用杰克逊不等式证明量子神经网络可高效逼近周期函数
Approximation rates of quantum neural networks for periodic functions via Jackson's inequality
- 通过构造三角多项式近似周期函数，利用量子神经网络实现逼近
- 参数数量减少一半，平滑函数所需参数更少，优于已有结果
- 适合研究量子机器学习逼近理论的学者参考
量子神经网络（QNN）是量子计算中类比经典神经网络的模型，由可训练参数的酉矩阵表示。受经典神经网络万能逼近性质启发，近期研究已建立单量子比特到多量子比特乃至混合经典-量子模型的类似结论。本文研究QNN在上确界范数下对周期函数的逼近能力，利用杰克逊不等式将目标函数近似为合适的三角多项式，并通过特定QNN实现。特别地，在周期函数类限制下，可实现参数数量的二次减少，获得优于文献的逼近效果；且函数越光滑，所需参数越少。
原文摘要 · Abstract (English)
Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical neural networks, ensuring that every continuous function can be arbitrarily well approximated uniformly on a compact set of a Euclidean space, some recent works have established analogous results for QNNs, ranging from single-qubit to multi-qubit QNNs, and even hybrid classical-quantum models. In this paper, we study the approximation capabilities of QNNs for periodic functions with respect to the supremum norm. We use the Jackson inequality to approximate a given function by implementing its approximating trigonometric polynomial via a suitable QNN. In particular, we see that by restricting to the class of periodic functions, one can achieve a quadratic reduction of the number of parameters, producing better approximation results than in the literature. Moreover, the smoother the function, the fewer parameters are needed to construct a QNN to approximate the function.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。