Expressibility-induced Concentration of Quantum Neural Tangent Kernels
Li-Wei Yu Weikang Li Qi Ye Zhide Lu Zizhao Han Dong-Ling Deng
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 trainability and
expressibility 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 pivotal
characteristic of quantum neural tangent kernels, offering valuable insights
for the design of wide quantum variational circuit models in practical
applications.