Lindbladian Homotopy Analysis Method to Solve Nonlinear Partial Differential Equations

TL;DR

This paper introduces the Lindbladian Homotopy Analysis Method (LHAM), a quantum approach for efficiently solving nonlinear PDEs by reducing dimensionality issues and enabling simulation of complex nonlinear dynamics.

Contribution

The paper proposes LHAM, a novel quantum differential equation solver that converts nonlinear PDEs into linear systems and embeds solutions in density matrices, reducing Hilbert space growth.

Findings

LHAM effectively solves nonlinear PDEs like Burgers' equation.

Hilbert space dimension in LHAM grows logarithmically with inverse error.

LHAM outperforms Carleman linearization and Koopman-von Neumann methods.

Abstract

Quantum scientific computing is to solve engineering and science problems such as simulation and optimization on quantum computers. Solving ordinary and partial differential equations (PDEs) is essential in simulations. However, existing quantum approaches to solve nonlinear PDEs suffer from the issues of curse of dimensionality and convergence during the linearization process. In this paper, a Lindbladian homotopy analysis method (LHAM) is proposed as a quantum differential equation solver to simulate non-unitary and nonlinear dynamics. The original nonlinear problem is first converted to a recursive sequence of linear PDEs with the homotopy analysis method and reformulated as a higher-dimensional lower block triangular linear homogeneous autonomous system. The solution is then embedded in the density matrix and obtained through the Lindbladian dynamics simulation. Compared to other methods such as Carleman linearization and the Koopman-von Neumann approach where the dimension of Hilbert space increases polynomially with the inverse of truncation error, the Hilbert space dimension in LHAM increases only logarithmically. LHAM is demonstrated with nonlinear PDEs including Burgers' equation and reduced magnetohydrodynamics equations.