Quantum machine learning (QML) is a discipline that seeks to transfer the advantages of quantum computing to data-driven tasks. However, many studies rely on toy datasets or heavy feature reduction, raising concerns about their scalability. Progress is further hindered by hardware limitations and the significant costs of encoding dense vector representations on quantum devices. To address these challenges, we propose an efficient approach called Hamiltonian classifier that circumvents the costs associated with data encoding by mapping inputs to a finite set of Pauli strings and computing predictions as their expectation values. In addition, we introduce two classifier variants with different scaling in terms of parameters and sample complexity. We evaluate our approach on text and image classification tasks, against well-established classical and quantum models. The Hamiltonian classifier delivers performance comparable to or better than these methods. Notably, our method achieves logarithmic complexity in both qubits and quantum gates, making it well-suited for large-scale, real-world applications. We make our implementation available on GitHub.
Demonstrating quantum advantage using conventional quantum algorithms remains challenging on current noisy gate-based quantum computers. Automated quantum circuit synthesis via quantum machine learning has emerged as a promising solution, employing trainable parametric quantum circuits to alleviate this. The circuit ansatz in these solutions is often designed through reinforcement learning-based quantum architecture search when the domain knowledge of the problem and hardware are not effective. However, the interpretability of these synthesized circuits remains a significant bottleneck, limiting their scalability and applicability across diverse problem domains. This work addresses the challenge of explainability in quantum architecture search (QAS) by introducing a novel genetic programming-based decompiler framework for reverse-engineering high-level quantum algorithms from low-level circuit representations. The proposed approach, implemented in the open-source tool DeQompile, employs program synthesis techniques, including symbolic regression and abstract syntax tree manipulation, to distill interpretable Qiskit algorithms from quantum assembly language. Validation of benchmark algorithms demonstrates the efficacy of our tool. By integrating the decompiler with online learning frameworks, this research potentiates explainable QAS by fostering the development of generalizable and provable quantum algorithms.
Geometric Machine Learning (GML) has shown that respecting non-Euclidean geometry in data spaces can significantly improve performance over naive Euclidean assumptions. In parallel, Quantum Machine Learning (QML) has emerged as a promising paradigm that leverages superposition, entanglement, and interference within quantum state manifolds for learning tasks. This paper offers a unifying perspective by casting QML as a specialized yet more expressive branch of GML. We argue that quantum states, whether pure or mixed, reside on curved manifolds (e.g., projective Hilbert spaces or density-operator manifolds), mirroring how covariance matrices inhabit the manifold of symmetric positive definite (SPD) matrices or how image sets occupy Grassmann manifolds. However, QML also benefits from purely quantum properties, such as entanglement-induced curvature, that can yield richer kernel structures and more nuanced data embeddings. We illustrate these ideas with published and newly discussed results, including hybrid classical -quantum pipelines for diabetic foot ulcer classification and structural health monitoring. Despite near-term hardware limitations that constrain purely quantum solutions, hybrid architectures already demonstrate tangible benefits by combining classical manifold-based feature extraction with quantum embeddings. We present a detailed mathematical treatment of the geometrical underpinnings of quantum states, emphasizing parallels to classical Riemannian geometry and manifold-based optimization. Finally, we outline open research challenges and future directions, including Quantum Large Language Models (LLMs), quantum reinforcement learning, and emerging hardware approaches, demonstrating how synergizing GML and QML principles can unlock the next generation of machine intelligence.
Quantum Machine Learning (QML) is the intersection of two revolutionary fields: quantum computing and machine learning. It promises to unlock unparalleled capabilities in data analysis, model building, and problem-solving by harnessing the unique properties of quantum mechanics. This research endeavors to conduct a comprehensive bibliometric analysis of scientific information pertaining to QML covering the period from 2000 to 2023. An extensive dataset comprising 9493 scholarly works is meticulously examined to unveil notable trends, impact factors, and funding patterns within the domain. Additionally, the study employs bibliometric mapping techniques to visually illustrate the network relationships among key countries, institutions, authors, patent citations and significant keywords in QML research. The analysis reveals a consistent growth in publications over the examined period. The findings highlight the United States and China as prominent contributors, exhibiting substantial publication and citation metrics. Notably, the study concludes that QML, as a research subject, is currently in a formative stage, characterized by robust scholarly activity and ongoing development.
