One needs d^N complex parameters to represent the state of a quantum system of N particles each with d levels accessible. This grows brutally fast. For problems involving more than a few particles, it makes it impossible to even just store the quantum state on a computer. Solving the dynamics is of course even harder. This, in a nutshell, is what limits our ability to simulate generic problems in nuclear physics, in quantum chemistry, and in strongly correlated electrons. We have the equations for everything we could ever realize on earth, but we cannot simulate them.

One strategy, outlined by Feynman is to just use the laws of Nature itself to simulate such systems. After all, Nature can very well solve quantum dynamics. One should "just" make a universal simulator of such many-body quantum dynamics. This motivates intense efforts nowadays on quantum computers.

An alternative option is to ask if, by chance, these d^N complex parameters couldn't be stored in a compressed form, with far fewer numbers. We might then solve the dynamics in compressed form on a standard computer, even for many particles. Tensor networks are the best candidate for such a sparse representation, and currently allow to solve the quantum many body problem in instances that were thought completely hopeless. There are of course difficulties to apply them in general, and the quantum many body problem is not yet solved classically. I'll explain what tensor networks are, and where we stand right now.

INDICO