Quantum Information and the simulation of quantum systems · 2007. 8. 3. · Quantum Information and the simulation of quantum systems José Ignacio Latorre Universitat de Barcelona

Quantum Informationand

the simulation of quantum systems

José Ignacio LatorreUniversitat de Barcelona

Buenos Aires, August 2007

I Simulation of quantum Mechanics

II Entanglement entropy

III Efficient representation of quantum systems

Physics

Theory 1 Theory 2

Exact solution

Approximated methods

Simulation

Classical Simulation

Quantum Simulation

Classical Theory

• Classical simulation• Quantum simulation

Quantum Mechanics

• Classical simulation• Quantum simulation

Classical simulation of Quantum Mechanics is related to our ability to supportlarge entanglement

Classical simulation may be enough to handle e.g. ground states

Quantum simulation needed for typical evolution of Quantum systems(linear entropy growth to maximum)

Classical computer

Quantum computer

?

IntroductionIntroduction

Is it possible to classically simulate faithfully a quantum system?

000 ψψ EH =

Heisenberg model0ψ

0)( ψtU

0210 ψψ OO∑ +⋅=i

ii SSH 1

represent

evolve

read

Introduction

Misconception: NO

• Exponential growth of Hilbert space

∑ ∑= =

〉=〉Ψd

i

d

inii

n

niic

1 11...

1

1...|...|

Classical representation requires dn complex coefficients

n

• A random state carries maximum entropy

ψψρ )( LnL Tr −=

( ) dLTrS LLL loglog)( ≈−≡ ρρρ


Refutation

• Realistic quantum systems are not random

• symmetries (translational invariance, scale invariance)• local interactions• little entanglement

• We do not have to work on the computational basis

• use an entangled basis


Plan

Measures of entanglement

Efficient description of slight entanglement

Entropy: physics vs. simulation

New ideas: MPS, PEPS, MERA


One qubit

∑=

=+=1,0

1110

i

i icβαψ

Quantum superposition

Two qubits

∑=

=+++=1,0,

2121

2111100100ii

ii iicδγβαψ

Quantum superposition + several parties = entanglement



Bii

Aii

ii

ii iiciic 21,0,

11,0,

2121

21

21

21 ∑∑==

==ψ

• Separable states

BABii

Aii iic ζξψ == ∑

=2

1,0,1

21

21

( ) ( ) ( )BBAA

102110

2111100100

41 ++=+++=ψe.g.

• Entangled states

BABii

Aii iic ζξψ ≠= ∑

=2

1,0,1

21

21

( )100121 −=−ψe.g.



∑= BiAiip ζξρ

• Classically correlated states

• Entangled states

∑≠ BiAiip ζξρ

Separability problem: given ρ find whether it is entangled or not


Pure states: Schmidt decomposition

BiAii

iAB p 〉〉=〉Ψ ∑=

ζξχ

|||1

BjA

B

ij

A

vuA iH

j

H

iAB 〉〉=〉Ψ ∑∑

==

|||dim

1

dim

1klkikij VUA

+= λ

A B

χ =min(dim HA, dim HB) is the Schmidt number

BA HHH ⊗=


1>χ Entanglement

Diagonalize A


BiAii

iAB p 〉〉=〉Ψ ∑=

ζξχ

|||1

Von Neumann entropy of the reduced density matrix

( ) Bi

iiAAA SppTrS =−=−= ∑=

χ

ρρ1

22 loglog

( ) ∑=

〉〈=Ψ〈〉Ψ=χ

ξξρ1

||||i

iiiABBA pTr

• χ=1 corresponds to a product state• Large χ implies large superpositions

• e-bit

ITrBA 21|| =Ψ〉〈Ψ== −−ρρ 1

21log

21

21log

21

22 =

+−== BA SS

Measures of entanglementMeasures of entanglement

Maximum Entropy for n-qubits

Strong subadditivity theorem

implies concavity on a chain of spins

nInn 221=ρ nS

n

innn ∑

=

=

−=

2

12 21log

21)(ρ

),(),()(),,( CBSBASBSCBAS +≤+

2MLML

LSSS −+ +≥

SL

SL-M

SL+M

Smax=n

Measures of entanglementMeasures of entanglement


Other measures of entanglement:• concurrence• negativity• purity• ….

