Exponentially Many Local Minima in Quantum Neural Networks
Exponentially Many Local Minima in Quantum Neural Networks
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发表时间：
2021-10
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通讯作者：
Xuchen You;Xiaodi Wu
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作者：
Xuchen You;Xiaodi Wu
量子神经网络（QNN），或者所谓的变分量子电路，是重要的量子应用，因为它们与经典神经网络有相似的前景，而且因为它们在近期中等大小的噪声量子机（NISQ）上实现的可行性。然而，QNN 的训练任务具有挑战性，而且了解较少。我们对 QNN 损失函数的情况进行了定量研究，并确定了一类简单但极其困难的 QNN 实例进行训练。具体来说，我们表明，对于典型的参数化不足的 QNN，存在一个数据集，该数据集会产生一个损失函数，其中虚假局部最小值的数量以指数方式依赖于参数的数量。此外，我们通过提供这种依赖性的几乎匹配的上限来展示我们的构造的最优性。经典神经网络中的局部最小值是由于非线性激活引起的，而在量子神经网络中，局部最小值是由于量子干涉现象而出现的。最后，我们凭经验证实，我们的构造在实践中确实可以成为典型的基于梯度的优化器的困难实例，这证明了我们的研究结果的实用价值。
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.