The impact of cost function globality and locality in hybrid quantum neural networks on NISQ devices

Quantum neural networks (QNNs) are often challenged with the problem of flat cost function landscapes during training, known as barren plateaus (BP). A solution to potentially overcome the problem of the BP has recently been proposed by Cerezo et al In this solution, it is shown that, for an arbitrary deep quantum layer(s) in QNNs, a global cost function (all qubits measured in an n-qubit system) will always experience BP, whereas a local cost function (single qubit measured in an n-qubit system) can help to alleviate the problem of BP to a certain depth ($\mathcal{O}(\mathrm {log}(n)$)). In this paper, we empirically analyze the locality and globality of the cost function in hybrid quantum neural networks. We consider two application scenarios namely, binary and multi-class classification, and show that for multiclass classification, the local cost function setting does not follow the claims of Cerezo et al; that is, the local cost function does not result in an extended quantum layer's depth. We also show that for multiclass classification, the overall performance in terms of accuracy for the global cost function setting is significantly higher than the local cost function setting. On the other hand, for binary classification, our results show that the local cost function setting follows the claims of Cerezo et al, and results in an extended depth of quantum layers. However, the global cost function setting still performs slightly better than the local cost function.