# Tensor Network / Matrix Product State methods for computational quantum many-body problems

Tensor Networks provide a representation of a quantum many-body wave-function (or a classical partition function) that is suitable for computational methods, mainly for low-dimensional problems (1D and 2D). The most mature branch of tensor networks is known as Matrix Product States (MPS), also known as the density-matrix renormalization-group (DMRG). These days this is probably the most widely used method to solve 1-dimensional quantum many-body problems. At UQ, we have developed the Matrix Product Toolkit, which is a comprehensive software toolkit for many kinds of MPS-based computations, including ground-states, real-time dynamics, and thermal and dissipative states. This software toolbox has been used in approximately 90 research publications in various fields of condensed matter and quantum science, ranging from ultra-cold atoms in optical lattices, ion traps, Josephson Junction arrays, strongly correlated systems, frustrated magnetism, topological order, and dynamical mean-field theory. See publications at ResearcherID for more details.

I am happy to discuss with you to find a suitable project at any level, from summer project to PhD. Possible short projects would be, for example, learning about the Matrix Product Toolkit and trying some calculations (depending on the details and rate of progress this could lead to a research publication), or implementing a simple MPS or tensor network algorithm in Python or Matlab.

Possible PhD projects would be based around one of several possible themes:

- Applications of the Toolkit to an area of current research. Current projects include frustrated magnetism, topologically ordered states in 1D and 2D, non-equilibrium dynamics of bose systems (including thermalization), dynamical phase transitions.
- Continued development of the Matrix Product Toolkit (new algorithms, or optimizing existing algorithms). The toolkit is written in C++, so experience in modern C++ highly desirable.
- New application areas of the Toolkit, for example neural networks and deep learning, compression algorithms, quantum chemistry, nuclear physics, lattice gauge theory. This would likely involve some computer programming
- Development of new Tensor Network algorithms outside the current scope of the Toolkit, eg PEPS, MERA, ... Good computational skills required, not necessarily in C++.