Entanglement Renormalization

1
Frontiers in Quantum Nanoscience A Sir Mark
Oliphant PITP Conference
Entanglement Renormalization
Noosa, January 2006
Guifre Vidal The University of Queensland
2
Introduction
3
Outline

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

4
Recent results

• Hastings

• Osborne

time
(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
16
4
8
small system to test 2D Heisenberg model
(High-T superconductivity)
Hilbert Space dimension 1,267,650,600,228,229,4
01,496,703,205,376
6
problems
solutions
7
simulation of time evolution in 1D quantum
lattices (spin chains, fermions, bosons,...)
8
Entanglement efficient simulations

• Schmidt decomposition

9
Entanglement in 1D systems

• Toy model I (non-critical chain)

correlation length

• Toy model II (critical chain)

10
summary
In DMRG, TEBD PEPS, the amount of entanglement
determines the efficiency of the simulation
Entanglement renormalization disentangle the
system
11
Entanglement renormalization
Examples
complete disentanglement
12
Entanglement renormalization
Multi-scale entanglement renormalization ansatz
(MERA)
13
Performance
system size (1D)
time
code
memory
greatest achievements in 13 years according to
S. White
DMRG ( )
Entanglement renormalization ( )
first tests at UQ
Extension to 2D work in progress
14
Conclusions