Dumb question: with all we know about theoretical quantum physics, couldn't we compute what happens in these extreme conditions? Or is it still computationally intractable? If so, why is that?
So instead we go the other way: use "close enough" experimental data and perturbation theory to approximate the results for more complex systems.
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.
I described the method we use and some of the computational difficulties in some previous threads
https://news.ycombinator.com/item?id=15780514
https://news.ycombinator.com/item?id=12048170
But, if you're comfortable with an argument from authority, suffice it to say that much of the computational power in the largest supercomputing centers all across the world are dedicated to quantum many-body problems.
If you want equilibrium properties at temperature T [non equilibrium properties, like real-time dynamics have additional exponentially-bad computational intractabilities, known as the sign problem], you want to evaluate a partition function
Z = tr[ exp( - H / T ) ]
We have importance-sampling Markov Chain Monte Carlo techniques for evaluating Z. For some problems it succeeds with flying colors, although demonstrating that you're in the thermodynamic [infinite number of particles and infinite volume] limit requires calculating at a variety of volumes and particle numbers, for extrapolation. If you just do random updates to your state vector, most updates will be rejected; use HMC to update the whole state at once [HMC = Hybrid Monte Carlo, or Hamiltonian Monte Carlo, if you're not a computational physicist, strangely].
This approach is not naively parallelizable, and the machines where HMC is performed requires extremely high-performance interconnects, think 10-100 times faster than what's available in a standard datacenter.
The method is inherently statistical: give me more computing time and I can shrink the error bars. But if the problem has certain technical difficulties (or is in real-time, for example, rather than equilibrium), you encounter the sign problem, where the partition function you want to evaluate now becomes
Z = tr[ exp( i H / T ) ]