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This paper presents QPI-DeepONet-MAC, a hybrid architecture that theoretically resolves key trainability issues like barren plateaus in quantum-enhanced physics-informed deep operator networks. By establishing rigorous bounds on gradients and sensor requirements, the work provides a scalable framework for solving parametric partial differential equations using combined classical and quantum representations. The results offer a concrete theoretical foundation for integrating quantum circuits into high-dimensional scientific computing tasks without sacrificing stability or universal approximation capabilities.
QPI-DeepONet-MAC: A Scalable and Stable Hybrid Classical-Quantum Architecture for Physics-Informed Deep Operator Networks
arXiv:2610.01824v1 Announce Type: new Abstract: General operator learning for parametric partial differential equations (PDEs) is a fundamental challenge at the intersection of artificial intelligence and physics-based modeling. Physics-informed Deep Operator Networks (PI-DeepONets) incorporate governing equations into learning, but face optimization difficulties in high dimensions. Quantum extensions offer additional representational capacity, yet their trainability can be hindered by barren plateaus. Here, we introduce QPI-DeepONet-MAC, a hybrid quantum-classical architecture that integrates parameterized quantum circuits into PI-DeepONets through a multiplicative-and-additive coupling (MAC) of classical representations and quantum expectation values. We prove analytically that the proposed architecture retains universal operator approximation capabilities. Furthermore, we derive bounds on quantum parameter gradients, establishing explicit conditions to avoid barren plateaus alongside a scaling criterion for trainability. Additionally, we derive a lower bound on the number of sensors required for a target accuracy, directly linking sampling requirements to input function regularity, domain dimension, and PDE dynamical growth. Finally, we propose an informational regularization scheme based on quantum coherence and state fidelity dynamics to prevent coherence loss and preserve quantum structure during training. Together, these results establish a theoretical foundation for quantum-enhanced physics-informed operator learning and position QPI-DeepONet-MAC as a scalable, stable framework for solving parametric PDEs.
Abstract:General operator learning for parametric partial differential equations (PDEs) is a fundamental challenge at the intersection of artificial intelligence and physics-based modeling. Physics-informed Deep Operator Networks (PI-DeepONets) incorporate governing equations into learning, but face optimization difficulties in high dimensions. Quantum extensions offer additional representational capacity, yet their trainability can be hindered by barren plateaus. Here, we introduce QPI-DeepONet-MAC, a hybrid quantum-classical architecture that integrates parameterized quantum circuits into PI-DeepONets through a multiplicative-and-additive coupling (MAC) of classical representations and quantum expectation values. We prove analytically that the proposed architecture retains universal operator approximation capabilities. Furthermore, we derive bounds on quantum parameter gradients, establishing explicit conditions to avoid barren plateaus alongside a scaling criterion for trainability. Additionally, we derive a lower bound on the number of sensors required for a target accuracy, directly linking sampling requirements to input function regularity, domain dimension, and PDE dynamical growth. Finally, we propose an informational regularization scheme based on quantum coherence and state fidelity dynamics to prevent coherence loss and preserve quantum structure during training. Together, these results establish a theoretical foundation for quantum-enhanced physics-informed operator learning and position QPI-DeepONet-MAC as a scalable, stable framework for solving parametric PDEs.
arXiv:2610.01824 [quant-ph]
(or arXiv:2610.01824v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.01824 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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