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Introduction to the many-body problem (1.5 hours)
1.1. Tensor Networks
1.1.0 Motivation
1.1.1 Why to use tensor networks -
- Building block
1.2.0 Vectors matrices tensors and their contractions in Penrose notation
1.2.1 Tensor contractions, vector norms matrix vector multiplaction, generalization
- Building block
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- The idea of data compression
1.3.0 Counting strings of...
- The idea of data compression
- Tensor networks for the many body problem
2.0 Defining many body systems
2.1 Without interaction, the product states
2.2 With interactions, strongly interacting problems in physics
2.3 The exponential scaling
2.4 Classes of states that can be represented efficiently
Exact diagonalization of local Hamiltonians in 1D and the entanglement of their ground states
- Tensor networks (1.5 hours)
1.0 Basic definitions, and examples of networks
1.1 Cost of tensor contractions, the pairwise strategy, finding the correct sequence
1.2 Matrix factorization, the SVD, QR and higher order SVDs and QR decompositions
1.3 Special tensors unitaries and Isometries
- Gauge freedom and gauge fixing
2.0 Gauge freedom in TN
2.1 The orthogonality center of a TN, gauge fixing
2.2 Standard forms for creating the orthogonality center, direct decomposition and pulling through
2.3 Tensor decompositions inside a tensor network
Using ncon to contract exemplary TN, e.g an MPS scalar product the expectation of a local operator in a MERA
- Matrix product states
1.0 Properties of MPS, correlation functions, reduced density matrices.
1.1 Standard MPS gauge fixing forms
1.2 Infinite matrix product states
1.3 The transfer matrix and its eigenvalues
- Connections with stat-mech
2.0 Matrix product operators
2.1 The 2D classical Ising partition function
2.2 Matrix product operators
2.3 Boundary contraction of the 2D Ising partition function and TEBD
Contracting the Ising partition function on an infinite plane, the boundary strategy and the corner transfer matrix
Tensor Networks are a Quantum-Inspired technology with utility far beyond basic research. Their great capacity to carry out complex calculations with few resources and emulate quantum systems make them one of the great tools for the current and future industry. From Ibermática, different innovative methods have been investigated and applied to real industrial cases. In this presentation we...
- Variational algorithms
1.0 DMRG as a variational algorithm on MPS.
1.1 DMRG as a RG algorithm
- Overview of time dependence and higher dimensions
2.0 Entanglement in time evolution, the exponential complexity
2.1. Time dependent DMRG
2.2 Higher dimensional TN, PEPS and MERA
2.3 Approximate contraction of PEPS wavefunctions
