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.

Figures (14)

• Figure 1: Workflow of Quantum Physics-Informed Neural Networks (QPINNs).
• Figure 2: Circuit of $U_{\boldsymbol{\theta}}(x)$. Since the circuit has a alternating structure, we consider it to comprise $L$ layers, each consisting of $R_x(-2 \arccos (x))$ and $R_z\left(\theta_j\right)$, followed by an additional $R_z\left(\theta_L\right)$.
• Figure 3: The quantum model of Proposition \ref{['prop:univariat_poly']} can be considered as the Hadamard test to estimate $p(x)=\langle+|^{\otimes2} (|0\rangle\langle0|\otimes U_{\boldsymbol{\theta_{1}}}(x)+|1\rangle\langle1|\otimes U_{\boldsymbol{\theta_{2}}}(x))|+\rangle^{\otimes2}$ where $U_{\boldsymbol{\theta_{1}}}(x),U_{\boldsymbol{\theta_{2}}}(x)$ are defined as Equation \ref{['eq:u_theta_main']} and shown in Figure \ref{['fig:single_qubit_QSP_main']}. $H$ and $M$ denote the Hadamard gate and measurement operation, respectively; this notation applies throughout this work.
• Figure 4: The circuit $U^p(\boldsymbol{x})$ retains exactly the same structure as $U_{\boldsymbol{\theta}}(x)$ in Figure \ref{['fig:single_qubit_QSP_main']}, except that it acts on multiple qubits via tensor products.
• Figure 5: The circuit of $W_p(\boldsymbol{x})$ consists of a single-qubit system, a $\lceil D\log L\rceil$-qubits system $\mathcal{A}$, and a $D$-qubits system $\mathcal{S}$. This quantum model can consider as the Hadamard test to estimate $p(\boldsymbol x)=\langle+|^{\otimes \lceil D\log L\rceil}_{\mathcal{A}} \langle+|^{\otimes D}_{\mathcal{S}} U_c(\boldsymbol{x}) |+\rangle^{\otimes \lceil D\log L\rceil}_{\mathcal{A}} |+\rangle^{\otimes D}_{\mathcal{S}}$ such that $U_c(\boldsymbol{x})=\sum_{i=1}^{(L+1)^D}\ket{i}_\mathcal{A}\bra{i}\otimes U^{\boldsymbol{n}^{(i)}}_\mathcal{S}(\boldsymbol{x})$ where $U^{\boldsymbol{n}^{(i)}}_\mathcal{S}(\boldsymbol{x})$ is defined in Equation \ref{['eq:monomial_main']} and shown in Figure \ref{['fig:U^p_main']}.

Theorems & Definitions (20)

• Definition 1: Quantum Physics-Informed Neural Networks
• Definition 2: Polynomial
• Definition 3: Tensor-Decomposed Polynomial
• Proposition 1: Quantum circuit for univariate polynomial
• Theorem 2: Quantum circuit for multivariate polynomial
• Theorem 3: Quantum circuit for tensor-decomposed polynomial
• Corollary 4: Quantum circuit for rank-1 tensor-decomposed polynomial
• Lemma 5
• Corollary 6
• proof