量子机器学习中的泛化：量子信息视角
Generalization in Quantum Machine Learning: a Quantum Information Perspective
摘要：
量子分类和假设检验是两个紧密相关的主题，主要区别在于前者是数据驱动的：如何将量子态$ρ(x)$分配给相应的类$c$（或假设）是在训练期间从示例中学习的，其中$x$可以是可调的实验参数或“嵌入”到量子态中的经典数据。该模型具有普适性吗？这是任何数据驱动策略的主要问题，即预测正确类的能力，即使是以前看不见的状态。在这里，我们建立了量子机器学习分类和量子假设检验（状态和通道鉴别）之间的联系，然后证明了量子分类器的准确性和泛化能力取决于量子状态空间$Q$和经典参数空间$X$或类空间$C$之间的（Rényi）互信息$I(C{:}Q)$和$I_2(X{:}Q)$。基于上述特征，然后我们展示了$Q$的不同属性如何影响分类准确性和泛化，例如希尔伯特空间的维度、噪声量以及来自$X$的被忽略的信息量，例如，池化层。此外，我们引入了量子版本的信息瓶颈原则，使我们能够探索准确性和泛化之间的各种权衡。最后，为了验证我们的理论预测，我们研究了伊辛自旋链的量子相位的分类，我们提出了变分量子信息瓶颈（VQIB）方法来优化经典数据的量子嵌入，以利于推广。
英文摘要：
Quantum classification and hypothesis testing are two tightly related subjects, the main difference being that the former is data driven: how to assign to quantum states $ρ(x)$ the corresponding class $c$ (or hypothesis) is learnt from examples during training, where $x$ can be either tunable experimental parameters or classical data "embedded" into quantum states. Does the model generalize? This is the main question in any data-driven strategy, namely the ability to predict the correct class even of previously unseen states. Here we establish a link between quantum machine learning classification and quantum hypothesis testing (state and channel discrimination) and then show that the accuracy and generalization capability of quantum classifiers depend on the (Rényi) mutual informations $I(C{:}Q)$ and $I_2(X{:}Q)$ between the quantum state space $Q$ and the classical parameter space $X$ or class space $C$. Based on the above characterization, we then show how different properties of $Q$ affect classification accuracy and generalization, such as the dimension of the Hilbert space, the amount of noise, and the amount of neglected information from $X$ via, e.g., pooling layers. Moreover, we introduce a quantum version of the Information Bottleneck principle that allows us to explore the various tradeoffs between accuracy and generalization. Finally, in order to check our theoretical predictions, we study the classification of the quantum phases of an Ising spin chain, and we propose the Variational Quantum Information Bottleneck (VQIB) method to optimize quantum embeddings of classical data to favor generalization.
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