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## Tensor network methods for quantum chemistry

#### CECAM-ETHZ HĂ¶nggerberg Campus, Zurich, Switzerland

#### Organisers

The key goal of quantum chemistry is the accurate calculation of geometrical and electronical ground state properties of molecules as well of their excited states. To this purpose, density functional theory is by far the most successful and flexible method. However, density functional theory can only be carried out by the use of an exchange-correlation functional which takes into account electronic correlation effects in the molecules. Unfortunately, this functional is only known approximately, and an important topic in quantum chemistry is the calculation of correlation effects in molecules („post-Hartree-Fock“ calculations). A (so-called) full configuration interaction (CI) calculation is limited to molecules with an extremely small number of orbitals due to the exponential explosion of quantum basis states; hence various approximate schemes to take into account subsets of quantum basis states have been proposed.

A similar situation is encountered in the study of strongly correlated quantum systems in condensed matter physics: in the study of lattice models such as the Hubbard or Heisenberg models, which are considered to capture the essentials of low-dimensional quantum magnetism, high-temperature superconductivity and other novel quantum states, the question of identifying relevant subsets of quantum basis states has been at the forefront of research for quite some time.

In the case of one-dimensional systems, the so-called density-matrix renormalization group method (DMRG) has emerged as the most powerful method to study correlation effects, both statically and dynamically. From an application point of view, it can be seen as an extension of exact diagonalization methods which are the counterpart of CI in physics. DMRG can therefore be used to extend the reach of CI in quantum chemistry, which has been successfully done by several groups worldwide. However, this is a complicated endeavour: whereas in physics the external one-dimensional lattice provides a natural ordering of sites (or orbitals), this is not the case in quantum chemistry, where the method must be optimized by a clever choice of orbital sets and ordering of orbitals on a pseudo-one-dimensional axis with long-ranged interactions. Nevertheless, impressive accuracies on the level of CI have been achieved.

Recently, a thorough reformulation of DMRG in terms of so-called matrix product states (MPS) has shown a profound connection of this method to quantum information theory and revealed that it is only one special method in a much more general set of methods that can give variationally optimal results for much more complicated „lattice“ arrangements: in physics, these would be quantum states on two- or even three-dimensional lattices (so-called tensor network states), and the current hope is that these methods will shed light on some of the big outstanding questions in condensed matter physics.

But again, there is a connection to quantum chemistry: these general networks of sites (or orbitals) are not restricted to regular lattices, but can be adapted to the complex arrangements and interactions between quantum chemistry orbitals. In the most naive approach, these arrangements would follow the expected geometrical structure of the molecule under study. However, this can be done more systematically by studying entanglement properties of quantum chemical states, as entanglement turns out to determine the efficiency and accuracy of these methods. Very little is known so far, but it is clear that the current restrictions of DMRG in quantum chemistry would be largely lifted due to the much more flexible setups allowed by tensor network states.

In this context, it would be highly fruitful to bring together quantum chemists with their expertise on molecular properties, orbital sets, and current quantum chemical methods based on low-rank tensor factorizations on the one hand and condensed matter physicists that are developing more general tensor network methods based on quantum information insights on the other hand, as their sets of expertise are clearly complementary and should be joined to promote this new and powerful approach. While there are many conferences on computational quantum chemistry on the one hand and also many conferences on the many varieties of tensor network methods, none of them to our knowledge has ever tried to bridge the aforementioned gap between practitioners of both fields, who mostly are not even aware of each other’s activities. In that sense, this workshop is highly likely to open up an entirely new field of research.

## References

**Belgium**

Frank Verstraete (University of Gent) - Organiser

**Germany**

Uli Schollwoeck (Ludwig-Maximilians-UniversitĂ¤t Munich) - Organiser & speaker

**Switzerland**

Markus Reiher (ETH Zurich) - Organiser

**United States**

Garnet Chan (Princeton University) - Organiser