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Computational approaches for quantum many-body systems

This course introduces computational approaches for quantum many-body systems such as tensor network states and variational methods. Topics include MPS ansatz, PEPS, MERA, neural quantum states, and more. Learn about ground state finding, tensor network extensions, and variational Monte Carlo. Study quantum critical ground states, ADS/CFT, and neural network quantum states. Access tensor network libraries like iTensor and ALPS for simulations. Explore references on density-matrix renormalization group, MERA, and neural network quantum state ansatz.

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Computational approaches for quantum many-body systems

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  1. Computational approaches for quantum many-body systems HGSFP Graduate Days SS2019 Martin Gärttner

  2. Course overview Lecture 1: Introduction to many-body spin systems Quantum Isingmodel,Bloch sphere, tensor structure, exact diagonalization Lecture 2: Collective spin models LMG model, symmetry, semi-classical methods,Monte Carlo Lecture 3: Entanglement Mixed states, partial trace, Schmidt decomposition Lecture 4: Tensor network states Area laws, matrix product states,tensor contraction, AKLT model Lecture 5: DMRG and other variational approaches Energy minimization, PEPS and MERA, neural quantum states

  3. Learning goals After today you will be able to … • … explain how to find ground states using the MPS ansatz (DMRG). • … dig deeper into tensor network states (PEPS and MERA) • … explain alternative variationalansätze inspired by neural networks.

  4. Tensor network states beyond MPS:Extensions and applications • Projected entangled pair states (PEPS): → extension to 2D • Problem: No efficient contraction:

  5. Tensor network states beyond MPS:Extensions and applications • Multiscale entanglement renormalization ansatz (MERA) • Treat quantum critical ground states ADS (2+1) CFT (1+1)

  6. Libraries for Tensor network states • iTensor: C++ library for tensor network state calculations. http://itensor.org/ • ALPS (Algorithms and Libraries for Physics Simulations). Contains many different numerical methods for quantum many-body systems, not only spins. Especially also quantum Monte Carlo methods. MPS: http://alps.comp-phys.org/static/mps_doc/ • Open Source MPS (OSMPS), Python frontend! https://openmps.sourceforge.io/

  7. Other variational approaches Variational Monte Carlo (VMC): Local energies: Sample states according to

  8. See also: Deng , Li, Das Sarma, PRX 2017, PRB 2017 Gao, Duan, Nat. Commun. 2017 Cirac et al., PRX 2018 Clark, J. Phys. A 2018 Moore, arXiv2017 Carleo, Nomura, Imada, arxiv 2018 Freitas, Morigi, Dunjko, arXiv2018 …… Neural-network quantum states [Carleo and Troyer, Science 2017] • Restricted Boltzmann machine Classical networks: probability

  9. Neural-network quantum states • Efficient evaluation . . . visible hidden

  10. Neural-network quantum states • Finding ground states: Stochastic reconfiguration method Minimize energy functional Learning rate Determine gradients by Monte Carlo sampling • Imaginary time evolution

  11. Neural-network quantum states • Time evolution: Time dependent variational Monte Carlo Minimize deviation from SE solution in each step Time-dependent variational principle time step Determine gradients by Monte Carlo sampling • Real time evolution

  12. References • Ulrich Schollwoeck: The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011) • Time dependent variational principle: Phys. Rev. Lett. 107, 070601 (2011) • MERA and AdS/CFT: e.g. Phys. Rev. D 86, 065007 (2012) • Neural network quantum state ansatz: Giuseppe Carleo, Matthias Troyer: Solving the Quantum Many-Body Problem with Artificial Neural Networks, Science 355, 602 (2017)

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