Yu Li-Wei, Li Weikang, Ye Qi, Lu Zhide, Han Zizhao, Deng Dong-Ling
Nankai University, Chern Institute of Mathematics, Tianjin, 300071, CHINA.
Tsinghua University, Center for Quantum Information, IIIS, Beijing, 100084, CHINA.
Rep Prog Phys. 2024 Oct 3. doi: 10.1088/1361-6633/ad82cf.
Quantum tangent kernel methods provide an efficient approach to analyzing the performance of quantum machine learning models in the infinite-width limit, which is of crucial importance in designing appropriate circuit architectures for certain learning tasks. Recently, they have been adapted to describe the convergence rate of training errors in quantum neural networks in an analytical manner. Here, we study the connections between the expressibility and value concentration of quantum tangent kernel models. In particular, for global loss functions, we rigorously prove that high expressibility of both the global and local quantum encodings can lead to exponential concentration of quantum tangent kernel values to zero. Whereas for local loss functions, such issue of exponential concentration persists owing to the high expressibility, but can be partially mitigated. We further carry out extensive numerical simulations to support our analytical theories. Our discoveries unveil a fundamental feature of quantum neural tangent kernels, indicating that the issue of their concentration cannot be bypassed merely by transitioning to a local encoding scheme while maintaining high expressibility. This offers valuable insights for the design of wide quantum variational circuit models in practical applications.
量子切核方法为分析量子机器学习模型在无限宽度极限下的性能提供了一种有效途径，这对于为特定学习任务设计合适的电路架构至关重要。最近，它们已被用于以解析方式描述量子神经网络中训练误差的收敛速度。在此，我们研究量子切核模型的可表达性和值集中性之间的联系。特别地，对于全局损失函数，我们严格证明全局和局部量子编码的高可表达性都可导致量子切核值指数级集中到零。而对于局部损失函数，由于高可表达性，这种指数级集中问题依然存在，但可以部分缓解。我们进一步进行了广泛的数值模拟以支持我们的解析理论。我们的发现揭示了量子神经切核的一个基本特征，表明仅通过在保持高可表达性的同时过渡到局部编码方案并不能回避其集中问题。这为实际应用中宽量子变分电路模型的设计提供了有价值的见解。
