"Solve the theory" is bit of a wish-washy expression. It depends on what you are interested in. What it really means is that it provides an approximation-free prescription for extracting observables from the QCD action.

Given some experimentally-measured observables (the pion mass, the kaon mass, etc. as long as you include 1 observable per parameter in QCD, meaning 1 gauge coupling + 1 per quark flavor you include in your simulation) you can calculate all other quantities---in a world that is pure QCD. We are now getting to an era where we can do lattice QCD+QED, but we cannot do lattice-standard-model because of the issue with chiral gauge theories mentioned in the article. However, since the weak sector is perturbative, we can combine pen-and-paper/Feynman-diagram-style calculations with LQCD calculations to get what we want: first-principles Standard Model predictions.

To discover new physics you need both an experimental signal and a precision prediction from the Standard Model. If they differ that difference is something not in the Standard Model.

What kinds of quantities can we calculate? The first major success was a calculation of the hadronic spectrum [spectrum]. We've also seen precision determinations of the QCD phase diagram (vacuum properties as a function of temperature) [phase diagram] and other properties that are important for the early universe, although there is a computational resources / scaling problem for nonzero density. With QED included there's a calculation of the proton-neutron mass splitting [mass splitting]. In terms of hadronic matrix elements we can compute the axial coupling of the neutron [gA], neutral kaon mixing, matrix elements needed for nuclear calculations of neutrinoless double beta decay [0vbb], all sorts of stuff. The FLAG review [flag] collects and evaluates individual results and tries to construct a community consensus for our "best values" for certain observables.

> Is it possible to state and solve initial value problem for, say, proton-proton scattering?

You picked one of the most difficult problems in lattice QCD (and one I spent a lot of time on as a postdoc): extracting baryon-baryon scattering. At high energy actually there is a lot of progress because QCD factorizes and you get a parton picture [PDFs]. But at low energy the situation is worse. There are a few issues:

1. We calculate in a finite box, typically with periodic boundary conditions (think PacMan). So there is no way to make protons "asymptotically far apart", and once they're heading apart, they're heading towards their next encounter. This conceptual difficulty was solved in the late 1980s: what you can do is take the energy of the finite-volume standing waves and turn that into phase shifts (quantum-mechanical scattering data).

2. Our calculations are done in Euclidean time, for reasons I have explained / touched on on hn before

https://news.ycombinator.com/item?id=15782932

https://news.ycombinator.com/item?id=23403883

So we don't have real Minkowski time; you can think: instead of things evolving like exp(i H t) things evolve like exp(- H t), where H is the Hamiltonian.

3. Baryons suffer from a signal-to-noise issue, where the longer we look in time the worse the variance gets. The source of this issue is understood and we have variational methods to get reliable signals earlier in time. The signal-to-noise problem is easier at heavier quark (or pion) masses (which we, as computational physicists can change even though experimentalists cannot).

For a long time we sharpened our tools at ~800 MeV pion masses. Current state of the art is down around 300 MeV.

This sounds dismal but I want to say: we have the whole pipeline working end-to-end. There are phenomenal results in the meson sector [meson scattering]. There's very-close-to-physical-parameters meson-baryon scattering [n-pi].

SO: is it possible to solve the initial value problem for proton-proton scattering using LQCD?

Not yet but work is in progress.

What these calculations give are phase shifts (and inelasticities) as a function of scattering momentum.

Once these are under control we can try to get a handle on the three-body sector; the three-body force is needed for accurate calculations in nuclear physics.

[spectrum] https://arxiv.org/abs/0906.3599 see especially figure 3

[phase diagram] https://arxiv.org/abs/1309.5258

[mass splitting] https://arxiv.org/abs/1406.4088

[gA] https://arxiv.org/abs/1805.12130

[0vbb] https://arxiv.org/abs/1805.02634

[flag] https://arxiv.org/abs/2111.09849

[PDFs] https://arxiv.org/abs/1711.07916 and a lot of progress has been made since then.

[meson scattering] https://arxiv.org/abs/1708.06667

[n-pi] https://arxiv.org/abs/2208.03867