  Variational quantum circuits are used in quantum machine learning and
variational quantum simulation tasks. Designing good variational circuits or
predicting how well they perform for given learning or optimization tasks is
still unclear. Here we discuss these problems, analyzing variational quantum
circuits using the theory of neural tangent kernels. We define quantum neural
tangent kernels, and derive dynamical equations for their associated loss
function in optimization and learning tasks. We analytically solve the dynamics
in the frozen limit, or lazy training regime, where variational angles change
slowly and a linear perturbation is good enough. We extend the analysis to a
dynamical setting, including quadratic corrections in the variational angles.
We then consider hybrid quantum-classical architecture and define a large-width
limit for hybrid kernels, showing that a hybrid quantum-classical neural
network can be approximately Gaussian. The results presented here show limits
for which analytical understandings of the training dynamics for variational
quantum circuits, used for quantum machine learning and optimization problems,
are possible. These analytical results are supported by numerical simulations
of quantum machine learning experiments.
