# Lecture: Analytical and Numerical Methods for Quantum Many-Body Systems from a Quantum Information Perspective (WS12/13)

Lecturer: Norbert Schuch

**Contents:**

Strongly correlated quantum many-body systems, i.e. those in which quantum correlations play an important role, exhibit many exciting phenomena such as superconductivity, the fractional quantum Hall effect, or topological order. In this lecture, we will discuss how to use quantum information concepts - in particular the theory of entanglement - to better understand those systems.

The focus of the lecture will be on Tensor Network methods (in particular Matrix Product States, PEPS, and MERA) which form a framework to describe correlated quantum many-body systems by capturing their entanglement properties, and which have proven to be very useful both analytically and numerically (e.g. in the Density Matrix Renormalization Group [DMRG] method). However, we will also discuss other quantum information aspects of quantum many-body systems, such as aspects of topological order, or the propagation of information and Lieb-Robinson bounds.

While there will be certain “core topics” in the lecture, the exact selection of topics - in particular also the balance between analytical and numerical aspects - will be adjusted depending on the interest of the audience.

**Prerequisites:**

A thorough background in quantum mechanics is required. Familiarity with topics in quantum many-body physics and quantum information theory is not necessary, but certainly useful.

**Material:**

Lecture notes |
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Lecture 1 (12.10.): Quantum many-body systems and models |

Lecture 2 (19.10.): Entanglement theory, the area law |

Lecture 3 (26.10.): Matrix Product States |

Lecture 4 (2.11.): Properties of Matrix Product States: Evaluation of expectation values and scaling of correlations |

Lecture 5 (9.11.): Variational minimization over MPS (the DMRG algorithm) |

Lecture 6 (16.11.): MPS simulations for periodic boundary conditions, excited states, and time evolution |

Lecture 7 (23.11.): more on time evolution, Schmidt decomposition and truncation of bond dimension, entanglement scaling and approximability |

Lecture 8 ( 30.11.): The AKLT model, parent Hamiltonians |

Lecture 9 (6.12.): Parent Hamiltonians for MPS: Uniqueness of the ground state and gap |

Lecture 10 (14.12.): Projected Entangled Pair States (PEPS) |

Lecture 11 (21.12.): The Multi-scale entanglement renormalization ansatz (MERA) |

Lecture 12 (10.1.): Measurement based quantum computation |

Lecture 13 (11.1.): Computational complexity and many-body systems: Classical complexity |

Lecture 14 (31.1.): Fermionic tensor networks |

Lecture 15 (1.2.): Quantum complexity of many-body systems |

Exercise Sheets |
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Sheet 1 (Lecture 1-3, discussed on 2.11.) |

Sheet 2 (Lecture 4 & 5, discussed on 16.11.), and a sample DMRG code (Disclaimer: This code is for pedagogical purposes, so some things are not implemented in the optimal way.) |

Sheet 3 (Lecture 6 & 7, discussed on 30.11.) |

Sheet 4 (Lecture 8 & 9, discussed on 14.12.) |

Sheet 5 (Lecture 10 & 11, to be discussed on 11.1.) |

Sheet 6 (Lecture 12 & 13, to be discussed on 1.2.) |

Further reading |
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U. Schollwöck: The density-matrix renormalization group in the age of matrix product states. |

F. Verstraete, J.I. Cirac, and V. Murg: Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems. |

D. Perez-Garcia, F. Verstraete, M.M. Wolf, and J.I. Cirac: Matrix Product State Representations. |

N. Schuch: Condensed-Matter Applications of Entanglement Theory (Lecture Notes for IFF spring school 2013.) |

R. Jozsa: An introduction to measurement-based quantum computation. |

T. Osborne: Hamiltonian complexity. |

**Organisatorial issues:**

The lecture takes place Friday 11:45-13:15 in 26C401.

The tutorials are bi-weekly Friday 15:15-16:45 in 26C401, starting Nov. 2nd.

Please feel free to contact me if you have any questions concerning the lecture.