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Quantum Tensor Networks and Entanglement

Periodic Reporting for period 5 - QUTE (Quantum Tensor Networks and Entanglement)

Reporting period: 2021-09-01 to 2022-02-28

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

During the project, we have achieved all envisioned objectives, but we also have been able to formulate and solve completely new problems, mainly related to the characterization of quantum topological phases of matter and to the construction of powerful numerical algorithms for contracting quantum tensor networks.
The main results 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 introduction of the concept of strange correlators for topological phases of matter, which has led to a very tangible relation between conformal field theories and topological field theories in terms of tensor networks and matrix product operators
- the construction of new conformal field theories based on categorical data, with the prime example a cft based on the Haagerup subfactor
- the construction of extremely efficient matrix product operator representations of thermal states and time evolution operators by means of cluster expansions
- the construction of novel entanglement scaling algorithms which enable the simulation of critical systems using MPS and PEPS with unprecendented precision
- the first proof of the existence of a quantum fault tolerance threshold for an error correcting code based on Turaev-Viro codes and hence universal for quantum computation
- the construction of matrix product operator representations of bimodule categories, opening up the possibility of describing dualities and boundaries of quantum spin systems in a unified way
- the construction of novel algorithms for simulating generic continuous matrix product states, leading to a far-reaching generalization of the Gross-Pitaevskii equation
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. Furthermore, tensor networks provide explicit lattice representations for some of the central unifying ideas in statistical physics and quantum field theory, namely the description of both topological field theories and conformal field theories in terms of modular tensor categories. Those ideas have very tangible applications in the context of quantum computation (by means of the construction of new powerful error correcting codes) and in the context of the numerical simulation of strongly correlated quantum systems.