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.

翻译：在通过梯度下降训练过参数化模型函数时，参数有时不会显著变化，而是保持接近初始值。这一现象被称为惰性训练，并促使我们考虑模型函数在初始参数附近的线性近似。在惰性区域中，这种线性近似模仿了参数化函数的行为，其相关核（称为切向核）决定了模型的训练性能。已知在（经典）大宽度神经网络中会出现惰性训练。本文表明，对于几何局域参数化量子电路，当量子比特数量较大时，其训练将进入惰性区域。具体而言，我们证明了此类几何局域参数化量子电路在训练过程中的参数变化速率，以及相关量子模型函数线性近似的精度界限；随着量子比特数增加，这些界限均趋于零。我们通过数值模拟支持了分析结果。