Quantum DeepONet: Neural operators accelerated by quantum computing

Pengpeng Xiao; Muqing Zheng; Anran Jiao; Xiu Yang; Lu Lu

doi:10.22331/q-2025-06-04-1761

Quantum DeepONet: Neural operators accelerated by quantum computing

Pengpeng Xiao¹, Muqing Zheng², Anran Jiao¹, Xiu Yang², and Lu Lu^1,3

¹Department of Statistics and Data Science, Yale University, New Haven, CT 06511, USA
²Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015, USA
³Wu Tsai Institute, Yale University, New Haven, CT 06510, USA

Published:	2025-06-04, volume 9, page 1761
Editor:	Jin-Peng Liu
Eprint:	arXiv:2409.15683v2
Doi:	https://doi.org/10.22331/q-2025-06-04-1761
Citation:	Quantum 9, 1761 (2025).

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Abstract

In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as deep operator network (DeepONet), which learn mappings between infinite-dimensional function spaces, promise efficient computation of PDE solutions for a new condition in a single forward pass. However, classical DeepONet entails quadratic complexity concerning input dimensions during evaluation. Given the progress in quantum algorithms and hardware, here we propose to utilize quantum computing to accelerate DeepONet evaluations, yielding complexity that is linear in input dimensions. Our proposed quantum DeepONet integrates unary encoding and orthogonal quantum layers. We benchmark our quantum DeepONet using a variety of PDEs, including the antiderivative operator, advection equation, and Burgers' equation. We demonstrate the method's efficacy in both ideal and noisy conditions. Furthermore, we show that our quantum DeepONet can also be informed by physics, minimizing its reliance on extensive data collection. Quantum DeepONet will be particularly advantageous in applications in outer loop problems which require exploring parameter space and solving the corresponding PDEs, such as uncertainty quantification and optimal experimental design.

Featured image: Architecture of quantum DeepONet. DeepONet consists of two subnetworks: the branch net and the trunk net. In quantum DeepONet, we replace these with QOrthoNN, which is composed of several quantum layers arranged sequentially. The addition of biases and nonlinear operations are performed on classical computers.

Popular summary

From climate modeling to aerospace engineering, many scientific and engineering tasks rely on solving partial differential equations (PDEs). These equations describe how physical quantities evolve in space and time, but solving them is notoriously expensive and time-consuming, especially under varying conditions.

Recent developments in machine learning have offered a new approach: neural operators, such as DeepONet, which can learn to solve entire families of PDEs efficiently. Once trained, DeepONet can produce solutions for new conditions in a single step. However, its evaluation cost grows quickly with input size, limiting its scalability in high-dimensional settings.

Our work introduces Quantum DeepONet, a new framework that leverages the power of quantum computing to accelerate this process. By integrating quantum circuit components into the DeepONet architecture, we reduce the evaluation complexity from quadratic to linear with respect to input dimension. This is achieved through a hybrid design: the network is trained classically but evaluated quantumly, combining the strengths of both paradigms.

Quantum DeepONet is not only fast. It can also be physics-informed, meaning it respects the underlying physical laws of the system, reducing the need for large datasets. We demonstrate its effectiveness on a range of PDEs, such as the advection and Burgers' equations, and analyze its robustness under realistic quantum noise. This approach opens a pathway to efficient scientific computing in the quantum era, especially for applications that require repeated PDE solves, such as uncertainty quantification, real-time control, and design optimization.

► BibTeX data

@article{Xiao2025quantumdeeponet,
  doi = {10.22331/q-2025-06-04-1761},
  url = {https://doi.org/10.22331/q-2025-06-04-1761},
  title = {Quantum {D}eep{ON}et: {N}eural operators accelerated by quantum computing},
  author = {Xiao, Pengpeng and Zheng, Muqing and Jiao, Anran and Yang, Xiu and Lu, Lu},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1761},
  month = jun,
  year = {2025}
}

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