Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number $N$ of training data points. We show that the generalization error of a quantum machine learning model with $T$ trainable gates scales at worst as $\sqrt{T/N}$. When only $K \ll T$ gates have undergone substantial change in the optimization process, we prove that the generalization error improves to $\sqrt{K / N}$. Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data.
翻译:现代量子机器学习(QML)方法涉及对培训数据集的参数化量子电路进行优化,并随后对测试数据集作出预测(即普遍化)。在这项工作中,我们在培训数量有限的培训数据点之后,对QML的通用性能进行全面研究。我们显示,以美元为最差的T$可训练门标尺的量子机器学习模型的通用错误,最差的为$@sqrt{T/N}。当只有$K\ll T$的门在优化过程中经历了实质性变化时,我们证明一般化错误已经改进到$\sqrt{K/N}。我们的结果意味着,将单位化成多数值的本地门,这是量子计算行业中通常使用指数级培训数据的关键应用,可以大大加快。我们还表明,在量子进量子神经网络的阶段过渡阶段,对量子状态的分类只需要非常小的培训数据集。其他潜在应用包括学习量性差校准代码或量性动态模拟。我们的工作从一般的磁场中保证将数据转化为希望。