杂度上限。 实验结果表明，在使用相干伊辛机（CIM）处理Fashion MNIST分类任务时，仅采用1.1位精度就达到了94.95%的准确率。
URL
https://arxiv.org/abs/2506.18240
