Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions

• URL: http://arxiv.org/abs/2604.09957v1
• Date: Fri, 10 Apr 2026 23:36:59 GMT
• Title: Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions
• Abstract summary: We show that partial differential equation (PDE) constraints into the VQC loss function provide a natural and effective mitigation mechanism against barren plateaus.<n>We find that PDE-constrained circuits exhibit favorable gradient variance scaling with system size, with the physics constraints creating a stabilizing effect that resists exponential gradient vanishing.<n>These results establish PDE constraints as a principled, physically motivated strategy for designing trainable variational quantum circuits.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The barren plateau phenomenon; where cost function gradients vanish exponentially with system size; remains a fundamental obstacle to training variational quantum circuits (VQCs) at scale. We demonstrate, both theoretically and numerically, that embedding partial differential equation (PDE) constraints into the VQC loss function provides a natural and effective mitigation mechanism against barren plateaus. We derive analytical gradient variance lower bounds showing that physics-constrained loss functions composed of local PDE residuals evaluated at spatial collocation points inherit the favorable polynomial scaling of local cost functions, while additionally benefiting from constraint-induced landscape narrowing that concentrates gradient information. Systematic numerical experiments on the one-dimensional heat equation, Burgers' equation, and the Saint-Venant shallow water equations quantify the gradient variance across 4-8 qubits and 1-5 layer depths, comparing global cost, local cost, PDE-constrained, and PDE-constrained with structured ansatz configurations. We find that PDE-constrained circuits exhibit favorable gradient variance scaling with system size, with the physics constraints creating a stabilizing effect that resists exponential gradient vanishing. Entanglement entropy analysis reveals that structured ansatze operate in a sub-maximal entanglement regime consistent with trainability. Convergence experiments confirm that physics-constrained VQCs achieve lower loss values in fewer epochs. These results establish PDE constraints as a principled, physically motivated strategy for designing trainable variational quantum circuits, with direct implications for quantum physics-informed neural networks and variational quantum simulation.

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