Entanglement Renormalization

Frontiers in Quantum NanoscienceA Sir Mark Oliphant & PITP Conference

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 1D quantum lattices (e.g. spin chains)

•Entanglement renormalization

Recent results

timeDMRG

1D ground state

•White1992

TEBD

1D timeevolution

2003

PEPS

2D•Verstraete•Cirac

2004

Entanglement renormalization

2005

2D

1D

•Hastings •Osborne

(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 = 24816

Hilbert Space dimension = 1,267,650,600,228,229,401,496,703,205,376

100N = 10 20 30 40

310 610 910 30101210dim(H) =

“small” system to test2D Heisenberg model (High-T superconductivity)

problems: solutions:

1

1

1n

n

i i ni i

i i

(i) state

(ii)1

1

1 1

1 1n

n

n n

i ij j n n

j j i i

U U j j i i

evolution

coefficients2n 1 ni i

i1 in…

i1 in…

U

j1 jn…

coefficients22 n 1

1

n

n

i ij jU

• Use a tensor network:

i1 in...i1 in...

(for 1D systemsMPS, DMRG)

• Decompose it into small gates:

(if , with )iHU e [ , 1]s s

s

H h i1 in...

j1 jn...

simulation of time evolution in 1D quantum lattices (spin chains, fermions, bosons,...)

•efficient update of ' U

'

U

=

operations

22 n

' U

i1 in...

•efficient description of

matrix product state

i1 in...

j1 jn...

Uand

Trotter expansion

BA

Entanglement & efficient simulations

coefficients2ni1 in…

2( )O n coefficients

tensor network(1D: matrix product state)

i1 in…

1, , 1, ,

1,2i

22coef

111 bap

222 bap

333 bap

bap

•Schmidt decomposition

1, ,

entanglementefficiency

/ 22 ?n

Entanglement in 1D systems

•Toy model I (non-critical chain):

•Toy model II (critical chain):

correlation length

#2 ( )S const #Snumber of

shared singletsA:B

#2 s pn # log( )S n

summary:

• non-critical 1D spins coefficientsn 0 ( )O n

• critical 1D spins coefficientsn pn ( )qO n

• arbitrary state spins coefficientsn / 22n ( 2 )nO n

In DMRG, TEBD & PEPS, the amount of entanglement determines the efficiency of the simulation

Entanglement renormalization

disentangle the systemchange of attitude

Examples:

U U

U

complete disentanglement

no disentanglement

partial disentanglement

Entanglement renormalization

Multi-scale entanglement renormalization ansatz (MERA)

Entanglement renormalization

Performance:

DMRG ( )

Entanglement renormalization( )

system size (1D)

greatest achievements in 13 yearsaccording to S. White

first tests at UQ

memory

time

5,000n 1000 days in “big” machine

C, fortran,highly optimized

16,000n 8eff a few hours in this laptop matlab

code

Extension to 2D: 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.

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

or simply...