Quantum machine learning holds the promise of combining the success of classical machine learning methods with the power of quantum computing, however one of the largest obstacles facing the field is the problem of barren plateaus. Parameterised quantum circuits offer a flexible framework for developing quantum machine learning models, but their practicality is constrained by a trade-off between trainability and classical simulability. In general, circuits that are sufficiently expressive to model complex behaviour often exhibit barren plateaus, where gradients vanish and optimisation fails.
In this work we investigate a compositional approach to mitigate this trade-off by assembling larger quantum models from smaller subcomponents. To ensure trainability of these subcomponents, we describe a framework for constructing group-invariant loss functions, which introduce symmetry-induced inductive bias and lead to improved gradient behaviour and generalisation. In particular, we use this framework to design permutation-equivariant quantum graph neural networks for identifying maximal cliques in graphs. The models we construct exhibit superior training gradients through symmetry-induced bias, and our experiments demonstrate that the trained models generalise to larger, more complex problem instances.
Finally, inspired by Quantum-Informed Recursive Optimisation Algorithms (arXiv:2308.13607), we implement a recursive hybrid quantum-classical heuristic using the learned quantum models to guide a classical search procedure, demonstrating improved inference accuracy and scalability. Together, these results suggest that compositional circuits could be a viable pathway towards scalable quantum learning models that remain challenging to reproduce classically.
Quantum machine learning holds the promise of combining the success of classical machine learning methods with the power of quantum computing, however one of the largest obstacles facing the field is the problem of barren plateaus. Parameterised quantum circuits offer a flexible framework for developing quantum machine learning models, but their practicality is constrained by a trade-off between trainability and classical simulability. In general, circuits that are sufficiently expressive to model complex behaviour often exhibit barren plateaus, where gradients vanish and optimisation fails.
In this work we investigate a compositional approach to mitigate this trade-off by assembling larger quantum models from smaller subcomponents. To ensure trainability of these subcomponents, we describe a framework for constructing group-invariant loss functions, which introduce symmetry-induced inductive bias and lead to improved gradient behaviour and generalisation. In particular, we use this framework to design permutation-equivariant quantum graph neural networks for identifying maximal cliques in graphs. The models we construct exhibit superior training gradients through symmetry-induced bias, and our experiments demonstrate that the trained models generalise to larger, more complex problem instances.
Finally, inspired by Quantum-Informed Recursive Optimisation Algorithms (arXiv:2308.13607), we implement a recursive hybrid quantum-classical heuristic using the learned quantum models to guide a classical search procedure, demonstrating improved inference accuracy and scalability. Together, these results suggest that compositional circuits could be a viable pathway towards scalable quantum learning models that remain challenging to reproduce classically.
Graph Neural Networks (QGNNs) offer a promising approach to combining quantum computing with graph-structured data processing. While classical Graph Neural Networks (GNNs) are scalable and robust, existing QGNNs often lack flexibility due to graph-specific quantum circuit designs, limiting their applicability to diverse real-world problems. To address this, we propose a versatile QGNN framework inspired by GraphSAGE, using quantum models as aggregators to generate quantum-native node embeddings. We integrate inductive representation learning techniques with parameterized quantum convolutional and pooling layers, bridging classical and quantum paradigms. The convolutional layer is flexible, allowing tailored designs for specific tasks. Benchmarked on a node regression task with the QM9 dataset, our framework, using a single minimal circuit for all aggregation steps, handles molecules with varying numbers of atoms without changing qubits or circuit architecture. While classical GNNs achieve higher training performance, our quantum approach remains competitive and often shows enhanced generalization as molecular complexity increases. We also observe faster learning in early training epochs. To mitigate trainability limitations of a single-circuit setup, we extend the framework with multiple quantum aggregators on QM9. Assigning distinct circuits to each hop substantially improves training performance across all cases. Additionally, we numerically demonstrate the absence of barren plateaus as qubit numbers increase, suggesting that the proposed model remains trainable to larger, more complex graph-based problems.
Abstract Leveraging the intrinsic probabilistic nature of quantum systems, generative quantum machine learning (QML) offers the potential to outperform classical learning models. Current generative QML algorithms mostly rely on general-purpose models that, while being very expressive, face several training challenges. One potential way to address these setbacks is by constructing problem-informed models that are capable of more efficient training on structured problems. In particular, probabilistic graphical models provide a flexible framework for representing structure in generative learning problems and can thus be exploited to incorporate inductive bias into QML algorithms. In this work, we propose a problem-informed quantum circuit Born machine Ansatz for learning the joint probability distribution of random variables, with independence relations efficiently represented by a Markov network (MN). We further demonstrate the applicability of the MN framework in constructing generative learning benchmarks and compare our model’s performance to previous designs, showing that it outperforms problem-agnostic circuits. Based on a preliminary analysis of trainability, we narrow down the class of MNs to those exhibiting favourable trainability properties. Finally, we discuss the potential of our model to offer quantum advantage in the context of generative learning, and present a practical scenario demonstrating the scalability of the approach.
