A major unsolved challenge in theoretical physics concerns the description of quantum many-body systems, i.e. a large number of interacting particles that require a quantum mechanical treatment. The dimension of the linear space of quantum states explodes exponentially with system size, and except for very small systems an exact solution is out of bounds with present-day computer designs.

And yet, in most physics domains, ranging from nuclear and atomic physics to condensed matter and quantum chemistry, an accurate determination of the ground state, and the excitations built upon it, is a necessary requirement to understand the physical properties at the microscopical quantum level. Many approximate quantum many-body methods have been devised with this goal in mind.

One of the most successful is the so-called DMRG (density matrix renormalization group) method. The underlying variational ansatz, called a matrix product state (MPS), is capable of extending the reach of exact diagonalization techniques to significantly larger active spaces. The success of DMRG is due to the very compact parametrization of relevant entanglement (quantum mechanical correlations) for physical systems, which is made clear by quantum information theory. DMRG is particularly efficient for one-dimensional systems, due to the linear nature of the MPS.

The MPS is a subset of the so-called tensor networks. In a tensor network, the quantum many body wave function is constructed with tensors as elementary building blocks. These tensors are connected to each other in a network. The network represents the wave function. Entanglement between the building blocks are encoded in the network. The magnitude and the particular structure of the entanglement heavily depends on the used network and the dimensions of the interconnecting bonds. As can be seen in fig. 1, the MPS is a one-dimensional tensor network, which explains its efficiency for one-dimensional systems. However, molecules are far from one-dimensional and the MPS is not always able to provide an efficient representation of their entanglement structure. For a more efficient representation of the physics of the system, we could use higher-dimensional tensor networks. These networks are gaining more and more momentum in condensed matter physics over the last years. Although other tensor networks have also been studied in quantum chemistry, the high efficiency of the MPS still ensures its dominant position.

Recently, a new type of tensor networks has been proposed and implemented by our research group, called three-legged tree tensor network states (T3NS) (see fig. 1). Interspersing two types of tensors, the so-called physical and branching tensors, allows us to construct a branching tensor network, resulting in a far richer collection of entanglement structures that can be encoded. By keeping the way of branching simple, only minimal trade-offs are made to computational efficiency. At our research group, the T3NS algorithm is actively being expanded and optimized.

Fig 1. left: an example of the MPS, a linear tensor network. Right: an example of the T3NS ansatz, blue circles are physical tensors, red circles are branching tensors.

Goal

Due to the large similarities between DMRG and T3NS (the MPS is a subset of the T3NS), a lot of techniques to improve DMRG calculations can be adapted and used for T3NS. Because of this, the student will first familiarize himself/herself with DMRG, the MPS wave function and a selection out of the vast collection of post-DMRG methods that have been developed in the past 20 years since its discovery. Post-DMRG methods are techniques that build upon the DMRG algorithm and are indispensable in improving calculations and are often needed to obtain a desired accuracy.

In a next step, the novel T3NS ansatz and its particular difficulties and intricacies will be studied. Once a solid basis is acquired, the application of a post-DMRG method for T3NS (a post-T3NS method if you will) will be studied and implemented. The student will use the already existing T3NS code and, this way, will contribute to the active expansion of the code.

Due to the novelty of the T3NS ansatz, post-T3NS methods are largely undiscovered domain. This gives freedom to the student in co-deciding the direction of the research in his or her particular interest, while still having the vast collection of post-DMRG methods as a potent guideline.
Tensor network optimizations are numerically intensive and next to algorithm development, some attention is also needed for the efficient implementation of the algorithm. If needed, the student will parallelize its own implementation.

Engineering and physics aspects
Engineering: The optimization of tensor networks is a complex numerical problem, and one needs sparse solvers, such as the Lanczos algorithm, to find ground states. Furthermore, since these are computationally demanding problems, parallelized implementations on HPCs are needed.

Physics: The efficiency of tensor networks can be theoretically shown by using quantum information theory. For a better understanding of systems with strong correlations, the development of efficient techniques for the simulation of these systems are needed.