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Quantum Tensor Networks and Entanglement

Periodic Reporting for period 4 - QUTE (Quantum Tensor Networks and Entanglement)

Reporting period: 2020-03-01 to 2021-08-31

The project is focused on developing novel techniques using quantum tensor networks for simulating, describing and classifying strongly correlated quantum many body systems. Tensor networks provide a novel language for tackling the ubiquitous quantum many body problem, and this project aims at exploring the full range of possibilities that this new "language" provides. In particular, the project consists of three subprojects:

1. the study of the manifold of tensor networks and its associated optimization algorithms
2. the study of post-tensor network methods, with the aim of describing dynamical properties
3. the study of gauge theories and the classification of strongly correlated states of matter

We have made significant progress towards all of those objectives.
We have made significant progress in tackling the problems envisioned in the proposal. The main results so far are:

- the characterization of distillable entanglement in gauge theories, which was made possible by exploiting the tensor network structure of gauge symmetries
- a systematic study of spontaneous symmetry breaking in relation to the convex structure of reduced density matrices; this was made possible due to the development of efficient matrix product state solvers
- the development of state of the art techniques for optimizing PEPS algorithms
- the application of the excitation ansatz to a multitude of interesting quantum spin systems
- the construction of a novel tensor product algorithm for simulating real time evolution in quantum spin systems respecting all symmetries, and the study of the ensuing symplectic quasi-classical equations of motion
- the study of scattering of elementary excitations in terms of S-matrices
- the classification of fermionic symmetry protected phases of matter in 1+1 dimensions
- the construction of string nets in the tensor network formalism, with a special emphasis on matrix product operator symmetries and the algebras of topological sectors and related anyons
- a classification of 2+1 D bosonic symmetry protected phases of matter
The biggest surprise so far has been the discovery of the enormous versatility of the tensor network formalism, and specifically the intriguing mathematical structures which arise when studying strongly correlated phases of matter. We are close now in understanding vast areas of research in theoretical physics, including quantum field theory and string theory, condensed matter physics, and quantum chemistry, in terms of tensor networks.