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Computational approaches for quantum many-body systems

Explore quantum many-body systems through lectures and programming exercises in Python with Jupyter notebooks. Covering topics like quantum spin systems, entanglement, tensor network states, DMRG, and variational approaches.

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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. Organizational matters • 90 min lecture + 90 min programming exercises • Materials: https://www.kip.uni-heidelberg.de/user/marting/teaching/ss19_hgsfp_graddays • Programming exercises: Python with Jupyter notebooks → Install Anaconda with Python 3 (https://jupyter.readthedocs.io/en/latest/install.html#id3) Alternative: https://jupyter.kip.uni-heidelberg.de → log in with your uni-id • Active participation and feedback is essential!

  3. 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

  4. Learning goals After today you will be able to … • … interpret the evolution of a single spin in the Bloch sphere picture. • … explain the complexity problem of quantum many-body systems and understand many-body spin Hamiltonians. • … apply the spin toolbox to build and diagonalizemany-body spin Hamiltonians. • … study a quantum phase transition in the transverse field Ising model.

  5. https://answergarden.ch/910798 What is a spin? • Intrinsic angular momentum • Electron spin • Nuclear spin • Polarizations of a photon • Ground and excited level of atom/ion… • States of a superconducting circuit… • States 0 and 1 • Unit of quantum information Physical spin Pseudo spin Two-level system Qubit

  6. Why care about spins? • Simple, but still shows fundamental physical phenomena • Analytically solvable many-body problems • Many condensed matter physics problems come in the form of spin models (magnetism, Hubbard models map so spin models in specific cases) • Quantum computers are just many-spin systems

  7. Quantum simulation • Special purpose quantum computers • Emulate (spin) model Hamiltonians in controlled experiments • Overcome problem of quantum complexity • Numerical methods for spin models • Benchmark quantum simulators in tractable regimes • Testing approximations using comparison to experiment • Examples: • Trapped ions (Bollinger, Monroe, Blatt) • Rydberg atoms (Lukin, Broways, Weidemüller) • Ultracold atoms in optical lattices (Greiner, Bloch) Nature 484, 489-492 (2012) Nature 551, 601-604(2017) Nat. Phys. 8, 277-284 (2012) Nature 551, 579-584 (2017) Nature 561, 79-82 (2018) Science  342, 954-956 (2013)Nature 545, 462-466 (2017) Science 349, 842 (2015)

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