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  1. Imagine you want to simulate a cubic meter down to the particle. By following the Standard Model and other basic physical equations, how much computing power would be required to do this, in say, a day?

  2. Would a quantum computer help you in this task?

  3. Could you somehow directly simulate the particles?

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This is clearly completely hopeless. Many particle systems in QFT or even in non-relativistic QM are computationally intractable. Just to store the many-body Schroedinger wave function of more than a few particles is out of the question (it is a function of $3^N$ variables, where $N$ for your problem would be $O(10^{23}))$.

In practice, lattice QCD is stuck at $N\sim 4$ (the Helium nucleus). First principles calculations in nuclear physics (starting from nuclear forces rather than QCD) are stuck at $N\sim 12$ (carbon). I'm not an expert in ab-initio quantum chemistry (which means ignoring QCD, only solving the QED part), but I think they can do on the order of $N\sim 100$.

There are some thoughts whether quantum computing might help with these problems. The answer is a resounding maybe, but again, even if it works, we are talking about $N\sim 100$ not $N\sim 10^{23}$, see for example here http://arxiv.org/pdf/1312.1695v3.pdf.

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Well, let's do some thinking here. You didn't specify what the cubic meter is composed of, but let's for sake of argument say that it is a gas of single, non-interacting particles (like, say, neutrons or something) at standard temperature and pressure. So we know we have 1 mole of gas in that cubic meter.

Let's further assume that we only needed a single, single-precision number to represent the particle (which is obviously not true, but just for sake of argument). So we have a single number that tells us everything we need to know about a particle. We have 6.022e23 particles in that cubic meter. If we assume the single precision number needs 4 bytes to represent it, that gives us 2.4e12 TB of RAM required to hold this single number!

To put that in some perspective, this article says that all of the devices in the world (in 2011) could store 295 exabytes of information. Our single cubic meter would require 2.5e6 exabytes! Or almost 10,000 times more storage than every storage medium on the planet. And this says nothing about the power needed to do anything with those numbers. It takes that much information just to store that single number for every particle, at a single instant in time.

Obviously, that's positively ridiculous. Never, ever, going to happen. I assumed we needed a single number but in reality we need at least 3 to represent velocity in each direction, probably need another to represent mass. If we assume them to be perfectly rigid billiard balls, that could be all we need. But we assumed the most basic kind of particle possible. As soon as you add in spin, orientations, moments of inertia, long-range forces, etc. then clearly the requirements are huge. And I also assumed we only need single precision. In all reality, we would likely need much more than that. Theoretically, we would need infinite precision because we need infinite resolution of the non-linear processes that occur.

So bottom line -- there is not likely going to be any way we can simulate a cubic meter of a material at STP. We can already do it if the material is very rarefied, or if we make many assumptions. Simulations can currently handle billions of atoms in a lattice, but only on tens of nanometer length scales and maybe, just maybe, tens of nanoseconds in time scales. And to do that, countless assumptions and simplifications are made to make the problem tractable.

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