Erwin Schrodinger Institute (ESI) Thu 13th Aug 2009

Workshop: Quantum Computation

and Quantum Spin Systems

Entanglement Renormalizationin two spatial dimensions:

frustrated antiferromagnets

Guifre Vidal

frustrated antiferromagnetsand interacting fermions

Entanglement Renormalization/MERA

Outline

• Quantum Computation

MERA = multi-scale entanglement renormalization ansatz

• Renormalization Group

transformation

• 1D systems • 2D systems

• spin systems(commuting variables)

• fermion systems(anticommuting variables)

Tensor Network representation of a many-body state

1 2

1 2

... 1 21 1 1

...N

N

d d d

GS i i i Ni i i

c i i i= = =

Ψ =∑∑ ∑⋯

1 2 ... N⊗ ⊗ ⊗H H H⋯d

N

1D TTNMPS (1D)

...1i

2i 3iNi

4i

Nd coefficients

1D TTN

PEPS

(2D)1D MERA

MPS (1D)

( )O N coefficients

Disclaimer

• No proofs (actually, no theorems) Merlin and Arthur

• Variational ansatz that is physically motivated (locality, entanglement)

• Numerical/analytical evidence that the MERA approximates ground states well in several particular cases.

• Counter-examples? Sure (but this is not the point)

Entanglement Renormalization/MERA

Outline

• Quantum Computation

MERA = multi-scale entanglement renormalization ansatz

• Renormalization Group

transformation

• 1D systems • 2D systems

• spin systems(commuting variables)

• fermion systems(anticommuting variables)

Tree Tensor Network (TTN)

• coarse-graining transformation:

=w

†w

I*( )ijk ijk

ijk

µ νµνδ=∑w w

LΨ

'L'Ψ

w

isometry:ijkµw

w

'L

LΨ

'Ψ

( 2 )L

(1)L

( 0 )L

( )0Ψ

( )1Ψ

( )2Ψ

Tree Tensor Network (TTN)

• quantum circuit:

LΨ

isometry0 0

unitary=

'L23' 0N⊗Ψ ⊗ 'L

LΨ

3' 0⊗Ψ ⊗

00 00 00 00 00

( 2 )L

(1)L

( 0 )L

00

0000 00 00 00 00 00 00 00

00 00

000

Ψ

0N⊗

Multi-scale Entanglement Renormalization Ansatz (MERA)

• Entanglement renormalization -- coarse-graining transformation:

LΨ

'Ψ 'L

=w

†w

Iw

isometry

=u

†u

unitary(disentangler)

u

Vidal, Phys. Rev. Lett. 99, 220405 (2007); ibid 101, 110501 (2008)

Ψ

'Ψ w

u

L

'L

( 2 )L

(1)L

( 0 )L

( )0Ψ

( )1Ψ

( )2Ψ

′L

L

L

w

u

w

′L

• why do we want disentanglers ?

without disentanglers: with disentanglers:

entangled

sites

unentangled

site

L

L u

Multi-scale Entanglement Renormalization Ansatz (MERA)

• quantum circuit

LΨ

isometry0 0

unitary=

( 2 )L

(1)L

( 0 )L

0000 00 00 00 00

0000

00

Ψ

0N⊗

Quantum Circuit as a Many-Body Ansatz

Ψ

0N⊗

“generic” quantum circuit ansatz

Ψ

0N⊗

tree quantum circuit ansatz

Quantum Circuit as a Many-Body Ansatz

• Entanglement ?

Ψ

0N⊗

NS ∼entanglement

(1)S O∼entanglement

Ψ

0N⊗

/2 2NNS ∼

entanglemententropy /2 (1)NS O∼

entanglemententropy

Quantum Circuit as a Many-Body Ansatz

Entanglement

Ψ

0N⊗

Ψ

0N⊗

NS ∼ (1)S O∼entanglement entanglement

Ψ

0N⊗

/2 2NNS ∼ /2 (1)NS O∼

entanglemententropy

entanglemententropy

/2 log( )NS N∼entanglemententropy

Quantum Circuit as a Many-Body Ansatz

• Causal cone/cost ?

