Learning PDEs for Portfolio Optimization with Quantum Physics-Informed Neural Networks

Abstract: 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 is represented as a polynomial that incorporates tensor rank decomposition. Despite using 80 times fewer parameters in experiments, our quantum models achieve higher accuracy and faster convergence than a classical fully connected PINN when solving the PDE for the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. Our quantum models further outperform a classical PINN constructed to share the same inductive bias, providing experimental evidence of quantum-induced improvement and highlighting a resource-efficient pathway toward classical and near-term quantum PDE solvers.

• The paper introduces a QPINN framework that leverages tensor-decomposed PQCs to efficiently solve high-dimensional PDEs in portfolio optimization.
• It provides rigorous theoretical guarantees while reducing quantum resource complexity from exponential to polynomial scaling via tensor rank constraints.
• Empirical results demonstrate faster convergence and higher accuracy using as few as 6-7 parameters compared to traditional PINN models.

Learning PDEs for Portfolio Optimization Using Quantum Physics-Informed Neural Networks

Problem Context and Motivation

Partial differential equations (PDEs) are the central analytic tools for modeling and optimization in financial mathematics, particularly for portfolio strategies under stochastic dynamics. The Merton portfolio problem, a paradigmatic example, translates the search for optimal portfolio allocation strategies into the solution of a high-dimensional Hamilton-Jacobi-Bellman (HJB) PDE. Traditional deterministic solvers (e.g., FDM, FEM) suffer from high computational cost in achieving precise solutions, especially as dimensionality or complexity grows. Physics-Informed Neural Networks (PINNs) provide a mesh-free, data-driven approach, but are inherently limited by slow convergence and representational inefficiency, particularly for solutions with separable or high-frequency structures.

The quantum computing paradigm promises exponential advantages in expressive capacity and state space manipulation, raising the question of whether quantum-inspired or hybrid quantum-classical neural approaches can deliver more efficient or accurate solutions to generalized PDEs in finance—especially within the constraints of noisy intermediate-scale quantum (NISQ) devices.

Technical Contributions

This work introduces a new Quantum Physics-Informed Neural Network (QPINN) framework, together with a quantum-inspired PINN variant, that leverages parameterized quantum circuits (PQCs) to represent solutions to PDEs with a focus on portfolio optimization. The central methodological innovation is the explicit construction of quantum circuits realizing polynomials based on tensor rank decomposition (referred to as tensor-decomposed polynomials). This decomposition exploits the separable structure typical of many financial PDE solutions, lowering quantum resource complexity from exponential—in the number of variables and polynomial degree—to polynomial, provided the tensor rank remains moderate.

Crucially, the hypothesis space induced by these QPINN models can be rigorously characterized, and it is guaranteed to contain tensor-decomposed polynomials of chosen degree and bounded rank that approximate the target PDE solution. This theoretical guarantee is missing from most prior quantum variational algorithms for PDEs, which often employ hardware-efficient ansatzes or re-uploading architectures with poorly understood expressivity.

Model Architecture and Quantum Resource Complexity

The proposed QPINN architecture consists of two major elements:

1. Tensor-Decomposed PQC Model: For a $D$-dimensional PDE, the solution is approximated as a sum over $R$ products of univariate polynomial components. This leads to PQCs with resource (width, depth, parameter count) scaling as $O(\mathrm{poly}(RDL))$, in stark contrast to the $O(L^D)$ scaling for general multivariate polynomial representations. The detailed quantum resource analysis is tailored both for platforms that support multi-controlled rotations natively (e.g., neutral atom systems—offering circuit depths $O(RDL)$), and for standard superconducting architectures, which decompose these operations into single- and two-qubit gates.
2. Entangling Layer for Enhanced Expressivity: For the full QPINN, an entangling unitary is appended after the tensor-decomposed block, enabling the hypothesis space to transcend strictly separable forms while preserving inclusion of the tensor-decomposed family. This allows the model to represent a broader class of PDE solutions and introduces quantum correlations inaccessible to classical simulators. In the quantum-inspired PINN variant, the entangling layer is omitted or set to identity, making the entire architecture classically simulable while maintaining the resource-efficient decomposition.

