Schrödingerization-based quantum simulation of some nonlinear scalar partial differential equations
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We consider the Schrödingerization method [S. Jin, N. Liu, and Y. Yu, PRA, 108, 032603, 2023] using a so-called warped phase transformation of linear partial differential equations (PDEs) for their quantum simulation. We provide a thorough analysis of the method when applied to nonlinear scalar PDEs by leveraging existing analytical mappings between the equation and some linear PDE. We investigate the properties of the method in terms of resolution and stability and propose ways to improve its efficiency. Specifically, we show that part of the information can be lost due to aliasing. To enhance the general accuracy of the simulation, we propose a method to choose the parameters of the simulation so as to impose the warped phase transport step on the discretization grid. We also provide a poly-depth circuit comprised of one-, two-, and three-qubit nonlocal Z-rotation gates. Numerical simulations on the Burgers' and Kardar-Parisi-Zhang equations support our analysis.