( )O Nwidth ~ (1)Owidth ~

Ψ

0N⊗

Ψ

0N⊗

exp( )Ncost ~

( )O Nwidth ~

cost ~

(1)Owidth ~

(log )O N

Quantum Circuit as a Many-Body Ansatz

Causal cone

Ψ

0N⊗

Ψ

0N⊗

( )O Nwidth ~ (1)Owidth ~

Ψ

0N⊗

( )O Nexp( )Ncost ~

width ~ (1)Ocost ~

width ~

(1)O

(1)O

cost ~

width ~

(log )O N

Summary: 1D MERA = quantum circuit + small causal cones = variational ansatz

Ψ

0N⊗

• efficient representation parameters4( log )O NχN

log N

• efficient representation parameters4( log )O Nχ(translation invariance)

• efficient simulationcomputational

costs

8( log )O Nχ(translation invariance)

• non-trivially entangled /2 logNS N∼entanglement entropy

• good approximation to ground states, including critical systems

( 3 )L

( 2 )L

(1)L

( 0 )L

⋯ ⋯

⋮

Example: scale invariant MERA

critical exponents

Vidal, Phys. Rev. Lett. 99, 220405 (2007) Vidal, Phys. Rev. Lett. 101, 110501 (2008)Evenbly, Vidal, arXiv:0710.0692 Evenbly, Vidal, arXiv:0801.2449Giovannetti, Montangero, Fazio, Phys. Rev. Lett. 101, 180503 (2008)Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)

Aguado, Vidal, Phys. Rev. Lett. 100, 070404 (2008)Koenig, Reichardt, Vidal, Phys. Rev. B 79, 195123 (2009)

critical systems(1D)

topologically ordered systems

(2D)

OPE, CFT

1D critical systems

Computation of ground state energy (1D scale invariant MERA)

Evenbly, Vidal, Phys. Rev. B 79, 144108 (2009)

10101010-5-5-5-5

10101010-4-4-4-4

10101010-3-3-3-3

1D MERA, Ground-Energies1D MERA, Ground-Energies1D MERA, Ground-Energies1D MERA, Ground-Energies

Ene

rgy

Err

or,

Ene

rgy

Err

or,

Ene

rgy

Err

or,

Ene

rgy

Err

or,

∆∆ ∆∆EE EE

IsingIsingIsingIsingXXXXXXXXHeisenbergHeisenbergHeisenbergHeisenbergPottsPottsPottsPotts

8888 16161616 24242424 32323232 40404040 48484848 56565656

10101010-9-9-9-9

10101010-8-8-8-8

10101010-7-7-7-7

10101010-6-6-6-6

χχχχ

Ene

rgy

Err

or,

Ene

rgy

Err

or,

Ene

rgy

Err

or,

Ene

rgy

Err

or,

Bulk

1D critical systems

Scaling dimensions/critical exponents

scaling

dimension

scaling

dimension error

0 0 ----

CFT∆MERA∆

(I)

Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), 040301(R) (2009)

2

1(0) ( )r

rφ φ ∆∼ 1

x x zr r r

r r

H σ σ σ+= − −∑ ∑Quantum Ising chain

Iσ

ε

1/ 8

1 1 / 8+

2 1/ 8+2

0

1

0

2

0 0 ----

0.125 0.124997 0.003%

1 0.99993 0.007%

1.125 1.12495 0.005%

1.125 1.12499 0.001%

2 1.99956 0.022%

2 1.99985 0.007%

2 1.99994 0.003%

2 2.00057 0.03%

(I)( )σ( )ε

Entanglement Renormalization/MERA

Outline

• Quantum Computation

MERA = multi-scale entanglement renormalization ansatz

• Renormalization Group

transformation

• 1D systems • 2D systems

• spin systems(commuting variables)

• fermion systems(anticommuting variables)

1D MERA

2D MERA

w

u

L

'L

2-site causal cone

Evenbly, Vidal, Phys. Rev. B 79, 144108 (2009)

wv uL 'L

2D MERA

2x2-site causal cone

• efficient representation parameters( log )qO Nχ(translation invariance)

• efficient simulationcomputational

costs

'( log )qO Nχ(translation invariance)

• Entropic area law LxLS L∼entanglement entropy

• good approximation to ground states

2D MERA

Vidal, Phys. Rev. Lett. 101, 110501 (2008)Evenbly, Vidal, arXiv:0710.0692 Evenbly, Vidal, arXiv:0801.2449Cincio, Dziarmaga, Rams, Phys. Rev. Lett. 100, 240603 (2008)Evenbly, Vidal, Phys. Rev. Lett. 102, 180406 (2009)

Aguado, Vidal, Phys. Rev. Lett. 100, 070404 (2008)Koenig, Reichardt, Vidal, Phys. Rev. B 79, 195123 (2009)

Evenbly, Vidal, arXiv:0904.3383

Corboz, Evenbly, Verstraete, Vidal, arXiv:0904.4151Pineda, Barthel, Eisert, arXiv:0905.0669 Corboz, Vidal, arXiv:0907.3184Barthel, Pineda, Eisert, arXiv:0907.3689

topological order(analytical)

scalable simulations

frustrated spins

interacting fermions

• good approximation to ground states

Example: Heisenberg antiferromagnet on Kagome latticeEvenbly, Vidal, arXiv:0904.3383

,i j

i j

H S S= ⋅∑Hamiltonian:

• Favours antiferromagnetic alignment

• Geometrically frustrated

• Ground state?• Ground state?

