Abstract 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 ( )). 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.
Abstract 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 ( )). 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.
The field of machine learning has experienced rapid development over the last decades and has found numerous applications across diverse domains ranging from industrial to personal settings. Quantum computing, on the other hand, is an emerging field offering a fundamentally different computational paradigm that can potentially surpass traditional computational approaches in certain domains.In this thesis, we explore the intersection between these two fields, commonly referred to as quantum machine learning (QML). In particular, we focus on Quantum neural networks (QNNs), which have gained attention in the field of QML due to the unparalleled success of their classical counterparts. However, the limitations of the current quantum computers have led to hybrid approaches combining classical and quantum devices. We, therefore, use hybrid quantum neural networks (HQNNs) to explore the potential advantages of QNNs in various practical applications.First, we explore the design space of HQNNs as there is no standardized methodology for designing quantum layers in HQNNs. Various quantum layer designs with different data encoding techniques are investigated, identifying the optimal set of quantum gates for different encoding strategies.We also investigate the impact of cost function globality and locality on the performance of HQNNs for real-world applications, namely: binary and multi-class classification. We exploit a single parameterized quantum circuit (PQC) structure with both single (local) and all-qubit (global) measurements. Our results show that the global cost function significantly outperforms the local cost function on multi-class classification. In contrast, both global and local cost functions have comparable performance on binary classification problems.Moreover, we propose a framework for analyzing the impact of various components of HQNNs with respect to each other on multi-class classification tasks, specifically focusing on the data encoding and quantum layers (expressibility and entanglement between qubits). Two ansatz structures, entangled and unentangled, are used to investigate the existence of the barren plateau (BP) in HQNNs. We further evaluate the impact of entanglement on trainability and provide recommendations for different constraint scenarios. The results show that the overall expressibility of the quantum layers impacts the presence of the BP phenomenon and that entanglement plays a role in the training landscapes of HQNNs, depending on the data encoding technique. Finally, we propose a new architecture for quantum neural networks (QNNs) called residual quantum neural networks (ResQNets), inspired by the classical residual neural networks (ResNets), to address the issue of BP in QNNs. ResQNets split the conventional QNN architecture into multiple quantum nodes (QNs), each with its own PQC, allowing residual connections between QNs. The results show that incorporating residual connections results in improved training performance, making ResQNets a promising solution for overcoming the problem of BP in QNNs.
We report an empirical study of how quantum neural networks (QNNs) train under global and local costfunctions, with layerwise training included as a third point of comparison. The main experiment is a set of 180 training runson binary MNIST (digits 3 and 6), using an 8-qubit hardware-efficient ansatz written in TensorFlow Quantum. A second,separate study measures gradient landscapes at a fixed input across 48 circuit configurations with 4 to 10 qubits. Wecompare three procedures: end-to-end training with a global product-Z readout (the baseline), a layerwise curriculum thatreuses trained parameters between stages, and training on a local averaged-Z cost. Test accuracy barely moves betweenmethods. The global baseline still reaches 87.2% at 8 layers, so we see no accuracy collapse in this regime, andmatched-seed tests at 8 layers find no significant difference (Friedman p = 0.654; every pairwise paired test isnon-significant, as are the tests at 4 and 6 layers). Gradient statistics tell the methods apart much more clearly. At depths 4,6 and 8, the local cost ends training with 1.5 to 2.4 times lower gradient variance than the global baseline (paired p <= 9.4x 10⁻³ after Holm correction; Cohen's d_z up to 2.3). It also costs far more to simulate at depth: 6.57 h per run at 8 layers,against 1.66 h for the baseline. In the landscape study, the local observable's gradient variance is 1.4 to 10 times lower thanthe global observable's over the whole configuration grid, and 4 to 10 times lower at depth 8. Layerwise training follows adifferent path, with gradient variance falling during optimization, yet it gives no
This work presents a systematic empirical study of techniques for mitigating the barren plateau phenomena in quantum neural networks (QNNs). We conduct 45 controlled experiments for binary MNIST classification of digits 3 and 6 using a computational benchmark built using TensorFlow Quantum. We compare three different training procedures: a standard baseline setup, an incremental layerwise learning curriculum and a protocol based on local cost functions. For 8 layers we see mean accuracies of 73.9% and 75.4% for layerwise and local-cost interventions (over five stochastic seeds). The baseline setting drops from about 74% at 4 and 6 layers to 52.7% at 8 layers. Statistical reconstruction of the 8-layer regime suggests that the difference of 1.5 percentage points between layerwise and local-cost methods is not statistically significant, but both strategies are statistically significantly different from the baseline (one-way ANOVA F = 138.90; Welch tests p < 1.5 × 10−5). These results also suggest that practical training degradation is possible even when gradient magnitudes are above a conventional barren plateau detection threshold, highlighting the significance of continuous indicators for trainability, rather than binary threshold-based diagnostics.
Effective training of Quantum Neural Networks (QNNs) is crucial in the Noisy Intermediate-Scale Quantum (NISQ) era, where noise accelerates the onset of barren plateaus (BPs) and limits scalability. This paper investigates how quantum noise impacts QNN trainability and demonstrates that careful selection of qubit measurement observables can mitigate these effects. We analyze PauliX, PauliY, PauliZ, and a customized Hermitian observable under both global (all-qubit measured) and local (single-qubit measured) cost functions. Our results show that with global cost function, PauliX and PauliY lead to flatter landscapes under noise, while PauliZ maintains training up to 8 qubits before encountering BPs. The customized Hermitian observable proves most robust, enabling training up to 10 qubits in noisy settings. For local cost function setting, PauliZ outperforms PauliX and PauliY, maintaining efficiency up to 10 qubits. These findings highlight the importance of noise-aware observable selection, offering a practical strategy to improve QNN performance and advance quantum machine learning in noisy environments.
Variational quantum algorithms (VQAs) optimize the parameters θ of a quantum neural network V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is a hardware-efficient ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to an exponentially vanishing gradient (i.e., a barren plateau) even when V(θ) is shallow. This implies that several VQAs in the literature must revise their proposed cost functions. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is O(logn). Taken together, our results establish a connection between locality and trainability. Finally, we illustrate these ideas with large-scale simulations, up to 100 qubits, of a particular VQA known as quantum autoencoders.
Recent advances in quantum computing and machine learning have brought about a promising intersection of these two fields, leading to the emergence of quantum machine learning (QML). However, the integration of quantum computing and machine learning poses several challenges. One of the prominent challenges lies in the presence of barren plateaus (BP) in QML algorithms, particularly in quantum neural networks (QNNs). Recent studies have successfully identified the fundamental causes underlying the existence of BP in QNNs. This paper presents a framework designed to explore the interplay of multiple factors contributing to the BP problem in quantum neural networks (QNNs), which poses a critical challenge for the practical applications of QML. We focus on the combined influence of data encoding, qubit entanglement, and ansatz expressibility in hybrid quantum neural networks (HQNNs) for multi-class classification tasks. Our framework aims to empirically analyze the joint impact of these factors on the training landscape of HQNNs. Our results show that the occurrence of the BP problem in HQNNs is contingent upon the expressibility of the underlying ansatz and the type of the adopted data encoding technique. Additionally, we observe that qubit entanglement also plays a role in exacerbating the BP problem. Leveraging various evaluation metrics for classification tasks, we systematically evaluate the performance of HQNNs and provide recommendations tailored to different constraint scenarios. Our findings emphasize the significance of our framework in addressing the practical success of QNNs.