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

TL;DR

This paper explores quantum linear programming algorithms to efficiently compute measure-valued solutions of nonlinear PDEs, showing potential polynomial advantages over classical methods in certain scenarios.

Contribution

It introduces quantum algorithms for solving the measure-valued formulation of nonlinear PDEs and analyzes their potential advantages over classical algorithms.

Findings

Quantum algorithms like the quantum central path method can have polynomial advantages over classical interior point methods.

No advantage is found for quantum algorithms when computing expected values of Young measures.

Polynomial advantage exists for quantum algorithms in solving random PDEs for Young measures.

Abstract

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.