



01673nas a2200145   4500008004100000245006300041210006300104260001400167300001600181490000800197520125000205100001601455700001501471856004101486       2021                        eng d00aExponentially Many Local Minima in Quantum Neural Networks0 aExponentially Many Local Minima in Quantum Neural Networks  c10/5/2021  a12144-121550 v1393 a<p>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.&nbsp;</p>1 aYou, Xuchen1 aWu, Xiaodi  uhttps://arxiv.org/pdf/2110.02479.pdf