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

2026-07-27

AI Summary

This work addresses the challenge of solving high-dimensional semilinear parabolic partial differential equations (PDEs) by introducing a novel quantum Transformer architecture. The proposed method reformulates the PDE as a backward stochastic differential equation (BSDE) and integrates, for the first time, causal self-attention with a fully connected variational quantum circuit (FC-VQC) to construct a hybrid quantum-classical model whose parameters reside entirely within the quantum module. By treating normalized state trajectories as spatiotemporal tokens, the model leverages causal attention to learn adaptive dynamic relationships inherent in the BSDE’s gradient process. Evaluated on three 36-dimensional PDE benchmarks, the QTransformer significantly outperforms an attention-free FC-VQC baseline and surpasses classical Transformers even with compact hidden dimensions, demonstrating its effectiveness and quantum advantage in modeling high-dimensional trajectories.

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.

Problem

• high-dimensional PDEs
• backward stochastic differential equations
• quantum computing
• parabolic partial differential equations
• stochastic control

Innovation

• Quantum Transformer
• Variational Quantum Circuits
• Backward Stochastic Differential Equations
• Causal Self-Attention
• High-Dimensional PDEs