Comparative Analysis on Two Quantum Algorithms for Solving the Heat Equation

As of now, an optimal quantum algorithm solving partial differential equations eludes us. There are several different methods, each with their own strengths and weaknesses. In past years comparisons of these existing methods have been made, but new work has emerged since then. Therefore, we conducted a survey on quantum methods developed post-2020, applying two such solvers to the heat equation in one spatial dimension. By analyzing their performance (including the cost of classical extraction), we explore their precision and runtime efficiency advancements between the two, identifying advantages and considerations.