Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum algorithms for Young measures -- applications to nonlinear partial differential equations

Nana Liu, Shi Jin, Yuhuan Yuan, Mária Lukáčová-Medviďová

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.2139/ssrn.6901772

Related papers

Quantum algorithms for nonlinear partial differential equations

Shi Jin, Nana Liu

Bulletin des Sciences Mathématiques•Jun 20, 2024•25 citations

Quantum algorithms for Young measures: applications to nonlinear partial differential equations

Shi Jin, Nana Liu, Maria Lukacova-Medvidova +1 more

Many nonlinear PDEs have singular or oscillatory solutions or may exhibit physical instabilities or uncertainties. This requires a suitable concept of physically relevant generalized solutions. Dissipative measure-valued solutions have been an effective analytical tool to characterize PDE behavior in such singular regimes. They have also been used to characterize limits of standard numerical schemes on classical computers. The measure-valued formulation of a nonlinear PDE yields an optimization problem with a linear cost functional and linear constraints, which can be formulated as a linear programming problem. However, this linear programming problem can suffer from the curse of dimensionality. In this article, we propose solving it using quantum linear programming (QLP) algorithms and discuss whether this approach can reduce costs compared to classical algorithms. We show that some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method. For problems where one is interested in the dissipative weak solution, namely the expected values of the Young measure, we show that the QLP algorithms offer no advantage over direct classical solvers. Moreover, for random PDEs, there can be up to polynomial advantage in obtaining the Young measure over direct classical PDE solvers. This is a significant advantage over standard PDE solvers, since the Young measure provides a more detailed description of the solution. We also propose some open questions for future development in this direction.

quant-pharXiv:2604.11825•Apr 14, 2026•0 citations

Quantum algorithms for uncertainty quantification: Applications to partial differential equations

François Golse, Shi Jin, Nana Liu

Science China Physics Mechanics and Astronomy•Aug 15, 2025•1 citation

Quantum homotopy analysis method with quantum-compatible linearization for nonlinear partial differential equations

Cheng Xue, Xiao‐Fan Xu, Xi‐Ning Zhuang +8 more

arXiv:2411.06759•Science China Physics Mechanics and Astronomy•Jun 18, 2025•6 citations

Solving Nonlinear Partial Differential Equations via a Hybrid Newton Method Using Quantum Linear System Solver

Maximilian Mandelt Buxadé, Stefan Langer, Philipp Bekemeyer

To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.

quant-pharXiv:2603.23258•Mar 25, 2026•0 citations

Finding Solutions of the Navier‐Stokes Equations through Quantum Computing—Recent Progress, a Generalization, and Next Steps Forward

Frank Gaitan

Abstract Efficient simulation of a quantum system's dynamics is expected to be an important application area for quantum computers as existing classical computers cannot do this. However, quantum systems are not unique in being hard to simulate. For example, classical nonlinear continuum systems and fields are governed by nonlinear partial differential equations whose solution is also hard for classical computers. Solving such equations is essential for many economically important industries/applications such as the aerospace industry, weather‐forecasting, fiber‐optics communication, and plasma magneto‐hydrodynamics. This raises the question: can a quantum computer speed‐up solving these equations? In this Review, a new quantum algorithm is described for solving nonlinear partial differential equations for which the answer is yes. First, a new quantum algorithm is discussed for solving the Navier‐Stokes nonlinear partial differential equations which govern the flow of a viscous fluid. Its construction, verification, and computational cost are described, and it is shown that a significant quantum speed‐up is possible. Its generalization to a quantum algorithm for solving nonlinear partial differential equations is described. The Review closes with a discussion of next steps forward. These new quantum algorithms open up a large new application area for quantum computing with substantial economic impact, including the trillion‐dollar aerospace industry.

Advanced Quantum Technologies•Sep 7, 2021•44 citations
Many nonlinear PDEs have singular or oscillatory solutions or may exhibit physical instabilities or uncertainties. This requires a suitable concept of physically relevant generalized solutions. Dissipative measure-valued solutions have been an effective analytical tool to characterize PDE behavior in such singular regimes. They have also been used to characterize limits of standard numerical schemes on classical computers. The measure-valued formulation of a nonlinear PDE yields an optimization problem with a linear cost functional and linear constraints, which can be formulated as a linear programming problem. However, this linear programming problem can suffer from the curse of dimensionality. In this article, we propose solving it using quantum linear programming (QLP) algorithms and discuss whether this approach can reduce costs compared to classical algorithms. We show that some QLP algorithms, such as the quantum central path algorithm, have up to polynomial advantage over the classical interior point method. For problems where one is interested in the dissipative weak solution, namely the expected values of the Young measure, we show that the QLP algorithms offer no advantage over direct classical solvers. Moreover, for random PDEs, there can be up to polynomial advantage in obtaining the Young measure over direct classical PDE solvers. This is a significant advantage over standard PDE solvers, since the Young measure provides a more detailed description of the solution. We also propose some open questions for future development in this direction.
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated computational fluid dynamic algorithms coupled with powerful supercomputers can not find approximate solutions for several design challenges in both adequate time and scale-resolving accuracy. One difficulty arises from solving high dimensional, strongly nonlinear partial differential equations, such as the Navier-Stokes equations, which capture the underlying physics. For nearly all classical algorithms, methods closely related to Newton's method are used to approximate a solution to the problem. Approximately solving the large-scale linear systems of equations occurring in this iterative scheme is generally a main contributor to the total computational complexity. In this paper a new quantum linear system solver supporting Newton's classical method to solve nonlinear partial differential equations is introduced. We present a new variant of the HHL algorithm, requiring less apriori information regarding the eigenvalues of the corresponding matrix. We apply this quantum linear system solver in a hybrid quantum-classical fashion to solve nonlinear partial differential equations. Moreover, a resource estimation for advanced use-cases of practical relevance is provided. Our results demonstrate how quantum computation may improve existing classical methodologies for solving nonlinear partial differential equations. This approach provides another promising application of quantum computers and presents a possible way forward for handling nonlinearities on inherently linear quantum systems.
Abstract Efficient simulation of a quantum system's dynamics is expected to be an important application area for quantum computers as existing classical computers cannot do this. However, quantum systems are not unique in being hard to simulate. For example, classical nonlinear continuum systems and fields are governed by nonlinear partial differential equations whose solution is also hard for classical computers. Solving such equations is essential for many economically important industries/applications such as the aerospace industry, weather‐forecasting, fiber‐optics communication, and plasma magneto‐hydrodynamics. This raises the question: can a quantum computer speed‐up solving these equations? In this Review, a new quantum algorithm is described for solving nonlinear partial differential equations for which the answer is yes. First, a new quantum algorithm is discussed for solving the Navier‐Stokes nonlinear partial differential equations which govern the flow of a viscous fluid. Its construction, verification, and computational cost are described, and it is shown that a significant quantum speed‐up is possible. Its generalization to a quantum algorithm for solving nonlinear partial differential equations is described. The Review closes with a discussion of next steps forward. These new quantum algorithms open up a large new application area for quantum computing with substantial economic impact, including the trillion‐dollar aerospace industry.