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Entanglement Renormalization Frontiers in Quantum Nanoscience A Sir Mark Oliphant & PITP Conference Noosa, January 2006 Guifre Vidal The University of Queensland

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Introduction Science and Technology of quantum many-body systems entanglement Quantum Information Theory simulation algorithms Computational Physics

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Outline Overview: new simulation algorithms for quantum systems Time evolution in 1D quantum lattices (e.g. spin chains) Entanglement renormalization

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Recent results time DMRG 1D ground state White 1992 TEBD 1D time evolution 2003 PEPS 2D Verstraete Cirac 2004 Entanglement renormalization 2005 2D 1D HastingsOsborne (Other tools: mean field, density functional theory, quantum Monte Carlo, positive-P representation...)

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Computational problem Simulating N quantum systems on a classical computer seems to be hard Hilbert Space dimension =2 48 16 Hilbert Space dimension = 1,267,650,600,228,229,401,496,703,205,376 100N = dim(H) = “small” system to test 2D Heisenberg model (High-T superconductivity)

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problems:solutions: (i) state (ii) evolution coefficients i1i1 inin … i1i1 inin … j1j1 jnjn … Use a tensor network: i1i1 inin... i1i1 inin (for 1D systems MPS, DMRG) Decompose it into small gates: (if, with ) i1i1 inin... j1j1 jnjn

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simulation of time evolution in 1D quantum lattices (spin chains, fermions, bosons,...) efficient update of = operations i1i1 inin... efficient description of matrix product state i1i1 inin... j1j1 jnjn and Trotter expansion

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BA Entanglement & efficient simulations coefficients i1i1 inin … tensor network (1D: matrix product state) i1i1 inin … coef Schmidt decomposition entanglement efficiency

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Entanglement in 1D systems Toy model I (non-critical chain): Toy model II (critical chain): correlation length number of shared singlets A:B

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summary: non-critical 1D spins coefficients critical 1D spinscoefficients arbitrary state spinscoefficients In DMRG, TEBD & PEPS, the amount of entanglement determines the efficiency of the simulation Entanglement renormalization disentangle the system change of attitude

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Examples: complete disentanglement no disentanglement partial disentanglement Entanglement renormalization

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Multi-scale entanglement renormalization ansatz (MERA) Entanglement renormalization

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Performance: DMRG ( ) Entanglement renormalization ( ) system size (1D) greatest achievements in 13 years according to S. White first tests at UQ memory time days in “big” machine C, fortran, highly optimized a few hours in this laptop matlab code Extension to 2D: work in progress

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Conclusions Understanding the structure of entanglement in quantum many-body systems is the key to achieving an efficient simulation in a wide range of problems. There are new tools to efficiently simulate quantum lattice systems in 1D, 2D,... or simply...

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