Quantum Transformer BSDE Solver via Multi-Layer Fully-Connected Variational Quantum Circuits

Abstract

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.