On Quantum BSDE Solver for High-Dimensional Parabolic PDEs

TL;DR

This paper introduces a quantum machine learning framework using Variational Quantum Circuits to solve high-dimensional parabolic PDEs reformulated as BSDEs, showing improved accuracy and robustness over classical neural networks in simulations.

Contribution

It presents a novel quantum BSDE solver that employs pure VQCs without classical neural networks, demonstrating advantages in high-dimensional stochastic control problems.

Findings

VQC-based solver achieves lower variance than DNNs.

VQC demonstrates greater robustness in nonlinear regimes.

Quantum approach shows potential scalability for high-dimensional PDEs.

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.