Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here, we investigate the potential role of quantum circuits for solving PDEs. We design a parameterized quantum circuit (PQC) for implementing a polynomial based on tensor rank decomposition, reducing the quantum resource complexity from exponential to polynomial when the corresponding tensor rank is moderate. Building on this circuit, we develop a Quantum Physics-Informed Neural Network (QPINN) and a Quantum-inspired PINN, both of which guarantee the existence of an approximation of the PDE solution, and this approximation can be represented as a polynomial that incorporates tensor rank decomposition. Numerical experiments are conducted on the Hamilton--Jacobi--Bellman (HJB) PDE arising from the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. The results show that our quantum models achieve lower losses and approximation errors than a classical fully connected PINN while using substantially fewer trainable parameters. Our quantum models further outperform a classical PINN constructed to share a similar inductive bias, providing experimental evidence of quantum-induced improvement in the tested settings and highlighting a resource-efficient pathway toward classical and near-term quantum solvers for PDEs with exploitable solution structure.
Variational quantum Monte Carlo (VMC) combined with neural-network quantum states offers a novel angle of attack on the curse-of-dimensionality encountered in a particular class of partial differential equations (PDEs); namely, the real- and imaginary time-dependent Schrödinger equation. In this paper, we present a simple generalization of VMC applicable to arbitrary time-dependent PDEs, showcasing the technique in the multi-asset Black-Scholes PDE for pricing European options contingent on many correlated underlying assets.
Quantum neural networks have emerged as powerful models for approximating nonlinear functions. Yet their use in solving integro-differential equations (IDEs) and fractional integro-partial differential equations (FIPDEs), which involve inherently nonlocal operators, remains unexplored. This work introduces a quantum physics-informed neural network (QPINN) framework that combines a quantum neural network with the governing equations of general nonlinear IDEs and FIPDEs. The proposed quantum network uses an affine feature map and variational quantum circuits to produce trial solutions with explicit trigonometric structure. We prove a quantitative
L2(μ) universal approximation theorem for this architecture, achieving a convergence rate of
O(n−1/2). This extends classical Fourier approximation theory to quantum circuits for physics-informed learning. We propose two QPINN variants: the numerical-quadrature QPINN (N-QPINN), which handles nonlocal integrals and fractional operators via high-order numerical quadrature while computing local derivatives through automatic differentiation of quantum trial solutions; and the auxiliary-function QPINN (A-QPINN), which eliminates numerical quadrature by introducing auxiliary variables that reformulate each integro-differential equation as an equivalent coupled system of partial differential equations, enabling a multi-output quantum neural network to simultaneously represent the solution and its associated variables. A series of numerical experiments demonstrates that the proposed QPINN framework accurately captures the behavior of nonlinear IDEs and FIPDEs and outperforms classical physics-informed neural networks.