标题： 振幅编码在量子分类上的局限性 显示英文标题

标题： Limitations of Amplitude Encoding on Quantum Classification

摘要： 目前尚不清楚量子机器学习（QML）在处理实际且有意义的任务时是否具有真正的优势。将经典数据编码为量子态是QML中的关键步骤之一。由于在同时将大量$2^{n}$个经典数据编码到$n$个量子比特方面的显著效率，振幅编码被广泛使用。然而，振幅编码对QML的理论影响尚未得到彻底研究。在这项工作中，我们证明，在一些广泛且典型的数据假设下，通过振幅编码编码的量子态的平均值往往会集中于一个特定的状态。这种集中现象严重限制了量子分类器的能力，因为它导致了损失屏障现象，即损失函数有一个无法通过任何优化算法改进的下限。此外，通过数值模拟，我们揭示了振幅编码的一个反直觉现象：随着训练数据量的增加，训练误差可能会增加而不是减少，从而导致新数据预测精度的降低。我们的结果突显了振幅编码在QML中的局限性，并表明应该投入更多努力寻找更有效的编码策略以充分发挥QML的潜力。 显示英文摘要

摘要： It remains unclear whether quantum machine learning (QML) has real advantages when dealing with practical and meaningful tasks. Encoding classical data into quantum states is one of the key steps in QML. Amplitude encoding has been widely used owing to its remarkable efficiency in encoding a number of $2^{n}$ classical data into $n$ qubits simultaneously. However, the theoretical impact of amplitude encoding on QML has not been thoroughly investigated. In this work we prove that under some broad and typical data assumptions, the average of encoded quantum states via amplitude encoding tends to concentrate towards a specific state. This concentration phenomenon severely constrains the capability of quantum classifiers as it leads to a loss barrier phenomenon, namely, the loss function has a lower bound that cannot be improved by any optimization algorithm. In addition, via numerical simulations, we reveal a counterintuitive phenomenon of amplitude encoding: as the amount of training data increases, the training error may increase rather than decrease, leading to reduced decrease in prediction accuracy on new data. Our results highlight the limitations of amplitude encoding in QML and indicate that more efforts should be devoted to finding more efficient encoding strategies to unlock the full potential of QML.