Comparative Analysis on Two Quantum Algorithms for Solving the Heat Equation

TL;DR

The paper surveys two post-2020 quantum PDE solvers for the 1D heat equation, comparing Costa et al.’s discrete-adiabatic-quantum-walk approach with Oz–San–Kara’s Chebyshev-QAEA pipeline. It analyzes discretization, conditioning, block-encoding or teleological problem framing, and the costs of extracting classical observables, concluding that in 1D the presumed quantum advantage is neutralized by read-out overhead when matching a single observable like total heat. The work highlights how κ scaling and state-preparation/read-out costs shape practical performance and emphasizes the influence of dimensionality, suggesting higher-dimensional settings may better realize quantum speedups. It provides guidance on where future gains could come from, including observable-aware solvers, variance reduction in read-out, and hybrid discretizations, while noting the FFT baseline remains a strong classical competitor in 1D.

Abstract

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.