3-party entanglement

∑=

=1,0,,

321321

321

iii

iii iiicψ

5 invariants under local unitaries:

2Atrρ

2Btrρ

2Ctrρ ( )ABBAtr ρρρ ⊗ ( )ψdetH

n-party: 2n measures of entanglement


3-party entanglement 5 invariants under local unitaries:


Von Neumann entropy has an asymptotic meaning

A B

p

A and B share p entangled states

A and B perform LOCC to distill q singlets

( )pqS p ∞→= limψ

ψ


Efficient description for slightly entangled states

BkAkk

kAB p 〉〉=〉Ψ ∑=

ζξχ

|||1

BA

H

i

H

iAB iic

B

ii

A

〉〉=〉Ψ ∑∑==

21

dim

1

dim

1

|||2

21

1

+=2121 kikkiii

VpUc

A BBA HHH ⊗=Schmidt decomposition

∑=

ΓΓ=χ

λ1

]2[]1[ 2121

k

ikk

ikiic

Efficient description

Retain eigenvalues and changes of basis


∑ ∑= =

〉=〉Ψd

i

d

inii

n

niic

1 11...

1

1...|...|

n

n

n

n

iniiiiic

][

...

]3[]2[]2[]1[]1[... 1

11

3

322

2

211

1

11....

−

−

ΓΓΓΓ= ∑ ααα

ααααααα λλ

Slight entanglement iff χ∼poly(n)

Matrix Product States

∑ ∑= =

〉=〉Ψd

i

d

inii

n

niic

1 11...

1

1...|...|

1

21

]1[ iA αα 232]2[ iA αα 343

]3[ iA αα 454]4[ iA αα 565

]5[ iA αα 676]6[ iA αα 787

]7[ iA αα

n

n

n

n

iniiiii AAAAc

][

...

