Yes, essentially that's right (regarding Child's paper), as far as I can tell. Child's article seems to be about a smart way to approximate sufficiently dissipa

Yes, essentially that's right (regarding Child's paper), as far as I can tell. Child's article seems to be about a smart way to approximate sufficiently dissipative, non-linear, n-dimensional ODEs (or spatially discretised PDEs for that matter), such that you can apply good old forward Euler. They do that in such a manner that you end up solving a linear system for time-stepping for which a log(n) quantum algo is known. This is pretty cool, because they basically use standard numerical analysis tools and manage to plug in the quantum computing part to show that, e.g., as you increase the spatial "resolution" with which you solve a PDE, the cost only increases logarithmically. This feels useful from a computational fluid dynamics perspective, say, if you're not interested in super turbulent flows. So this is more about "how do we make solving hard PDEs faster" than "finally cracking non-linear PDEs".