Solving high-dimensional parabolic partial differential equations (PDEs) is important in engineering, physics, and stochastic control. Deep BSDE methods reformulate semilinear PDEs as backward stochastic differential equations and admit a model-based reinforcement learning interpretation, where trajectories are generated from known stochastic dynamics while a trainable model learns the gradient-related control process. We propose a Quantum Transformer BSDE solver based on Multi-Layer Fully-Connected Variational Quantum Circuits (FC-VQC). The method treats the normalized state trajectory as time--coordinate tokens and applies causal self-attention to learn interactions in the adapted BSDE gradient process. All trainable model parameters are contained within the FC-VQC embedding, projection, feed-forward, and decoder modules, while attention and structural operations remain classical and parameter-free. Experiments on three d=36 PDE benchmarks show that QTransformer consistently improves over the non-attentive FC-VQC baseline and outperforms the classical Transformer at compact hidden widths, while the wider classical Transformer achieves the best overall accuracy. These results demonstrate that combining causal attention with FC-VQC provides an effective quantum architecture for high-dimensional BSDE trajectory learning.
Solving high-dimensional parabolic partial differential equations (PDEs) is important in engineering, physics, and stochastic control. Deep BSDE methods reformulate semilinear PDEs as backward stochastic differential equations and admit a model-based reinforcement learning interpretation, where trajectories are generated from known stochastic dynamics while a trainable model learns the gradient-related control process. We propose a Quantum Transformer BSDE solver based on Multi-Layer Fully-Connected Variational Quantum Circuits (FC-VQC). The method treats the normalized state trajectory as time--coordinate tokens and applies causal self-attention to learn interactions in the adapted BSDE gradient process. All trainable model parameters are contained within the FC-VQC embedding, projection, feed-forward, and decoder modules, while attention and structural operations remain classical and parameter-free. Experiments on three d=36 PDE benchmarks show that QTransformer consistently improves over the non-attentive FC-VQC baseline and outperforms the classical Transformer at compact hidden widths, while the wider classical Transformer achieves the best overall accuracy. These results demonstrate that combining causal attention with FC-VQC provides an effective quantum architecture for high-dimensional BSDE trajectory learning.
We propose a quantum machine learning framework for approximating solutions to high-dimensional parabolic partial differential equations (PDEs) that can be reformulated as backward stochastic differential equations (BSDEs). In contrast to popular quantum-classical network hybrid approaches, this study employs the pure Variational Quantum Circuit (VQC) as the core solver without trainable classical neural networks. The quantum BSDE solver performs pathwise approximation via temporal discretization and Monte Carlo simulation, framed as model-based reinforcement learning. We benchmark VQCbased and classical deep neural network (DNN) solvers on two canonical PDEs as representatives: the Black-Scholes and nonlinear Hamilton-Jacobi-Bellman (HJB) equations. The VQC achieves lower variance and improved accuracy in most cases, particularly in highly nonlinear regimes and for out-of-themoney options, demonstrating greater robustness than DNNs. These results, obtained via quantum circuit simulation, highlight the potential of VQCs as scalable and stable solvers for highdimensional stochastic control problems.
Variational quantum circuits (VQCs) are a leading approach to quantum machine learning on near-term devices, yet it remains unclear which circuit architecture yields the best accuracy-parameter trade-off on classical tabular data. We present a systematic empirical comparison of four VQC families -- multi-layer fully-connected (FC-VQC), residual (ResNet-VQC), hybrid quantum-classical transformer (QT), and fully quantum transformer (FQT) -- across five regression and classification benchmarks. Our key findings are: \textbf{(i)}~FC-VQCs achieve 90-96\% of the R2 of attention-based VQCs while using 40-50\% fewer parameters, and consistently outperform equal-capacity MLPs (mean R2=0.829 vs.\ MLP720's 0.753 on Boston Housing, 3-seed average); \textbf{(ii)}~FC-VQC's Type~4 inter-block connectivity provides partial cross-token mixing that approximates the role of attention -- explicit quantum self-attention yields only marginal gains on most datasets while significantly increasing parameter count; \textbf{(iii)}~expressibility saturates at circuit depth~≈3, explaining why shallow VQCs already cover the Hilbert space effectively; \textbf{(iv)}~LayerNorm on the fully quantum transformer improves classification accuracy, suggesting normalization is important when all operations are quantum; \textbf{(v)}~in our noise study on Boston Housing, FQT degrades gracefully under depolarizing noise while QT collapses. All results are validated across three random seeds. These findings provide practical architectural guidance for deploying VQCs on near-term quantum hardware.
