Grover's Oracle for the Shortest Vector Problem and Its Application in Hybrid Classical–Quantum Solvers

Microscopic thermal machines promise to play an important role in future quantum technologies. Making such devices widely applicable will require effective strategies to channel their output into easily accessible storage systems like classical degrees of freedom. Here, we develop a self-consistent theoretical framework that makes it possible to model such quantum-classical hybrid devices in a thermodynamically consistent manner. Our approach is based on the assumption that the quantum part of the device is subject to strong decoherence and dissipation induced by a thermal reservoir. Due to the ensuing separation of time scales between slowly evolving classical and fast relaxing quantum degrees of freedom, the dynamics of the hybrid system can be described by means of adiabatic-response theory. We show that, upon including fluctuations in a minimally consistent way, the resulting equations of motion can be equipped with a first and second law, both on the ensemble level and on the level of individual trajectories of the classical part of the system, where thermodynamic quantities like heat and work become stochastic variables. As an application of our theory, we work out a physically transparent model of a quantum-classical hybrid engine, whose working system consists of a chain of Rydberg atoms, which is confined in an optical cavity and driven by periodic temperature variations. We demonstrate through numerical simulations that the engine can sustain periodic oscillations of a movable mirror, which acts as a classical load, against external friction and extract the full distributions of input heat and output work. By making the statistics of thermodynamic processes in quantum-classical hybrid systems accessible without the need to further specify a measurement protocol, our work contributes to bridging the long-standing gap between classical and quantum stochastic thermodynamics.

We address the limitations of variational quantum circuits (VQCs) in hybrid classical-quantum transfer learning by introducing post-variational strategies, which reduce training overhead and mitigate optimization issues. Our approach Post Variational Classical Quantum Transfer Learning (PVCQTL) includes three designs: (1) modified observable construction, (2) a hybrid approach, and (3) a variational-post-variational combination. We evaluate these on pre-trained models (VGG19, ResNet50, ResNet18, MobileNet) for 4 and 8 qubits, with ResNet50 performing best in deepfake detection. Compared to classical models (MLP, ResNet50) and quantum baselines hybrid quantum classical neural network (HQCNN), classical-quantum transfer learning (CQTL). PVCQTL consistently achieves better accuracy. The modified observable variant reaches 85% accuracy for Deepfake dataset with lower computational cost. To evaluate generalizability, we tested PVCQTL on three additional binary classification datasets, observing improved accuracy on each. We conducted ablation studies to assess the effects of architectural choices on quantum component variations, including the choice of quantum gates, use of fixed ansatz circuits, and observable measurements. Robustness to input noise and sensitivity of the PVCQTL models were examined through ablation studies on learning rate, batch size, and number of qubits. These results demonstrate that PVCQTL offers a measurable improvement over traditional hybrid classical-quantum approaches.

Deep Neural Networks have offered numerous innovative solutions to brain-related diseases including Alzheimer’s. However, there are still a few standpoints in terms of diagnosis and planning that can be transformed via quantum Machine Learning (QML). In this study, we present a hybrid classical–quantum machine learning model for the detection of Alzheimer’s using 6400 labeled MRI scans with two classes. Hybrid classical–quantum transfer learning is used, which makes it possible to optimally pre-process complex and high-dimensional data. Classical neural networks extract high-dimensional features and embed informative feature vectors into a quantum processor. We use resnet34 to extract features from the image and feed a 512-feature vector to our quantum variational circuit (QVC) to generate a four-feature vector for precise decision boundaries. Adam optimizer is used to exploit the adaptive learning rate corresponding to each parameter based on first- and second-order gradients. Furthermore, to validate the model, different quantum simulators (PennyLane, qiskit.aer and qiskit.basicaer) are used for the detection of the demented and non-demented images. The learning rate is set to 10−4 for and optimized quantum depth of six layers, resulting in a training accuracy of 99.1% and a classification accuracy of 97.2% for 20 epochs. The hybrid classical–quantum network significantly outperformed the classical network, as the classification accuracy achieved by the classical transfer learning model was 92%. Thus, a hybrid transfer-learning model is used for binary detection, in which a quantum circuit improves the performance of a pre-trained ResNet34 architecture. Therefore, this work offers a method for selecting an optimal approach for detecting Alzheimer’s disease. The proposed model not only allows for the automated detection of Alzheimer’s but would also speed up the process significantly in clinical settings.

Cardiovascular diseases are among the major health problems that are likely to benefit from promising developments in quantum machine learning for medical imaging. The chest X-ray (CXR), a widely used modality, can reveal cardiomegaly, even when performed primarily for a non-cardiological indication. Based on pre-trained DenseNet-121, we designed hybrid classical-quantum (CQ) transfer learning models to detect cardiomegaly in CXRs. Using Qiskit and PennyLane, we integrated a parameterized quantum circuit into a classic network implemented in PyTorch. We mined the CheXpert public repository to create a balanced dataset with 2436 posteroanterior CXRs from different patients distributed between cardiomegaly and the control. Using k-fold cross-validation, the CQ models were trained using a state vector simulator. The normalized global effective dimension allowed us to compare the trainability in the CQ models run on Qiskit. For prediction, ROC AUC scores up to 0.93 and accuracies up to 0.87 were achieved for several CQ models, rivaling the classical-classical (CC) model used as a reference. A trustworthy Grad-CAM++ heatmap with a hot zone covering the heart was visualized more often with the QC option than that with the CC option (94% vs. 61%, p < 0.001), which may boost the rate of acceptance by health professionals.