Theoretical Guarantees and Inductive Bias

One of the primary theoretical claims is that for any continuous PDE solution that admits a tensor-decomposed polynomial approximation (a property that holds for many financial, physical, and engineering problems), the QPINN's hypothesis space will necessarily contain a sufficiently good approximation. This guarantee is explicit and quantifiable in terms of the polynomial degree and tensor rank. The result establishes a lower bound on expressivity and addresses a critical gap in the literature on quantum variational algorithms, namely, the lack of guarantees that the circuit hypothesis space can in principle represent the true solution.

The tensor-decomposed structure imposes a strong inductive bias, exploiting the known separability of many portfolio optimization PDEs, notably the Merton HJB. The inductive bias leads to increased efficiency and accuracy compared to fully connected architectures with orders of magnitude more parameters.

Numerical Results and Empirical Claims

The models are empirically validated on the classical Merton portfolio optimization HJB PDE:

• The QPINN and quantum-inspired PINN (with only 7 and 6 parameters, respectively) achieve higher accuracy and faster convergence than both a classical PINN constrained to the same inductive (tensor product) bias, and a standard fully connected PINN with 481 parameters.
• The QPINN achieves the lowest final loss, outperforming the quantum-inspired PINN, classical tensor-decomposed PINN, and FC PINN, demonstrating practical expressivity benefits attributed to the entangling quantum layer.
• All models with tensor-decomposed bias exhibit significant training and generalization improvement over classical black-box approaches, confirming the practical value of well-aligned hypothesis design in PINN architectures.

Notably, the QPINN and quantum-inspired PINN converge faster and more accurately, even with drastically fewer trainable parameters, underlining the crucial role of hypothesis bias over sheer network width or depth.

Practical and Theoretical Implications

The study supports several substantive implications:

• Advancement Toward NISQ-Era Quantum PDE Solvers: By curtailing the required circuit complexity for a useful class of PDEs, this approach offers a resource-aware route to deployable quantum-assisted PDE solvers on near-term hardware.
• Classic-Quantum Continuum: The demonstrable advantage of "quantum-inspired" decompositions, even when simulated classically, suggests that importing quantum architectural design into classical deep learning can unlock new solution spaces for complex PDEs without hardware limitations.
• Rigorous Hypothesis Space Characterization: By furnishing lower-bounded expressivity guarantees rooted in polynomial approximation theory, this work advances the theoretical understanding of VQAs and QML for scientific computing.
• Extension Beyond Finance: Since tensor-decomposed PDE solutions are ubiquitous (e.g., in heat, Schrödinger, and Laplace equations), the methodology is applicable to a broad range of problems beyond financial mathematics.

Future Research Directions

Outstanding questions remain regarding trainability and optimization landscape analysis, particularly in relation to barren plateau phenomena for expressive quantum ansatzes. Further, the effect of entanglement depth on hypothesis space geometry and gradient concentration warrants detailed investigation, as does the explicit characterization of approximation error bounds in finite-noise, limited-qubit regimes. Additionally, more sophisticated choices of entangling Hamiltonians and adaptive tensor rank mechanisms could enhance both expressivity and trainability, potentially ushering in practical quantum advantages as hardware matures.

Conclusion

This paper introduces a QPINN architecture for portfolio optimization PDEs, underpinned by tensor rank decomposition and polynomial circuit construction, achieving theoretically justified lower resource complexity and superior empirical performance compared to standard classical PINNs. The work bridges quantum resource theory, numerical PDE analysis, and neural architecture design, with significant implications for scalable and expressive physics-informed learning on both near-term quantum and classical hardware (2604.03346).