Variational quantum circuits (VQCs) face an expressivity-trainability dilemma and scalability challenges. We propose Multi-Layer Fully-Connected Variational Quantum Circuits (FC-VQC), a general-purpose quantum machine learning framework that connects local VQC blocks through measurement, deterministic parameter-free routing, and re-encoding. All trainable model parameters reside within the quantum blocks, without trainable classical neural components. We study fully connected, sliding-window, and parallel block mixing and analyze computational costs, conditional error propagation, and block information exchange. Using classical simulation, we evaluate tabular regression and classification, with our main comparison addressing spatio-temporal function approximation for the Black--Scholes, Burgers, and time-dependent oscillatory PDEs with up to 144 spatial dimensions. Comparisons include neural-network, explicit angle-feature, tensor-network, and gradient-boosted-tree baselines. The results show that FC-VQC achieves the lowest trajectory relative MAE for all PDEs at the higher dimensions d∈{81,144}. Gradient-dynamics and depolarizing-noise experiments provide complementary empirical diagnostics of trainability and preliminary noise sensitivity.
Partial differential equations frequently appear in the natural sciences and related disciplines. In this work, we explore the potential for enhancing a classical deep-learning-based method for solving high-dimensional nonlinear partial differential equations with suitable quantum subroutines. In a first approach, we construct a deep-learning architecture based on variational quantum circuits without provable guarantees. In a second approach, tailored towards fault-tolerant quantum computers, find that quantum-accelerated Monte Carlo methods offer the potential to speed up the estimation of the loss function. In addition, we identify and analyze the trade-offs when using quantum-accelerated Monte Carlo methods to estimate the gradients with different methods, including a recently developed backpropagation-free forward gradient method. Finally, we discuss the usage of a suitable quantum algorithm for accelerating the training of feed-forward neural networks. Hence, this work provides different avenues with the potential for polynomial speedups for deep-learning-based methods for nonlinear partial differential equations.
Abstract The combination of Transformer architectures with Quantum Computing has opened new directions in Quantum Natural Language Processing (QNLP) by leveraging the Transformer’s representational capacity together with the computational parallelism and high-dimensional feature space of quantum systems. However, existing approaches remain constrained by partial designs and limited integration, hindering scalability and trainability. This study introduces FitQT—a novel, fully integrated, and trainable quantum transformer that unifies three essential components: (1) a quantum multi-head attention mechanism employing a scaled Gaussian kernel, (2) a quantum feed-forward network, and (3) quantum residual connections. The residual connections stabilize gradient flow and facilitate the training of deeper quantum circuits, while the attention and feed-forward components jointly enhance representational capacity. Evaluations on sentiment analysis datasets demonstrate FitQT’s effectiveness. On the IMDb dataset, FitQT improves accuracy by 14.89% over the baseline Quantum Self-Attention Network and achieves competitive task-level performance relative to BERT-Base, despite being trained from scratch without large-scale pretraining, while requiring only 2,595 parameters and substantially reduced training time. Moreover, FitQT achieves state-of-the-art performance compared to prior quantum-based transformer architectures such as QSANN and QMSAN. These results highlight the importance of incorporating classical deep learning principles into quantum architectures and provide a practical foundation for scalable, resource-efficient QNLP systems suitable for the noisy intermediate-scale quantum era.
The recent exploding growth in size of state-of-the-art machine learning models highlights a well-known issue where exponential parameter growth, which has grown to trillions as in the case of the Generative Pre-trained Transformer (GPT), leads to training time and memory requirements which limit their advancement in the near term. The predominant models use the so-called transformer network and have a large field of applicability, including predicting text and images, classification, and even predicting solutions to the dynamics of physical systems. Here we present a variational quantum circuit architecture named Self-Attention Sequential Quantum Transformer Channel (SASQuaTCh), which builds networks of qubits that perform analogous operations of the transformer network, namely the keystone self-attention operation, and leads to an exponential improvement in parameter complexity and run-time complexity over its classical counterpart. Our approach leverages recent insights from kernel-based operator learning in the context of predicting spatiotemporal systems to represent deep layers of a vision transformer network using simple gate operations and a set of multi-dimensional quantum Fourier transforms. To validate our approach, we consider image classification tasks in simulation and with hardware, where with only 9 qubits and a handful of parameters we are able to simultaneously embed and classify a grayscale image of handwritten digits with high accuracy.