The advancements in the field of quantum machine learning and the ability of quantum-enhanced networks to acquire a higher effective dimension are promising for remote sensing applications and synthetic aperture radar (SAR) data. This paper aims to incorporate variational quantum circuits with existing classification-based deep learning state-of-the-art architectures. Tests are proposed by using ResNets, DenseNets, and a SqueezeNet-based convolutional neural network. Various dressed quantum circuits associated with these encoders are created and tested for novel applicability analysis. A famous publicly available X-Band SAR image-based dataset called moving and stationary target acquisition and recognition (MSTAR) is utilized to demonstrate image-centric target classification. The utility of quantum-enhanced and hybrid classical-quantum transfer learning models is provided.

The automatic detection of Alzheimer’s disease (AD) using 3D volumetric MRI data is a complex, multi-domain challenge that has traditionally been addressed by training classical convolutional neural networks (CNNs). With the rise of quantum computing and its potential to replace classical systems in the future, there is a growing need to: (i) develop automated systems for AD detection that run on quantum computers, (ii) explore the capabilities of current-generation classical-quantum architectures, and (iii) identify their potential limitations and advantages. To reduce the complexity of multi-domain expertise while addressing the emerging demands of quantum-based automated systems, our contribution in this paper is twofold. First, we introduce a simple preprocessing framework that converts 3D MRI volumetric data into 2D slices. Second, we propose CQ-CNN, a parameterized quantum circuit (PQC)-based lightweight hybrid classical-quantum convolutional neural network that leverages the computational capabilities of both classical and quantum systems. Our experiments on the OASIS-2 dataset reveal a significant limitation in current hybrid classical-quantum architectures, as they face difficulties converging when class images are highly similar, such as between moderate dementia and non-dementia classes of AD, which leads to gradient failure and optimization stagnation. However, when convergence is achieved, the quantum model demonstrates a promising quantum advantage by attaining state-of-the-art accuracy with far fewer parameters than classical models. For instance, our -3-qubit model achieves 97.5% accuracy using only 13.7K parameters (0.05 MB), which is 5.67% higher than a classical model with the same parameter count. Nevertheless, our results highlight the need for improved quantum optimization methods to support the practical deployment of hybrid classical-quantum models in AD detection and related medical imaging tasks.

Hybrid classical–quantum Convolutional Neural Network for stenosis detection in X-ray coronary angiography

This paper presents a system for solving binary-valued linear equations using quantum computers. The system is called Mod2VQLS, which stands for Modulo 2 Variational Quantum Linear Solver. As far as we know, this is the first such proposal. The design is a classical–quantum hybrid. The quantum components are a new circuit design for implementing matrix multiplication modulo 2, and a variational circuit to be optimized. The classical components are the optimizer, which measures the cost function and updates the quantum parameters for each iteration, and the controller that runs the quantum job and classical optimizer iterations. We propose two alternative ansatze or templates for the variational circuit and present results showing that the rotation ansatz designed specifically for this problem provides the most direct path to a valid solution. Numerical experiments in low dimensions indicate that Mod2VQLS, using the custom rotations ansatz, is on par with the block Wiedemann algorithm, which is the best-known to date solution for this problem.

Blockchain has revolutionized numerous fields, which include financial services, health care, the Internet of things, academia and supply chain management. Blockchain technology enables us to have an immutable, distributed ledger for managing the transactions of untrusted users. However, the technology has many open challenges, such as privacy leaks, scalability, and energy consumption. User identity can be easily tracked using network analysis, as transactions are accessible to everyone, which is a serious concern of blockchain. In this paper, we propose a new efficient, privacy-preserving, and quantum-resistant key generation algorithm, namely, lattice-based hierarchical deterministic key generation (LB-HDKG), for maintaining user privacy in the public blockchain. The LB-HDKG scheme generates many cryptographic keys in a tree-like structure from a single seed to hide the links between transactions of the same user. Our proposal uses the lattice NTRU cryptosystem, the security of which relies on the shortest vector problem (SVP) and closest vector problem (CVP) over the polynomial ring. Operations on the lattice NTRU cryptosystem are efficient and secure against classical computers and quantum computers. Security and performance analyses of our scheme show that the model is more secure and efficient and should replace current models to safeguard data from quantum computers.

Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman–von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical–quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect—the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical–quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.