Abstract Despite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine learning (GQML) has emerged as a potential solution to some of those issues. The key insight of GQML is that one should design architectures, such as equivariant QNNs, encoding the symmetries of the problem at hand. Here, we focus on problems with permutation symmetry (i.e., symmetry group Sn), and show how to build Sn-equivariant QNNs We provide an analytical study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides theoretical guarantees for equivariant QNNs, thus indicating the power and potential of GQML.
Quantum machine learning methods, including quantum kernels and variational quantum algorithms (VQAs), are poised to demonstrate quantum advantages in machine learning, chemistry, condensed matter physics, and materials science. However, these methods face critical challenges hindering practical quantum advantates: quantum kernels require meticulous design to outperform classical alternatives, while VQA training is obstructed by barren plateaus (BPs) in the optimization landscape. To address these issues, this thesis proposes two novel frameworks to guide quantum kernel design and tackle quantum neural network (QNN) training challenges in VQAs. Specifically, to resolve quantum kernel design challenges, we propose a data-driven method for automatic quantum kernel design, leveraging classical machine learning to learn from input data and identify optimal kernel structures. Experiments show this method finds suitable quantum kernels that avoid vanishing similarity, a common flaw in traditional designs. To improve VQA trainability, we propose an encoder-decoder parameter prediction framework, where a graph neural network (GNN) acts as the encoder to process information based on the QNN’s directed acyclic graph (DAG). Experiments confirm the parameters from this framework significantly enhance VQAs’ robustness, convergence speed, and trainability. In conclusion, we propose two schemes, QuKerNet and QuGHN, for automatic quantum kernel design and enhanced VQA trainability. QuKerNet considers optimal circuit layouts and variational parameters, and is more resource-efficient for contemporary quantum hardware by accounting for qubit topology and count. QuGHN is versatile, applicable to circuits with varying qubits, ansatzes, and tasks; its parameters enable faster convergence and more stable VQA solutions. These advancements enhance quantum machine learning practicality on near-term machines and its utility in large-scale quantum circuit simulations.
Realizing successful quantum algorithms on current generation of quantum hardware depends on developing appropriate parametrized quantum circuits, which are the quantum computing counterpart to classical neural networks. A key consideration in this context is the algorithm’s trainability that can be hindered by the existence of so-called barren plateaus. In this case, the algorithm suffers from exponentially vanishing gradients as the number of qubits increases, resulting in a flat optimization landscape and making an efficient training intractable. Due to its serious impact on the success of such algorithms, this problem has gained substantial attention in recent years of quantum computing research. Theoretical and heuristic approaches have been proposed to investigate, avoid and mitigate this issue. In this work, we explore this phenomenon in detail and study the underlying mathematical concepts in depth, investigating the factors that influence the structure of the optimization landscape. Using corresponding mathematical tools, we diagnose whether a given variational circuit will exhibit barren plateaus when going beyond tens of qubits. Additionally, a review of mitigation strategies will be provided. The theory will then be applied to a use case that involves image classification to evaluate whether the model is affected by this phenomenon and which hyperparameters improve or degrade the overall performance.
Quantum neural network architectures that have little to no inductive biases are known to face trainability and generalization issues. Inspired by a similar problem, recent breakthroughs in machine learning address this challenge by creating models encoding the symmetries of the learning task. This is materialized through the usage of equivariant neural networks the action of which commutes with that of the symmetry. In this work, we import these ideas to the quantum realm by presenting a comprehensive theoretical framework to design equivariant quantum neural networks (EQNNs) for essentially any relevant symmetry group. We develop multiple methods to construct equivariant layers for EQNNs and analyze their advantages and drawbacks. Our methods can find unitary or general equivariant quantum channels efficiently even when the symmetry group is exponentially large or continuous. As a special implementation, we show how standard quantum convolutional neural networks (QCNNs) can be generalized to group-equivariant QCNNs where both the convolution and pooling layers are equivariant to the symmetry group. We then numerically demonstrate the effectiveness of a S U ( 2 ) -equivariant QCNN over symmetry-agnostic QCNN on a classification task of phases of matter in the bond-alternating Heisenberg model. Our framework can be readily applied to virtually all areas of quantum machine learning. Lastly, we discuss about how symmetry-informed models such as EQNNs provide hopes to alleviate central challenges such as barren plateaus, poor local minima, and sample complexity. Published by the American Physical Society 2024