Title: Enhancing Expressivity of Quantum Neural Networks Based on the SWAP test Show Chinese title

Title: 基于SWAP测试的量子神经网络表达能力增强

Abstract: Parameterized quantum circuits represent promising architectures for machine learning applications, yet many lack clear connections to classical models, potentially limiting their ability to translate the wide success of classical neural networks to the quantum realm. We examine a specific type of quantum neural network (QNN) built exclusively from SWAP test circuits, and discuss its mathematical equivalence to a classical two-layer feedforward network with quadratic activation functions under amplitude encoding. Our analysis across classical real-world and synthetic datasets reveals that while this architecture can successfully learn many practical tasks, it exhibits fundamental expressivity limitations due to violating the universal approximation theorem, particularly failing on harder problems like the parity check function. To address this limitation, we introduce a circuit modification using generalized SWAP test circuits that effectively implements classical neural networks with product layers. This enhancement enables successful learning of parity check functions in arbitrary dimensions which we analytically argue to be impossible for the original architecture beyond two dimensions regardless of network size. Our results establish a framework for enhancing QNN expressivity through classical task analysis and demonstrate that our SWAP test-based architecture offers broad representational capacity, suggesting potential promise also for quantum learning tasks. Show Chinese abstract

Abstract: 参数化量子电路代表了机器学习应用中有前途的架构，但许多电路与经典模型缺乏明确的联系，这可能限制了它们将经典神经网络的广泛成功转化为量子领域的能力。 我们研究了一种特定类型的量子神经网络（QNN），该网络仅由SWAP测试电路构建，并讨论了其在幅度编码下的数学等价性，即与具有二次激活函数的经典两层前馈网络的等价性。 我们在经典现实世界和合成数据集上的分析显示，尽管这种架构可以成功学习许多实际任务，但由于违反了通用逼近定理，它表现出基本的表达能力限制，特别是在像奇偶校验函数这样的困难问题上表现不佳。 为了解决这一限制，我们引入了一种使用广义SWAP测试电路的电路修改方法，该方法有效地实现了具有乘积层的经典神经网络。 这种改进使我们在任意维度上成功学习奇偶校验函数成为可能，我们通过分析论证，原架构在超过两个维度的情况下，无论网络大小如何，都无法实现这一点。 我们的结果建立了一个通过经典任务分析增强QNN表达能力的框架，并表明我们的基于SWAP测试的架构具有广泛的表示能力，这表明它在量子学习任务中也具有潜在的前景。