Quantum feature maps are a key component of quantum machine learning, encoding classical data into quantum states to exploit the expressive power of high-dimensional Hilbert spaces. Despite their theoretical promise, designing quantum feature maps that offer practical advantages over classical methods remains an open challenge. In this work, we propose an agentic system that autonomously generates, evaluates, and refines quantum feature maps using large language models. The system consists of five component: Generation, Storage, Validation, Evaluation, and Review. Using these components, it iteratively improves quantum feature maps. Experiments on the MNIST dataset show that it can successfully discover and refine feature maps without human intervention. The best feature map generated outperforms existing quantum baselines and achieves competitive accuracy compared to classical kernels across MNIST, Fashion-MNIST, and CIFAR-10. Our approach provides a framework for exploring dataset-adaptive quantum features and highlights the potential of LLM-driven automation in quantum algorithm design.
A central question in machine learning is how reliable the predictions of a trained model are. Reliability includes the identification of instances for which a model is likely not to be trusted based on an analysis of the learning system itself. Such unreliability for an input may arise from the model family providing a variety of hypotheses consistent with the training data, which can vastly disagree in their predictions on that particular input point. This is called the underdetermination problem, and it is important to develop methods to detect it. With the emergence of quantum machine learning (QML) as a prospective alternative to classical methods for certain learning problems, the question arises to what extent they are subject to underdetermination and whether similar techniques as those developed for classical models can be employed for its detection. In this work, we first provide an overview of concepts from Safe AI and reliability, which in particular received little attention in QML. We then explore the use of a method based on local second-order information for the detection of underdetermination in parameterized quantum circuits through numerical experiments. We further demonstrate that the approach is robust to certain levels of shot noise. Our work contributes to the body of literature on Safe Quantum AI, which is an emerging field of growing importance.
This paper introduces the quantum deep sets model, expanding the quantum machine learning tool-box by enabling the possibility of learning variadic functions using quantum systems. A couple of variants are presented for this model. The first one focuses on mapping sets to quantum systems through state vector averaging: each element of the set is mapped to a quantum state, and the quantum state of the set is the average of the corresponding quantum states of its elements. This approach allows the definition of a permutation-invariant variadic model. The second variant is useful for ordered sets, i.e., sequences, and relies on optimal coherification of tristochastic tensors that implement products of mixed states: each element of the set is mapped to a density matrix, and the quantum state of the set is the product of the corresponding density matrices of its elements. Such variant can be relevant in tasks such as natural language processing. The resulting quantum state in any of the variants is then processed to realise a function that solves a machine learning task such as classification, regression or density estimation. Through synthetic problem examples, the efficacy and versatility of quantum deep sets and sequences (QDSs) is demonstrated.
Parameterized Quantum Circuits (PQCs) with fixed structures severely degrade the performance of Quantum Machine Learning (QML). To address this, a Hybrid Quantum-Classical Classifier (HQCC) is proposed. It opens a practical way to advance QML in the Noisy Intermediate-Scale Quantum (NISQ) era by adaptively optimizing the PQC through a Long Short-Term Memory (LSTM) driven dynamic circuit generator, utilizing a local quantum filter for scalable feature extraction, and exploiting architectural plasticity to balance the entanglement depth and noise robustness. We realize the HQCC on the TensorCircuit platform and run simulations on the MNIST and Fashion MNIST datasets, achieving up to 97.12\% accuracy on MNIST and outperforming several alternative methods.
We present an extension of K-P time-optimal quantum control solutions using global Cartan $KAK$ decompositions for geodesic-based solutions. Extending recent time-optimal \emph{constant-$\theta$} control results, we integrate Cartan methods into equivariant quantum neural network (EQNN) for quantum control tasks. We show that a finite-depth limited EQNN ansatz equipped with Cartan layers can replicate the constant-$\theta$ sub-Riemannian geodesics for K-P problems. We demonstrate how for certain classes of control problem on Riemannian symmetric spaces, gradient-based training using an appropriate cost function converges to certain global time-optimal solutions when satisfying simple regularity conditions. This generalises prior geometric control theory methods and clarifies how optimal geodesic estimation can be performed in quantum machine learning contexts.
This work focuses on the limitations about the insufficient fitting capability of current quantum machine learning methods, which results from the over-reliance on a single data embedding strategy. We propose a novel quantum machine learning framework that integrates multiple quantum data embedding strategies, allowing the model to fully exploit the diversity of quantum computing when processing various datasets. Experimental results validate the effectiveness of the proposed framework, demonstrating significant improvements over existing state-of-the-art methods and achieving superior performance in practical applications.
In classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong data-processing inequality. Here, we investigate quantum Doeblin coefficients as a generalization of the classical concept. In particular, we define various new quantum Doeblin coefficients, one of which has several desirable properties, including concatenation and multiplicativity, in addition to being efficiently computable. We also develop various interpretations of two of the quantum Doeblin coefficients, including representations as minimal singlet fractions, exclusion values, reverse max-mutual and oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual and oveloH informations. Our interpretations of quantum Doeblin coefficients as either entanglement-assisted or unassisted exclusion values are particularly appealing, indicating that they are proportional to the best possible error probabilities one could achieve in state-exclusion tasks by making use of the channel. We also outline various applications of quantum Doeblin coefficients, ranging from limitations on quantum machine learning algorithms that use parameterized quantum circuits (noise-induced barren plateaus), on error mitigation protocols, on the sample complexity of noisy quantum hypothesis testing, on the fairness of noisy quantum models, and on mixing times of time-varying channels. All of these applications make use of the fact that quantum Doeblin coefficients appear in upper bounds on various trace-distance contraction coefficients of a channel. Furthermore, in all of these applications, our analysis using Doeblin coefficients provides improvements of various kinds over contributions from prior literature, both in terms of generality and being efficiently computable.
Quantum Machine Learning (QML) presents as a revolutionary approach to weather forecasting by using quantum computing to improve predictive modeling capabilities. In this study, we apply QML models, including Quantum Gated Recurrent Units (QGRUs), Quantum Neural Networks (QNNs), Quantum Long Short-Term Memory(QLSTM), Variational Quantum Circuits(VQCs), and Quantum Support Vector Machines(QSVMs), to analyze meteorological time-series data from the ERA5 dataset. Our methodology includes preprocessing meteorological features, implementing QML architectures for both classification and regression tasks. The results demonstrate that QML models can achieve reasonable accuracy in both prediction and classification tasks, particularly in binary classification. However, challenges such as quantum hardware limitations and noise affect scalability and generalization. This research provides insights into the feasibility of QML for weather prediction, paving the way for further exploration of hybrid quantum-classical frameworks to enhance meteorological forecasting.
The Transformer model, renowned for its powerful attention mechanism, has achieved state-of-the-art performance in various artificial intelligence tasks but faces challenges such as high computational cost and memory usage. Researchers are exploring quantum computing to enhance the Transformer's design, though it still shows limited success with classical data. With a growing focus on leveraging quantum machine learning for quantum data, particularly in quantum chemistry, we propose the Molecular Quantum Transformer (MQT) for modeling interactions in molecular quantum systems. By utilizing quantum circuits to implement the attention mechanism on the molecular configurations, MQT can efficiently calculate ground-state energies for all configurations. Numerical demonstrations show that in calculating ground-state energies for H2, LiH, BeH2, and H4, MQT outperforms the classical Transformer, highlighting the promise of quantum effects in Transformer structures. Furthermore, its pretraining capability on diverse molecular data facilitates the efficient learning of new molecules, extending its applicability to complex molecular systems with minimal additional effort. Our method offers an alternative to existing quantum algorithms for estimating ground-state energies, opening new avenues in quantum chemistry and materials science.
Self-attention has revolutionized classical machine learning, yet existing quantum self-attention models underutilize quantum states' potential due to oversimplified or incomplete mechanisms. To address this limitation, we introduce the Quantum Complex-Valued Self-Attention Model (QCSAM), the first framework to leverage complex-valued similarities, which captures amplitude and phase relationships between quantum states more comprehensively. To achieve this, QCSAM extends the Linear Combination of Unitaries (LCUs) into the Complex LCUs (CLCUs) framework, enabling precise complex-valued weighting of quantum states and supporting quantum multi-head attention. Experiments on MNIST and Fashion-MNIST show that QCSAM outperforms recent quantum self-attention models, including QKSAN, QSAN, and GQHAN. With only 4 qubits, QCSAM achieves 100% and 99.2% test accuracies on MNIST and Fashion-MNIST, respectively. Furthermore, we evaluate scalability across 3-8 qubits and 2-4 class tasks, while ablation studies validate the advantages of complex-valued attention weights over real-valued alternatives. This work advances quantum machine learning by enhancing the expressiveness and precision of quantum self-attention in a way that aligns with the inherent complexity of quantum mechanics.
With the growing interest in quantum machine learning, the perceptron -- a fundamental building block in traditional machine learning -- has emerged as a valuable model for exploring quantum advantages. Two quantum perceptron algorithms based on Grover's search, were developed in arXiv:1602.04799 to accelerate training and improve statistical efficiency in perceptron learning. This paper points out and corrects a mistake in the proof of Theorem 2 in arXiv:1602.04799. Specifically, we show that the probability of sampling from a normal distribution for a $D$-dimensional hyperplane that perfectly classifies the data scales as $\Omega(\gamma^{D})$ instead of $\Theta({\gamma})$, where $\gamma$ is the margin. We then revisit two well-established linear programming algorithms -- the ellipsoid method and the cutting plane random walk algorithm -- in the context of perceptron learning, and show how quantum search algorithms can be leveraged to enhance the overall complexity. Specifically, both algorithms gain a sub-linear speed-up $O(\sqrt{N})$ in the number of data points $N$ as a result of Grover's algorithm and an additional $O(D^{1.5})$ speed-up is possible for cutting plane random walk algorithm employing quantum walk search.
