Exponentially Many Local Minima in Quantum Neural Networks
This addresses the training challenge in QNNs for quantum computing researchers, revealing fundamental limitations due to quantum interference, which is incremental as it builds on prior work but provides new quantitative insights.
The paper identifies a class of Quantum Neural Network (QNN) instances that are extremely hard to train, showing that for typical under-parameterized QNNs, there exists a dataset inducing a loss function with an exponential number of spurious local minima in the number of parameters, and provides an almost matching upper bound on this dependence.
Quantum Neural Networks (QNNs), or the so-called variational quantum circuits, are important quantum applications both because of their similar promises as classical neural networks and because of the feasibility of their implementation on near-term intermediate-size noisy quantum machines (NISQ). However, the training task of QNNs is challenging and much less understood. We conduct a quantitative investigation on the landscape of loss functions of QNNs and identify a class of simple yet extremely hard QNN instances for training. Specifically, we show for typical under-parameterized QNNs, there exists a dataset that induces a loss function with the number of spurious local minima depending exponentially on the number of parameters. Moreover, we show the optimality of our construction by providing an almost matching upper bound on such dependence. While local minima in classical neural networks are due to non-linear activations, in quantum neural networks local minima appear as a result of the quantum interference phenomenon. Finally, we empirically confirm that our constructions can indeed be hard instances in practice with typical gradient-based optimizers, which demonstrates the practical value of our findings.