Full disclosure, I don't work on processors but am in a tangentially related field.
However, I don't feel that you need to have an analytical solution to the Schrödinger equation. In fact even in chemistry we don't do analytical solutions instead using fancy basis sets which allow us to do approximations.
Regardless, I don't think even that is particularly necessary, as quantum at that level means you basically have some level of leakage where the electrons can just tunnel through the barrier created by the transistor when off. So if I had to guess most of it is just ways to rectify this leakage so it doesn't effect calculations, probably similar to a form of error correcting.
(This ignores that you may have to do some initial quantum calculations using Density Functional Theorem to get a guess at how much leakage based off the materials you are using, though if I had to guess most of that work was done a while ago.)
However, I don't feel that you need to have an analytical solution to the Schrödinger equation. In fact even in chemistry we don't do analytical solutions instead using fancy basis sets which allow us to do approximations.
Regardless, I don't think even that is particularly necessary, as quantum at that level means you basically have some level of leakage where the electrons can just tunnel through the barrier created by the transistor when off. So if I had to guess most of it is just ways to rectify this leakage so it doesn't effect calculations, probably similar to a form of error correcting.
(This ignores that you may have to do some initial quantum calculations using Density Functional Theorem to get a guess at how much leakage based off the materials you are using, though if I had to guess most of that work was done a while ago.)