Lindbladian Homotopy Analysis Method to Solve Nonlinear Partial Differential Equations
For quantum computing researchers, LHAM offers a more efficient quantum algorithm for nonlinear PDEs, addressing the curse of dimensionality and convergence issues.
The paper proposes the Lindbladian homotopy analysis method (LHAM) to solve nonlinear PDEs on quantum computers, achieving logarithmic growth in Hilbert space dimension with respect to inverse truncation error, compared to polynomial growth in existing methods. Demonstrated on Burgers' equation and magnetohydrodynamics equations.
Quantum scientific computing is to solve engineering and science problems such as simulation and optimization on quantum computerss. 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 nonunitary and nonlinear dynamics. The original nonlinear problem is first converted to a recursive sequence of linear PDEs with the homotopy analysis and reformulated as a higher-dimensional lower block triangular linear homogeneous system. The solution is then embedded in the density matrix and obtained through the Lindbladian dynamics simulation. Compared to other methods such as the Carleman linearization and Koopman-von Neumann approach where the dimension of Hilbert space increases polynomially with the inverse of truncation error, the Hilbert space in LHAM increases only logarithmically. LHAM is demonstrated nonlinear PDEs including Burgers' equation and magnetohydrodynamics equations.