The procedure is essentially an evaluation of a millions-of-dimensions-large integral via Markov-chain Monte Carlo. Each step in the Markov chain requires inverting extremely large ill-conditioned matrices, which we can make better-conditioned by moving away from physical parameters (and then trying to extrapolate). People are glad to have only hundreds or thousands of independent samples in an ensemble. You need ensembles for multiple lattice spacings and spatial volumes in order to ensure you can control your discretization and finite-volume artifacts when you extrapolate to the continuum and to infinite volume.
What is generated in the Markov chain is essentially the configurations of the gluons and quark loops. With these configurations in hand we can measure physical observables on each.
Observables with baryons have an exponentially bad signal-to-noise problem in the temporal direction (time on the grid, not Monte Carlo time) and you either need an exponentially large computer, an exponentially long time, or an exponentially wealthy funding agency. There are attempts to circumvent this issue, we'll see how successful they prove...
I know a few QFT types - they tend to be decently intelligent and mathematically capable. Ok, they're more like wickedly smart...
So, basically a lack of computing resources. QCs aren't generally that much faster than classical ones, but in this particular case they are--understandably so, given the universe runs on quantum mechanics. Turing machines are not good enough for this, and no number of Intel CPUs will let us overcome that barrier.
Physicists try really hard anyway, but the approximations are... Well. As I said, awful.
Usually we only bring up Turing machines for decidability, ie when the only thing that matters is whether it takes finite or infinite amounts of time.
(Yes, I know, you can also calculate exact runtimes with Turing machines, but nobody really cares, I think?)
I guess you mean to say classical computer or so?
> [...] and no number of Intel CPUs will let us overcome that barrier.
Or at least that's everybody's best guess. On similar footing to the empirical evidence for P != NP. It's still theoretically possible that BQP = BPP, if memory serves right. (https://en.wikipedia.org/wiki/BQP vs https://en.wikipedia.org/wiki/BPP_(complexity))
Most of the time the different ways to model computing only have a polynomial slowdown to emulate each other. In that sense, quantum computers are different from Turing machines: the best way we know to emulate quantum computers on Turing machines has exponential slow downs.
Solving chess (in the strict sense) is a similar example. E.g. all 6-piece endgames were known in 2005; by 2012 we had the complete 7-piece endgame tablebase, but the 8-piece endgame tablebase is such a hard problem that we're nowhere close even with Moore's law; the sheer size of that tablebase is estimated to be ~10 petabytes.
I believe the core difference here between QCD and QED (the classical perturbative QFT) is that QED has a very small coupling constant (it's the fine structure constant, so ~1/137) whereas QCD has a large one (comparable to or greater than one.) It turns out that a perturbative expansion is effectively in powers of the coupling constant, so QED expansions converge rapidly...and QCD expansions converge not at all.
At high (collider) energies, it is okay to work perturbatively.
One way of calculating observables in quantum field theory is by performing a path integral. That is, an integral over all configurations of the fields in your system. We start with a partition function
Z = ∫ D[fields] e^{i S[fields]}
S[fields] = ∫ dSpacetime L[fields[spacetime]]
We can calculate expecation values of an observable O, which is a functional of the fields, by taking averages.
< O[fields] > = 1/Z ∫ D[fields] e^{i S[fields]} O[fields]
< O[fields] > = 1/Z ∫ D[fields] e^{-S[fields]} O[fields]
S_lattice[fields] = ∑[Spacetime] L[fields[spacetime]]
This produces an ensemble of N configurations. Because our configurations are generated according to the weight e^{-S} we can estimate simply:
< O[fields] > = 1/N ∑[field configuration] O[fields]
+ error of order 1/sqrt(N)
Once we have an ensemble of field configurations, we can measure all sorts of observables on them. Typically one measures order parameters or correlation functions, but it depends on what physics you're interested in.
What approximations have I made? First, I calculate on a finite, discrete spacetime volume. Second, I don't have an infinite budget, so I have statistical uncertainty in my result, much like an experimentalist. In fact, we usually say we "measure" the observable O. Of course we're not interrogating nature, but a system of axioms (or something). For our purposes, a computer is like a telescope, but it lets you see into the Platonic world.
To control the continuum limit, you need to redo the above Monte Carlo procedure a few times with different lattice spacings (holding the hadronic spectrum fixed). To control the infinite-volume extrapolation you need to redo the above MC procedure a few times with different numbers of lattice sites but otherwise fixed parameters.
I have neglected some details. For example, there are different ways to discretize, and some ways are better than others (but typically that corresponds to a more expensive method). Depending on your discretization you may have different input parameters you can fiddle with. In QCD, the QCD coupling constant and the quark masses are the ones you can fiddle with. Take some set of parameters and compute, for example, the hadronic spectrum. If the ratios of the masses match those we see in nature, we rejoice. We can then compare the mass computed from the lattice (which is a dimensionless number) and compare it to the masses in the real world (which are measured in GeV). The conversion factor is the lattice spacing (usually on the scale of .1 to 0.05 fm, which we convert to GeV using hbar=1), m_lattice = m_physical * a_lattice. If you are near the continuum limit, anything dimensionful you need to measure on the lattice will simply be converted by the correct powers of the lattice spacing, and any errors will scale like some higher power of the lattice spacing. That this is true is essentially a reflection of the fact that QCD has a negative beta function and becomes weakly coupled / perturbative at large energies / short distances. In theories without a UV fixed point you cannot take a continuum limit---in perturbative language, this is the Landau pole coming to bite you.
As an aside, from this perspective, Wilson's renormalization program is much less mysterious: if you want to hold physics fixed and change your lattice spacing, you'd better change the bare, dimensionless parameters you stuck into your action.
Calculations with baryons suffer from a severe signal-to-noise problem, and the latest calculation of a substantially-easier-to-compute quantity, the nucleon isovector axial coupling, is only at ~1% precision: https://arxiv.org/abs/1710.06523 (full disclosure: one author of said calculation).
I don't know of a calculation of the proton magnetic moment with physical parameters.