  In the training of over-parameterized model functions via gradient descent,
sometimes the parameters do not change significantly and remain close to their
initial values. This phenomenon is called lazy training, and motivates
consideration of the linear approximation of the model function around the
initial parameters. In the lazy regime, this linear approximation imitates the
behavior of the parameterized function whose associated kernel, called the
tangent kernel, specifies the training performance of the model. Lazy training
is known to occur in the case of (classical) neural networks with large widths.
In this paper, we show that the training of geometrically local parameterized
quantum circuits enters the lazy regime for large numbers of qubits. More
precisely, we prove bounds on the rate of changes of the parameters of such a
geometrically local parameterized quantum circuit in the training process, and
on the precision of the linear approximation of the associated quantum model
function; both of these bounds tend to zero as the number of qubits grows. We
support our analytic results with numerical simulations.