]3[]2[]1[1... 1

12

3

43

2

32

1

21.... α

ααααααα∑

−

=

i

α

Approximate physical states with a finite χ MPS

IAA ii

i =+∑ ][][ ][][]1[][ iiii

i AA Λ=Λ −+∑canonical form PVWC06

λΓ=A

Efficient descriptionEfficient description

Graphic representation of a MPS χα ,,1=

di ,,1=jjj

ijA ][1+αα

ψ

Efficient computation of scalar products

operations2χd

3χnd


Local action on MPS

U

lkjiklijU δγδαδβγβαβ λλ ΓΓ=ΓΓ

~~~


n

n

n

n

iniiiii AAAAc

][

...

]3[]2[]1[1... 1

12

3

43

2

32

1

21.... α

ααααααα∑

−

=

Intelligent way to represent and manipulateentanglement

Classical analogy:I want to send 16,24,36,40,54,60,81,90,100,135,150,225,250,375,625Instruction: take all 4 products of 2,3,5 MPS= compression algorithm

n

n

n

n

iniiiiic

][

...

]3[]2[]2[]1[]1[... 1

11

3

322

2

211

1

11....

−

−

ΓΓΓΓ= ∑ ααα

ααααααα λλ


〉ΓΓ=

〉=〉

∑

∑

=

=

11,...,

)()1(

1

4

1...,...

...|....

...||

1

1

1

21

,1

1

ii

iic

nini

nii

iiimage

n

n

n

n

n

χ

αααααα

ψ

i1=1 i1=2

i1=3 i1=4

| i1 〉i2=1 i2=2

i2=3 i2=4

| i2 i1 〉105| 2,1 〉

Crazy ideas: Image compression

pixel addresslevel of grey

RG addressing


....

χ = 1PSNR=17

χ = 4PSNR=25

χ = 8PSNR=31

Max χ = 81

QPEG

• Read image by blocks• Fourier transform• RG address and fill• Set compression level: χ• Find optimal• gzip (lossless, entropic compression) •(define discretize Γ’s to improve gzip)• diagonal organize the frequencies and use 1d RG• work with diferences to a prefixed table

}{ )(aΓ


0),...,,(),...,,( 2121 =∂∂∂ nn xxxfO

)()...()()...(),...,,( 2121 2121

niiiiii

n xhxhxhAAAtrxxxf nn=

2}{min OfA

Constructed: adder, multiplier, multiplier mod(N)

Note: classical problems with a direct product structure!

Crazy ideas: Differential equations

Crazy ideas: Shor’s algorithm with MPS

Efficient descriptionEfficient description

Success of MPS will depend on how much entanglement is present in the physical state

Physics

exactS

Simulation

)(χS

If nSexact log>> MPS is in very bad shape

Back to the central idea: entanglement support

Physics vs. simulationPhysics vs. simulation

Exact entropy for a reduced block in spin chains

LcS LL 2log3 → ∞→ |1|log6 22/

λ−=∞→=cS NL

At Quantum Phase Transition Away from Quantum Phase Transition


Maximum entropy support for MPS

α

χ

αα λλ∑

=

−=1

logS

Maximum supported entanglement

χλα

1== ct

χlogmax, =≤ MPSSS


Faithfullness = Entanglement support

LcS LL 2log3 → ∞→

Spin chainsMPS

χχ

λα log1

max =→= S

Spin networks

LS LLxL → ∞→

Area law

Computations of entropies are no longer academic exercises but limits on simulations

PEPS


Physics

LcSL 2log3= VLRK02-03

LSL = For 3-SAT OL04

Simulation

0)(

S ~ .1 nNP-complete problems3-SAT Exact Cover

S ~ n log2 nFermionic systems?

S ~ r ~ nShor Factorization

S ~ nd-1/dSpin chains in d-dimensions

S ~ log2 nCritical spin chains

S ~ ctNon-critical spin chains

Local (12 levels), nearest neighbor H is QMA-complete!! AGK07


New ideas

MPS using Schmidt decompositions

Arbitrary manipulations of 1D systems

PEPS

2D, 3D systems

MERA

Scale invariant 1D, 2D, 3D systems

New ideas

New ideas

MPS for translational invariant spin chains (iTEBD)

0ψψε →− He

∑∑∑ +++ ⋅+⋅=⋅=iodd

iiieven

iii

ii SSSSSSH 111

commute commute

All even gates can be performe simultaneouslyAll odd gates can be performe simultaneouslyUse Trotter decomposition to combine them

ψ

New ideas

Bλ Aλ BλAΓ BΓBBjAAiBijβγβγαγααβ λλλ ΓΓ=Θ

ijklij

kl U αβαβ Θ=Θ~

la

Aa

ka

kl WV βααβ λ~~ =Θ

iB

Ai Vαβα

αβ λ1~ =Γ B

iBi Wβ

αβαβ λ1~ =Γ

Bλ Aλ~ BλAΓ~ BΓ~

New ideasNew ideas

Heisenberg model

060.443147182ln41

,0 −=−=exactE

S=.994-.44276223χ=8

S=1.26-.443094χ=16

S=.919-.44249501χ=6

S=.764-.44105813χ=4S=.486-.42790793χ=2

Trotter 2 order, δ=.001

New ideasNew ideas

New ideas

PEPS: Projected Entangled Pairs

iAγ

δβ

α

physical index

ancillae

Good: PEPS support an area law!!

Bad: Contraction of PEPS is #P

New results beat Monte Carlo simulations

New ideas

New ideas

MERA: Multiscale Entanglement Renormalization Ansatz

Intrinsic support for scale invariance!!

If MPS, PEPS, MERA are a good representation of QM

• Approach new problems

• PrecisionCan we do any better than DMRG?e.g.: Faithfull numbers for entropy? Exact solutions? Smaller errors?

• Can we simulate better than Monte Carlo?

• Are MPS, PEPS and MERA the best simulation solution?

Keep in mind:

• scaling of entropy: Area law

Documents

Quantum Information and the simulation of quantum systems · 2007. 8. 3. · Quantum Information and the simulation of quantum systems José Ignacio Latorre Universitat de Barcelona