On Quantum BSDE Solver for High-Dimensional Parabolic PDEs

ArXiv ID: 2506.14612 “View on arXiv”

Authors: Howard Su, Huan-Hsin Tseng

Abstract

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.

Keywords: Variational Quantum Circuit (VQC), Backward Stochastic Differential Equations (BSDE), Quantum Machine Learning, Hamilton-Jacobi-Bellman (HJB), Black-Scholes, Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is densely packed with advanced mathematics, including deep stochastic calculus, PDE theory, and quantum circuit formulations, with rigorous theoretical derivations. However, the empirical results are limited to numerical simulations without backtesting, real market data, or implementation details for live trading, placing it firmly in the theoretical research domain.

  flowchart TD
    A["Research Goal<br/>Develop a pure Variational Quantum Circuit<br/>solver for high-dimensional PDEs"] --> B["Methodology: Quantum BSDE Solver"]
    
    B --> C["Data: PDE Benchmarks"]
    C --> C1["Black-Scholes Eq.<br/>Linear"]
    C --> C2["Nonlinear HJB Eq.<br/>High-dim control"]
    
    B --> D["Key Processes"]
    D --> D1["Pathwise Approx.<br/>via Temporal Discretization"]
    D --> D2["Model-Based RL<br/>with Monte Carlo Paths"]
    
    B --> E["Computational Core<br/>Pure VQC (No Classical NNs)"]
    
    E --> F["Outcomes vs Classical DNN"]
    
    F --> G["Lower Variance<br/>(Better Stability)"]
    F --> H["Improved Accuracy<br/>(Nonlinear & Out-of-Money)"]
    
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        L2["Methodology"]
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        L5["Outcome"]
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