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

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

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

4. 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...)

5. 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)

6. 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

7. 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

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

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

10. 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

11. Examples: complete disentanglement no disentanglement partial disentanglement Entanglement renormalization

12. Multi-scale entanglement renormalization ansatz (MERA) Entanglement renormalization

13. 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

14. 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...