The classical neural network has provided remarkable results to diagnose neurological disorders against neuroimaging data. However, in terms of efficient and accurate classification, some standpoints need to be improved by utilizing high-speed computing tools. By integrating quantum computing phenomena with deep neural network approaches, this study proposes an AlexNet–quantum transfer learning method to diagnose neurodegenerative diseases using magnetic resonance imaging (MRI) dataset. The hybrid model is constructed by extracting an informative feature vector from high-dimensional data using a classical pre-trained AlexNet model and further feeding this network to a quantum variational circuit (QVC). Quantum circuit leverages quantum computing phenomena, quantum bits, and different quantum gates such as Hadamard and CNOT gate for transformation. The classical pre-trained model extracts the 4096 features from the MRI dataset by using AlexNet architecture and gives this vector as input to the quantum circuit. QVC generates a 4-dimensional vector and to transform this vector into a 2-dimensional vector, a fully connected layer is connected at the end to perform the binary classification task for a brain disorder. Furthermore, the classical–quantum model employs the quantum depth of six layers on pennyLane quantum simulators, presenting the classification accuracy of 97% for Parkinson’s disease (PD) and 96% for Alzheimer’s disease (AD) for 25 epochs. Besides this, pre-trained classical neural models are implemented for the classification of disorder and then, we compare the performance of the classical transfer learning model and hybrid classical–quantum transfer learning model. This comparison shows that the AlexNet–quantum learning model achieves beneficial results for classifying PD and AD. So, this work leverages the high-speed computational power using deep network learning and quantum circuit learning to offer insight into the practical application of quantum computers that speed up the performance of the model on real-world data in the healthcare domain.

Simulating quantum dynamics is expected to be performed more easily on a quantum computer than on a classical computer. However, the currently available quantum devices lack the capability to implement fault-tolerant quantum algorithms for quantum simulation. Hybrid classical quantum algorithms such as the variational quantum algorithms have been proposed to effectively use current term quantum devices. One promising approach to quantum simulation in the noisy intermediate-scale quantum (NISQ) era is the diagonalisation based approach, with some of the promising examples being the subspace variational quantum simulator (SVQS), variational fast forwarding (VFF), fixed-state variational fast forwarding (fs-VFF), and the variational Hamiltonian diagonalisation (VHD) algorithms. However, these algorithms require a feedback loop between the classical and quantum computers, which can be a crucial bottleneck in practical application. Here, we present the classical quantum fast forwarding (CQFF) as an alternative diagonalisation based algorithm for quantum simulation. CQFF shares some similarities with SVQS, VFF, fs-VFF and VHD but removes the need for a classical-quantum feedback loop and controlled multi-qubit unitaries. The CQFF algorithm does not suffer from the barren plateau problem and the accuracy can be systematically increased. Furthermore, if the Hamiltonian to be simulated is expressed as a linear combination of tensored-Pauli matrices, the CQFF algorithm reduces to the task of sampling some many-body quantum state in a set of Pauli-rotated bases, which is easy to do in the NISQ era. We run the CQFF algorithm on existing quantum processors and demonstrate the promise of the CQFF algorithm for current-term quantum hardware. We compare CQFF with Trotterization for a XY spin chain model Hamiltonian and find that the CQFF algorithm can simulate the dynamics more than 105 times longer than Trotterization on current-term quantum hardware. This provides a 104 times improvement over the previous record.

We describe how multidimensional linear diffusion with application to image processing could be carried out on a hybrid classical–quantum computer. We present the quantum-lattice-gas-based algorithms, their effective finite difference approximations and representative simulation results. The methods for two-dimensional diffusion processing are particularly relevant to image enhancement tasks. We additionally demonstrate an extension to constrained linear diffusion that provides for non-uniform image smoothing. The simulation results and accompanying analysis support the utility of both classical and classical–quantum lattice gases for image enhancement. The diffusion modelling has applicability to many other fields including biological and physical science.

Generalized algebraic transformations and exactly solvable classical-quantum models

This article presents a first end-to-end application of a quantum support vector machine (QSVM) algorithm for a classification problem in the financial payment industry using the IBM Safer Payments and IBM Quantum Computers via the Qiskit software stack. Based on real card payment data, a thorough comparison is performed to assess the complementary impact brought in by the current state-of-the-art quantum machine-learning algorithms with respect to the classical approach. A new method to search for best features is explored using the QSVM's feature map characteristics. The results are compared using fraud-specific key performance indicators, i.e., accuracy, recall, and false positive rate, extracted from analyses based on human expertise (such as rule decisions), classical machine-learning algorithms (such as random forest and XGBoost), and quantum-based machine-learning algorithms using QSVM. In addition, a hybrid classical–quantum approach is explored by using an ensemble model that combines classical and quantum algorithms to better improve the fraud prevention decision. We found, as expected, that the results highly depend on feature selections and algorithms that are used to select them. The QSVM provides a complementary exploration of the feature space that led to an improved accuracy of the mixed quantum-classical method for fraud detection, on a drastically reduced dataset to fit current state of quantum hardware.