Natansh Mathur

Natansh Mathur, Panagiotis Kl. Barkoutsos, Masako Yamada et al.

Training quantum neural networks (QNNs) on quantum hardware is currently bottlenecked by the cost of gradient estimation: standard parameter-shift methods require a number of circuit evaluations that grows quadratically with the number of trainable parameters, making hardware-based optimisation impractical beyond small system sizes. In this work, we introduce a training framework that reduces this cost to logarithmic in the number of qubits, making gradient-based QNN optimisation feasible on near-term hardware at increasing scales. Our framework combines three co-designed ingredients: (i) a structured, subspace-preserving Butterfly circuit architecture with $O(n \log n)$ parameters and logarithmic depth; (ii) a layer-wise training strategy that confines on-hardware optimisation to one small, well-structured layer at a time; and (iii) a parallelised parameter-shift rule that exploits the commuting structure within each Butterfly layer to extract all gradients in a constant number of circuit executions. Together these reduce the number of distinct circuit evaluations per optimisation step from $O(n^2)$ to $O(\log n)$. We validate the framework on clinical data imputation using the MIMIC-III electronic health record dataset, a demanding benchmark sensitive to optimisation instability and model variance. Hybrid classical-quantum models are trained directly on IonQ Forte Enterprise trapped-ion hardware at 16 qubits without performance degradation relative to ideal or noisy simulation and via tensor-network simulation at 32 qubits, with 32-qubit inference executed on hardware. The resulting models match or exceed strong classical neural baselines in downstream patient survival prediction while exhibiting reduced variance across runs, demonstrating that the proposed framework enables practical, scalable QNN training under realistic hardware constraints.

El Amine Cherrat, Iordanis Kerenidis, Natansh Mathur et al.

In this work, quantum transformers are designed and analysed in detail by extending the state-of-the-art classical transformer neural network architectures known to be very performant in natural language processing and image analysis. Building upon the previous work, which uses parametrised quantum circuits for data loading and orthogonal neural layers, we introduce three types of quantum transformers for training and inference, including a quantum transformer based on compound matrices, which guarantees a theoretical advantage of the quantum attention mechanism compared to their classical counterpart both in terms of asymptotic run time and the number of model parameters. These quantum architectures can be built using shallow quantum circuits and produce qualitatively different classification models. The three proposed quantum attention layers vary on the spectrum between closely following the classical transformers and exhibiting more quantum characteristics. As building blocks of the quantum transformer, we propose a novel method for loading a matrix as quantum states as well as two new trainable quantum orthogonal layers adaptable to different levels of connectivity and quality of quantum computers. We performed extensive simulations of the quantum transformers on standard medical image datasets that showed competitively, and at times better performance compared to the classical benchmarks, including the best-in-class classical vision transformers. The quantum transformers we trained on these small-scale datasets require fewer parameters compared to standard classical benchmarks. Finally, we implemented our quantum transformers on superconducting quantum computers and obtained encouraging results for up to six qubit experiments.

Paolo Marcandelli, Natansh Mathur, Stefano Markidis et al.

Solving partial differential equations remains a central challenge in scientific machine learning. Neural operators offer a promising route by learning mappings between function spaces and enabling resolution-independent inference, yet they typically require supervised data. Physics-informed neural networks address this limitation through unsupervised training with physical constraints but often suffer from unstable convergence and limited generalization capability. To overcome these issues, we introduce a multi-stage physics-informed training strategy that achieves convergence by progressively enforcing boundary conditions in the loss landscape and subsequently incorporating interior residuals. At each stage the optimizer is re-initialized, acting as a continuation mechanism that restores stability and prevents gradient stagnation. We further propose the Physics-Informed Spline Fourier Neural Operator (PhIS-FNO), combining Fourier layers with Hermite spline kernels for smooth residual evaluation. Across canonical benchmarks, PhIS-FNO attains a level of accuracy comparable to that of supervised learning, using labeled information only along a narrow boundary region, establishing staged, spline-based optimization as a robust paradigm for physics-informed operator learning.

Natansh Mathur, Brian Coyle, Nishant Jain et al.

Identification of defects or anomalies in 3D objects is a crucial task to ensure correct functionality. In this work, we combine Bayesian learning with recent developments in quantum and quantum-inspired machine learning, specifically orthogonal neural networks, to tackle this anomaly detection problem for an industrially relevant use case. Bayesian learning enables uncertainty quantification of predictions, while orthogonality in weight matrices enables smooth training. We develop orthogonal (quantum) versions of 3D convolutional neural networks and show that these models can successfully detect anomalies in 3D objects. To test the feasibility of incorporating quantum computers into a quantum-enhanced anomaly detection pipeline, we perform hardware experiments with our models on IBM's 127-qubit Brisbane device, testing the effect of noise and limited measurement shots.

Sohum Thakkar, Skander Kazdaghli, Natansh Mathur et al.

Quantum algorithms have the potential to enhance machine learning across a variety of domains and applications. In this work, we show how quantum machine learning can be used to improve financial forecasting. First, we use classical and quantum Determinantal Point Processes to enhance Random Forest models for churn prediction, improving precision by almost 6%. Second, we design quantum neural network architectures with orthogonal and compound layers for credit risk assessment, which match classical performance with significantly fewer parameters. Our results demonstrate that leveraging quantum ideas can effectively enhance the performance of machine learning, both today as quantum-inspired classical ML solutions, and even more in the future, with the advent of better quantum hardware.