Analytic theory for the dynamics of wide quantum neural networks
This work addresses a foundational gap in quantum machine learning theory by providing analytic insights into training dynamics, which could benefit researchers developing quantum algorithms, though it is incremental as it builds on existing variational models.
The paper tackles the problem of understanding the convergence rate for training quantum neural networks, which has been largely heuristic, by analyzing gradient descent dynamics for a class of variational quantum machine learning models and deriving an analytic formula that predicts an exponential decay of residual training error, validated with numerical experiments.
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.