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There are a number of simulators, which simulate the macroscopic world around us (Space Engine: http://en.spaceengine.org/ comes to mind, but there is also Universe Sandbox).

My question, does anything like this exist for the quantum world ? (I know the formulation of QFT is exactly saying that this isn't possible, but maybe an approximation is).

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Sure thing. People do simulations in quantum chemistry all the time, as well as lattice QCD, although I don't know of any accessible ones. I used to enjoy watching these simple QM simulations when I was in high school. The problem with simulating a "quantum world" is a matter of computing requirements.

Suppose you wanted to simulate just the spin of one particle, which is about the simplest quantum mechanical thing there is. Classically, the spin can either be up or down, so we need one bit of information to represent it's state (1 or 0). A quantum spin can be in a superposition of those states ($\alpha \uparrow + \beta \downarrow$), so we need a single complex number to store that amount of information (something like the quantity $\tfrac{\alpha}{\beta}$ would suffice). You need infinitely many bits to store a single complex number, whereas classically you only needed one. Even for an approximation, using double precision numbers you would need something like 128 bits to store a single complex number.

Taken on their own, these storage requirements aren't impressive at all. The point is just how much harder the problem has become for such a simple system (spin). To keep track of a particle's position in space, classically you would need only 3 numbers. To store the particle's wavefunction, you would now need to store a single complex number for each point in space (!).

This is only for a single particle. To store the data for $N$ particles, you would raise whatever number you got for one particle to the power of $N$. People have been making all sorts of approximations to make storage and computing requirements manageable, but classical computers really have a handicap here.

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