Parameterized quantum circuits can be used as quantum neural networks and have the potential to outperform their classical counterparts when trained for addressing learning problems. To date, much of the results on their performance on practical problems are heuristic in nature. In particular, the convergence rate for the training of quantum neural networks is not fully understood. Here, we analyze the dynamics of gradient descent for the training error of a class of variational quantum machine learning models. We define wide quantum neural networks as parameterized quantum circuits in the limit of a large number of qubits and variational parameters. We then find a simple analytic formula that captures the average behavior of their loss function and discuss the consequences of our findings. For example, for random quantum circuits, we predict and characterize an exponential decay of the residual training error as a function of the parameters of the system. We finally validate our analytic results with numerical experiments.

翻译：参数化量子电路可用作量子神经网络，在训练解决学习问题时具有超越经典对应模型的潜力。迄今为止，关于这类模型在实际问题中性能的结论大多基于启发式方法。具体而言，量子神经网络训练的收敛速率尚未得到充分理解。本文针对一类变分量子机器学习模型，分析了梯度下降法在训练误差上的动力学特性。我们将宽量子神经网络定义为在大量量子比特和变分参数极限下的参数化量子电路，并提出了一个简单解析公式来描述其损失函数的平均行为，同时讨论了相关结论的影响。例如，对于随机量子电路，我们预测并刻画了残余训练误差随系统参数呈指数衰减的特性。最后通过数值实验验证了理论分析结果。