Lindbladian Homotopy Analysis Method to Solve Nonlinear Partial Differential Equations

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.