Frontiers in Quantum Nanoscience A Sir Mark Oliphant

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

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

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

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

Computational problem • Simulating N quantum systems on a classical computer seems to be hard Hilbert Space dimension = 16 2 8 4 N= 100 dim(H) = “small” system to test 2 D Heisenberg model (High-T superconductivity) Hilbert Space dimension = 1, 267, 650, 600, 228, 229, 401, 496, 703, 205, 376

problems: (i) solutions: state • Use a tensor network: coefficients i 1 (ii) … (for 1 D systems MPS, DMRG) in evolution i 1 . . . in • Decompose it into small gates: (if coefficients i 1 … in j 1 … jn , with ) i 1 . . . in j 1 . . . jn

simulation of time evolution in 1 D quantum lattices (spin chains, fermions, bosons, . . . ) • efficient description of i 1 . . . and i 1 . . . in Trotter expansion in matrix product state • efficient update of = operations j 1 . . . jn

Entanglement & efficient simulations tensor network (1 D: matrix product state) i 1 … in i 1 coefficients • Schmidt decomposition … in coefficients efficiency coef entanglement A B

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

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

Examples: Entanglement renormalization complete disentanglement no disentanglement partial disentanglement

Entanglement renormalization Multi-scale entanglement renormalization ansatz (MERA)

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

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. or simply. . . • There are new tools to efficiently simulate quantum lattice systems in 1 D, 2 D, . . .