The power of available computers has now grown exponentially for many decades. The ability to discover numerically the implications of equations and models has opened our eyes to previously hidden aspects of physics.

Many exciting phenomena observed in condensed matter systems, such as superconductivity and the quantum Hall effect, emerge due to the quantum mechanical interplay of many electrons. The laws of quantum physics are governed by the Schrödinger equation, whose complexity grows exponentially with the number of particles it describes. Hence, even an approximate numerical solution of the Schrödinger equation is impossible for only just a few particles, not to mention for the millions of particles that are present in real materials. This talk focuses on a new approximation scheme in terms of so-called Tensor Network States, which allow for an arbitrarily accurate description of realistic quantum solid state systems at merely a polynomial overhead in the particle number, thus enabling efficient simulations of such systems on today's computers.