Strongly correlated quantum systems are among the most intriguing and fundamental systems in physics. One such example is the Tonks-Girardeau gas, proposed about 40 years ago, but until now lacking experimental realization; in such a gas, the repulsive interactions between bosonic particles confined to one dimension dominate the physics of the system. In order to minimize their mutual repulsion, the bosons are prevented from occupying the same position in space. This mimics the Pauli exclusion principle for fermions, causing the bosonic particles to exhibit fermionic properties. However, such bosons do not exhibit completely ideal fermionic (or bosonic) quantum behaviour; for example, this is reflected in their characteristic momentum distribution. Here we report the preparation of a Tonks-Girardeau gas of ultracold rubidium atoms held in a two-dimensional optical lattice formed by two orthogonal standing waves. The addition of a third, shallower lattice potential along the long axis of the quantum gases allows us to enter the Tonks-Girardeau regime by increasing the atoms' effective mass and thereby enhancing the role of interactions. We make a theoretical prediction of the momentum distribution based on an approach in which trapped bosons acquire fermionic properties, finding that it agrees closely with the measured distribution.

This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.

The treatment of high-dimensional problems such as the Schr€ odinger equation can be approached by concepts of tensor product approximation. We present general techniques that can be used for the treatment of high-dimensional optimization tasks and time-dependent equations, and connect them to concepts already used in many-body quantum physics. Based on achievements from the past decade, entanglement-based methods-developed from different perspectives for different purposes in distinct communities already matured to provide a variety of tools-can be combined to attack highly challenging problems in quantum chemistry. The aim of the present paper is to give a pedagogical introduction to the theoretical background of this novel field and demonstrate the underlying benefits through numerical applications on a text book example. Among the various optimization tasks, we will discuss only those which are connected to a controlled manipulation of the entanglement which is in fact the key ingredient of the methods considered in the paper. The selected topics will be covered according to a series of lectures given on the topic "New wavefunction methods and entanglement optimizations in quantum

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