Amplitude embedding (AE) is essential in quantum machine learning (QML) for encoding classical data onto quantum circuits. However, conventional AE methods suffer from deep, variable-length circuits that introduce high output error due to extensive gate usage and variable error rates across samples, resulting in noise-driven inconsistencies that degrade model accuracy. We introduce EnQode, a fast AE technique based on symbolic representation that addresses these limitations by clustering dataset samples and solving for cluster mean states through a low-depth, machine-specific ansatz. Optimized to reduce physical gates and SWAP operations, EnQode ensures all samples face consistent, low noise levels by standardizing circuit depth and composition. With over 90% fidelity in data mapping, EnQode enables robust, high-performance QML on noisy intermediate-scale quantum (NISQ) devices. Our open-source solution provides a scalable and efficient alternative for integrating classical data with quantum models.
Quantum technologies are increasingly pervasive, underpinning the operation of numerous electronic, optical and medical devices. Today, we are also witnessing rapid advancements in quantum computing and communication. However, access to quantum technologies in computation remains largely limited to professionals in research organisations and high-tech industries. This paper demonstrates how traditional neural networks can be transformed into neuromorphic quantum models, enabling anyone with a basic understanding of undergraduate-level machine learning to create quantum-inspired models that mimic the functioning of the human brain -- all using a standard laptop. We present several examples of these quantum machine learning transformations and explore their potential applications, aiming to make quantum technology more accessible and practical for broader use. The examples discussed in this paper include quantum-inspired analogues of feedforward neural networks, recurrent neural networks, Echo State Network reservoir computing and Bayesian neural networks, demonstrating that a quantum approach can both optimise the training process and equip the models with certain human-like cognitive characteristics.
This paper proposes a single-qudit quantum neural network for multiclass classification, by using the enhanced representational capacity of high-dimensional qudit states. Our design employs an $d$-dimensional unitary operator, where $d$ corresponds to the number of classes, constructed using the Cayley transform of a skew-symmetric matrix, to efficiently encode and process class information. This architecture enables a direct mapping between class labels and quantum measurement outcomes, reducing circuit depth and computational overhead. To optimize network parameters, we introduce a hybrid training approach that combines an extended activation function -- derived from a truncated multivariable Taylor series expansion -- with support vector machine optimization for weight determination. We evaluate our model on the MNIST and EMNIST datasets, demonstrating competitive accuracy while maintaining a compact single-qudit quantum circuit. Our findings highlight the potential of qudit-based QNNs as scalable alternatives to classical deep learning models, particularly for multiclass classification. However, practical implementation remains constrained by current quantum hardware limitations. This research advances quantum machine learning by demonstrating the feasibility of higher-dimensional quantum systems for efficient learning tasks.
The recent advancements in noisy intermediate-scale quantum (NISQ) devices implementation allow us to study their application to real-life computational problems. However, hardware challenges are not the only ones that hinder our quantum computation capabilities. Software limitations are the other, less explored side of this medal. Using satellite image segmentation as a task example, we investigated how difficult it is to run a hybrid quantum-classical model on a real, publicly available quantum device. We also analyzed the costs of such endeavor and the change in quality of model.
Recent advancements in Single-Image Super-Resolution (SISR) using deep learning have significantly improved image restoration quality. However, the high computational cost of processing high-resolution images due to the large number of parameters in classical models, along with the scalability challenges of quantum algorithms for image processing, remains a major obstacle. In this paper, we propose the Quantum Image Enhancement Transformer for Super-Resolution (QUIET-SR), a hybrid framework that extends the Swin transformer architecture with a novel shifted quantum window attention mechanism, built upon variational quantum neural networks. QUIET-SR effectively captures complex residual mappings between low-resolution and high-resolution images, leveraging quantum attention mechanisms to enhance feature extraction and image restoration while requiring a minimal number of qubits, making it suitable for the Noisy Intermediate-Scale Quantum (NISQ) era. We evaluate our framework in MNIST (30.24 PSNR, 0.989 SSIM), FashionMNIST (29.76 PSNR, 0.976 SSIM) and the MedMNIST dataset collection, demonstrating that QUIET-SR achieves PSNR and SSIM scores comparable to state-of-the-art methods while using fewer parameters. These findings highlight the potential of scalable variational quantum machine learning models for SISR, marking a step toward practical quantum-enhanced image super-resolution.