Entanglement Renormalization

KITP

12th of February 2007

Guifre Vidal University of Queensland (Brisbane, Australia)

• Simulation of Quantum Many-Body Systems

University of Queensland

G.V.Roman Orus (postdoc)Glen Evenbly (PhD)Jacob Jordan (PhD)Sukhi Singh (PhD)

Collaborators

Frank Verstraete (Univ. Vienna)Ignacio Cirac (MPQ Garching)Miguel Aguado (MPQ Garching)Lluis Masanes (Univ. Cambridge)Huan-Qiang Zhou (Chongqing Univ.)

• Efficient description of n-qubit states

1

1

1n

n

i i ni i

i iΨ = Ψ∑ coefficients2n1 ni iΨ

i1 in…

Ψmatrix product state MPS1D

tree tensor network TTN1D,tree

multi-scale entanglement renormalization ansatz

MERA 1D, 2D

projected entangled pair statePEPS 2D

DMRG (White 92)MPS 1D

time evolution (Vidal 03)

1D Infinite(Vidal 06)

2D Infinite(CVV 07)

Summary of recent developments

PEPS 2D(Verstraete, Cirac 04)

[ 20x20 sites ]MERA 1D,2D

topologicalorder

ΛN

Ψ

0 00 0 0 00 0 0 00 0 0 0 00

00 00 000 0

0000

00

0 0

θ

012

Χ

Multi-Scale Entanglement Renormalization Ansatz (MERA)GV, cond-mat/0512165, quant-ph/0610099

ΛN

Ψ

0 00 0 0 00 0 0 00 0 0 0 00

00 00 000 0

0000

00

0 0

θ

012

Χ

Multi-Scale Entanglement Renormalization Ansatz (MERA)

s

• Causal cone has bounded width !!!Local reduced density matrices can be computed efficiently

GV, cond-mat/0512165, quant-ph/0610099

ΛN

Ψ

0 00 0 0 00 0 0 00 0 0 0 00

00 00 000 0

0000

00

0 0

θ

Multi-Scale Entanglement Renormalization Ansatz (MERA)

s

• Causal cone has bounded width !!!Local reduced density matrices can be computed efficientlyRenormalization group (or coarse-graining) transformations

τ 0Ψ

1Ψ

2Ψ

3Ψ

4Ψ

GV, cond-mat/0512165, quant-ph/0610099

τΨ

1τ+Ψ

τΨ

isometry

disentangler

Entanglement Renormalization

coarse-grained lattice

Λτ

Λ 1τ+

GV, cond-mat/0512165, quant-ph/0610099

Entanglement Renormalization

• Ground state of a local Hamiltonian:

• Gapped systems(finite correlation length)

• Critical systems(quantum phase transition)

0Ψ

0Ψ

1 0Ψ ≈ Ψ

2 0Ψ ≈ Ψ

3 0Ψ ≈ Ψ

2 ϕ ϕ ϕΨ ≈

1ΨUnentangled state after several coarse-graining transformations

Self-similar MERA(ground state always remains entangled)

Glen Evenbly, GV

GV, cond-mat/0512165, quant-ph/0610099

Entanglement Renormalization

State

EffectiveHamiltonian

RG transformation

RG transformation

(discrete!) scale invariance in critical systems

Entanglement Renormalization

• Ground states with topological order:

• 2D MERA

• Kitaev’s toric code(contains topological order)

Topological informationis stored in the top tensor of the MERA

More complicated structure(2+1) dimensional quantum circuitbounded causal cone

A B C

CBAθx y

τ

“torus”

top tensor

00 01 10 11{ , , , }χ χ χ χ

{00 , 01 , 10 , 11}

ground states

Miguel Aguado, Ignacio Cirac, GV

GV, cond-mat/0512165, quant-ph/0610099

• Transformation of – stabilizers- topological charges

• Excitations• Fixed point of RG flow?

X

XX

X

X

XX

X

Z ZZ

ZZ ZZ

Z

Kitaev’s code on a torus

X

XX

X

X

XX

X

Z ZZ

ZZ ZZ

Z

Disentanglers

X

X

X

X

X

X

XX

Z ZZ ZZ

Z

Disentanglers

X

X

X

X

X

X

ZZZZ ZZ

Levin-Wen model on a torus

X

X

X

X

X

X

ZZZZ ZZ

Disentanglers

Z

Z ZZ

ZZ

X

X

X

X

Disentanglers

ZZ

ZZ

X

X

X

X

Kitaev’s code on a twisted torus

Kitaev’s code on a twisted torus

Disentanglers

Levin-Wen model on a torus

Disentanglers

Kitaev’s code on a coarsed-grained torus

• Transformation of – stabilizers- topological charges

• Excitations• Fixed point of RG flow?

Charges

Z

Z

Z

Z

Z

Z

Z

Z

Z Z Z Z Z Z Z Z

Charges

Charges

Charges

Charges

Charges

• Transformation of – stabilizers- topological charges

• Excitations• Fixed point of RG flow?

Excitations and ER: annihilation of a pair of magnetic charges(similar for magnetic vortices)

-1 XX

XX X XXX

-1

X

XX

X

X

XX

X

XX

XX X XXX

-1 -1

X

XX

X

X

XX

X

Excitations and ER: annihilation of a pair of magnetic charges(similar for magnetic vortices)

X

X

X

X

X

X

XX

-1 XX

XXX

XX-1

Excitations and ER: annihilation of a pair of magnetic charges(similar for magnetic vortices)

X

X

X

X

X

X

+1

Excitations and ER: annihilation of a pair of magnetic charges(similar for magnetic vortices)

• Transformation of – stabilizers- topological charges

• Excitations• Fixed point of RG flow?

toric code toric codeRG transformation

toric code+ dilute, short-ranged pairs

?

• MERA (multi-scale entanglement renormalization ansatz) Quantum circuit with bounded width causal cones

- Renormalization group transformation- States can be efficiently prepared

• Good description of

Critical ground states Quantum phase transitions

Kitaev’s toric code Topological order

Classical systems Classical statistical mechanics

Conclusions

key insight from quantum information