Solving in the sense of an analytic, closed form solution to the non-relativistic Schrodinger equation (itself an approximation to QFT) using commonly accepted elementary functions. Even this requires actually evaluating the elementary functions to some degree of numerical accuracy if you want digits.

I often see this claim that only hydrogen is solvable, but I find it misleading. We have algorithms that will give solutions to the non-relativistic Schrodinger equation for helium to whatever degree of decimal precision you like (see FCI QMC methods) — how long these algorithms take to run is a different matter (see the fermion sign problem), but for helium it’s not too bad. The algorithms are unbiased, which I consider to be an exact solution.

You can think of it in the same way that we have Monte Carlo solutions to the rendering equation for global illumination. They are not closed form in terms of arbitrary elementary functions, but they converge to the exact solution over time.