Introductory Course to Tensor Networks

25 Sept 2023, 09:30 → 28 Sept 2023, 18:00 Europe/Madrid

Aula de Seminarios (IGFAE - CESGA)

Irais Bautista Guzman (CESGA)

Among quantum-inspired classical computing approaches to Quantum Simulation, Tensor Networks stand out as a promissing avenue to handle problems whose solutions involve a bounded amount of entanglement 

This course provides an introduction to the use of Tensor Newtorks.  The course will be in-person, and include both theory presentations and python labs.  There is a limited number of seats available. Registration is needed. Final admission will be subject to approval. The course is charge free. 

The pre-requisites involve knowledge and understanding of quantum mechanics of several constituents, like spin 1/2 systems, the meaning of Hamiltonian evolution, and the properties of mixed states and density matrices.  At the computational level, python skills to write and debug codes made of several functions with numpy and scipy will do. 

The main speaker will be Luca Tagliacozzo, from Instituto de Física Fundamental, IFF-CSIC, who has significant contributions in the field as well as expertise in teaching this course. 

Contributional seminars from industry experts will also be programmed. 

 

Monday 25 September

1 Introduction to the Many-body problem

1. 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

2. 1. 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
3. 1. The idea of data compression
      1.3.0 Counting strings of bits
      1.3.1 Counting strings of bits with constraints
      1.3.2 From strings of bits to probability distributions of classical spins

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

2 Entanglement

1. 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

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

3 Python tutorial 1

Exact diagonalization of local Hamiltonians in 1D and the entanglement of their ground states

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

Tuesday 26 September

4 Tensor networks

1. 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

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

5 Gauge freedom and gauge fixing

1. 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

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

6 Python tutorial 2

Using ncon to contract exemplary TN, e.g an MPS scalar product the expectation of a local operator in a MERA

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

Wednesday 27 September

7 Matrix product states

1. 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

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

8 Connections with stat-mech

1. 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

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

9 Python tutorial 3

Contracting the Ising partition function on an infinite plane, the boundary strategy and the corner transfer matrix

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

10 Tensor Networks for the Industry: Ibermática.

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 will talk about some of these cases, with applied examples that can also be extrapolated to basic research.

Speaker: Alejandro Mata Ali (IBERMATICA)

Thursday 28 September

11 Variational algorithms

1. Variational algorithms
   1.0 DMRG as a variational algorithm on MPS.
   1.1 DMRG as a RG algorithm

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

12 Overview of time dependence and higher dimensions

1. 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

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)

13 Python tutorial 4,

Quench in the Ising model with TEBD.

Speaker: LUCA TAGLIACOZZO (IFF-CSIC)