Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917400304156672 |
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| author | Hewage, Prasad Nimantha Madusanka Ukwatta Chakkravarthy, Midhun Abeysekara, Ruvan Kumara |
| author_facet | Hewage, Prasad Nimantha Madusanka Ukwatta Chakkravarthy, Midhun Abeysekara, Ruvan Kumara |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_09957 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions Hewage, Prasad Nimantha Madusanka Ukwatta Chakkravarthy, Midhun Abeysekara, Ruvan Kumara Quantum Physics 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. |
| title | Mitigating Barren Plateaus in Variational Quantum Circuits through PDE-Constrained Loss Functions |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2604.09957 |