(Quantum Monte Carlo techniques suffer from sign problem)

Example: Heisenberg antiferromagnet on Kagome lattice

- Valence Bond Crystal ? - Spin Liquid ?

Singh, Huse, Phys. Rev. B 76, 180407 (2007)

series expansion

Marston, Zeng (1991), Syromyatnikov, Maleyev (2002) Nikolic, Senthil (2003), Budnik, Auerbach (2004)

Jiang, Weng, Sheng, Phys. Rev. Lett. 101, 117203 (2008)

DMRG

Ran, Hermele, Lee, Wen, Phys. Rev. Lett. 98, 117205 (2007)

Gutzwiller ansatz

2D MERA (!!?)

Example: Heisenberg antiferromagnet on Kagome latticeEvenbly, Vidal, arXiv:0904.3383

Example: Heisenberg antiferromagnet on Kagome lattice

MERA solution (infinite lattice with 36-spin unit cell):

Order VBC

0 -0.375

1 -0.375

2 -0.42187

3 -0.42578

Singh, Huse, Phys. Rev. B 76, 180407 (2007)

series expansion

Evenbly, Vidal, arXiv:0904.3383

3 -0.42578

4 -0.43155

5 -0.43208

VBC

8 -0.4298

14 -0.4307

20 -0.4314

26 -0.4319

1-layer MERA

χ-0.420

-0.426

-0.432

-0.423

-0.429

5th4th

3rd

2nd

8χ =14χ =20χ =

26χ =energy

series exp.

MERA

Entanglement Renormalization/MERA

Outline

• Quantum Computation

MERA = multi-scale entanglement renormalization ansatz

• Renormalization Group

transformation

• 1D systems • 2D systems

• spin systems(commuting variables)

• fermion systems(anticommuting variables)

Fermionic MERA

( 2 )L

(1)L

( 0 )L

0000 00 00 00 00

0000

00

Ψ

0N⊗

Corboz, Evenbly, Verstraete, Vidal, arXiv:0904.4151Corboz, Vidal, arXiv:0907.3184

Pineda, Barthel, Eisert, arXiv:0905.0669 Barthel, Pineda, Eisert, arXiv:0907.3689

( 0 )L

Ψ

quantum wire

1 2 2 1a a a a=Bosons: b =

Fermions:1 2 2 1a a a a= − f = swap

Fermionic MERA

quantum wire

1 2 2 1a a a a=Bosons:

Fermions: 1 2 2 1a a a a= −

b

f

=

= swap

parity sectors + + + + + − + −( ) ( )V V V+ −≅ ⊕

parity sectors

swap =+ + + +

swap =+ − + −

swap =+ − +−

swap = −−− − −

Fermionic MERA

Benchmark results1) free spinless fermions on 6x6 lattice (periodic BC)

† † † †

,

[ ( )] 2free r s r s r s r rr s r

H c c c c c c c cγ λ= − + −∑ ∑pairing potential

chemical potential

0 0

†( ) r r rC r c c+=0 0r r r+

Fermionic MERA

Benchmark results

† † † †free

,

[ ( )] 2r s r s r s r rr s r

H c c c c c c c cγ λ= − + −∑ ∑

2) interacting spinless fermions on a 6x6 lattice (PBC)

pairing potential

chemical potential

† †int free

,r r s s

r s

H H V c c c c= + ∑

interaction

Fermionic MERA

Benchmark results

† † † †free

,

[ ( )] 2r s r s r s r rr s r

H c c c c c c c cγ λ= − + −∑ ∑pairing potential

chemical potential

† †int free

,r r s s

r s

H H V c c c c= + ∑

2) interacting spinless fermions on a 6x6 lattice (PBC)

interaction

† †( ) k kP k c c−=

| ( ) |totk

P P k=∑

pairing amplitude

Fermionic MERA

† † † †

,

[ ( )] 2free r s r s r s r rr s r

H c c c c c c c cγ λ= − + −∑ ∑pairing potential

chemical potential

3) scalability: free spinless fermions on a LxL lattice (PBC)with L up to 162

Benchmark results

MERA = quantum circuit + small causal cone

Conclusions

• 1D: quantum criticality

• 2D: frustrated antiferromagnetsinteracting fermions

• cost of simulations entanglement

What can be simulated efficiently?