Entanglement entropy and the simulation of quantum systems Open discussion with pde2007

Entanglement entropyand

the simulation of quantum systems

Open discussion with pde2007

José Ignacio LatorreUniversitat de Barcelona

Benasque, September 2007

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: MPS, PEPS, MERA

Quantum simulation needed for time evolution of quantum systemsand for non-local Hamiltonians

Classical computer

Quantum computer

?

IntroductionIntroduction

Is it possible to classically simulate faithfully a quantum system?

000 EH

Quantum Ising model

0

0)( tU

00 xj

xi

n

i

zi

n

i

xi

xiH

111

represent

evolve

read

IntroductionIntroduction

Introduction

n

i

zi

n

i

xi

xiH

111

10

01

0

0

01

10 zyx

i

i

1

1

2

1

0

1

The lowest eigenvalue state carries a large superposition of product states

ccc Ex. n=3

Naïve answer: NO

• Exponential growth of Hilbert space

d

i

d

inii

n

niic

1 11...

1

1...|...|

Classical representation requires dn complex coefficients

n

• A random state carries maximum entropy

)( LnL Tr

dLTrS LLL loglog)(

IntroductionIntroduction

computational basis

Is it possible to classically simulate faithfully a quantum system?

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

IntroductionIntroduction

Plan

Measures of entanglement

Efficient description of slight entanglement

Entropy: physics vs. simulation

New ideas: MPS, PEPS, MERA

Measures of entanglement

One qubit

1,0

1110

i

i ic

Quantum superposition

Two qubits

1,0,

211,0,

21

21

21

21

2111100100ii

BA

ii

ii

ii iiciic

Quantum superposition + several parties = entanglement

Measures of entanglement

2CH

22 CCH

Measures of entanglement

Bii

A

ii

ii

ii iiciic 21,0,

11,0,

21

21

21

21

21

• Separable states

BABii

A

ii iic

21,0,

1

21

21

BBAA

102

110

2

111100100

2

1e.g.

• Entangled states

BABii

A

ii iic

21,0,

1

21

21

10012

1e.g.

Measures of entanglement

Local realism is droppedQuantum non-local correlations

Measures of entanglement

Pure states: Schmidt decomposition = Singular Value Decomposition

BiAii

iAB p

|||1

BjA

B

ij

A

vuA i

H

j

H

iAB

|||dim

1

dim

1klkikij

VUA

A B

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

BA HHH

Measures of entanglement

1 Entangled state

Diagonalise A

Measures of entanglement

1 Separable state

BiAii

iAB p

|||1

Von Neumann entropy of the reduced density matrix

Bi

iiAAA SppTrS

1

22 loglog

1

||||i

iiiABBA pTr

ITrBA 2

1|| 1

2

1log

2

1

2

1log

2

122

BA SS

Measures of entanglementMeasures of entanglement

1 Product state0S large S large Very entangled state

e-bit

Maximum Entropy for n-qubits

Strong subadditivity theorem

implies entropy concavity on a chain of spins

nInn 22

1 nS

n

innn

2

12 2

1log

2

1)(

),(),()(),,( CBSBASBSCBAS

2MLML

L

SSS

SL

SL-M

SL+M

Smax=n

Measures of entanglementMeasures of entanglement

222 CCCH

Efficient description for slightly entangled states

BkAkk

kAB p

|||1

BA

H

i

H

iAB iic

B

ii

A

21

dim

1

dim

1

|||2

21

1

2121 kikkiii VpUc

A BBA HHH Schmidt decomposition

1

]2[]1[ 21

21k

ikk

ikiic

Efficient description

Retain eigenvalues and changes of basis

Efficient description

d

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....

Slight entanglement iff poly(n)<< dn

• Representation is efficient• Single qubit gates involve only local updating• Two-qubit gates reduces to local updating• Readout is efficient

Vidal 03: Iterate this process

ndndparameters 2#

efficient simulation

Efficient description

),(|

Graphic representation of a MPS ,,1

di ,,1j

jj

ijA ][

1

Efficient computation of scalar products

operations2d

3nd

Efficient description

Efficient computation of a local action

U

lklk

jiklij MU

~~~))((

Efficient description

3d

Matrix Product States

d

i

d

inii

n

niic

1 11...

1

1...|...|

1

21

]1[ iA 2

32

]2[ iA 3

43

]3[ iA 4

54

]4[ iA 5

65

]5[ iA 6

76

]6[ iA 7

87

]7[ iA

n

n

n

n

iniiiii AAAAc ][

...

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

12

3

43

2

32

1

21....

i

α

Approximate physical states with a finite MPS

IAA i

i

i ][][ ][][]1[][ iii

i

i AA canonical form PVWC06

A

Efficient descriptionEfficient description

n

n

n

n iniiiii AAAAc ][

...

]3[]2[]1[1 1

12

3

43

2

32

1

2

1 ....

Intelligent way to represent, manipulate, read-outentanglement

Classical simplified 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

1

1 ....

Efficient description

Adaptive representation for correlations among parties

11,...,

)()1(

1

4

1...,...

...|....

...||

1

1

1

21

,1

1

ii

iic

nini

nii

iiimage

n

n

n

n

n

i1=1 i1=2

i1=3 i1=4

| i1 i2=1 i2=2

i2=3 i2=4

| i2 i1 105| 2,1

Spin-off: Image compression

pixel addresslevel of grey

RG addressing

Efficient description

....

= 1PSNR=17

= 4PSNR=25

= 8PSNR=31

Max = 81

QPEG

• Read image by blocks• Fourier transform• RG address and fill• Set compression level: • 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

}{ )(a

Efficient description

0),...,,(),...,,( 2121 nn xxxfO

)()...()()...(),...,,( 2121 21

21niii

iiin xhxhxhAAAtrxxxf

n

n2

}{min OfA

Note: classical problems with a direct product structure!

Spin-off: Differential equations

Efficient descriptionEfficient description

0),,(),,( 11 nn xxxx

D D DD

D

2222

2

2

Efficient description

Matrix Product States for continuous variables

)(2

1 21

1

222

aa

n

aapm

dxdtH

)()()(.... 21][1

...

]2[]1[1 21

12

2

32

1

2 niiiinii xxxAAA

n

n

n

n

Harmonic chains

MPS handles entanglement Product basis

di ,,1

Truncate tr d tr

2,,1n

d

Efficient description

][][ AHA

i

iiHH 1,Nearest neighbour interaction

][AH

][A

0][][

][][][

AA

AHA

A i

Minimize by sweeps

Choose Hermite polynomials for local basis )()exp()( 2 xhaxx ii

optimize over a

Efficient description

Results for n=100 harmonic coupled oscillators(lattice regularization of a quantum field theory)

dtr=3 tr=3

dtr=4 tr=4

dtr=5 tr=5

dtr=6 tr=6

Newton-raphson on a

Efficient description

Errorin Energy

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

Physics

exactS

Simulation

)(S

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

Lc

SLL 2log

3

|1|log6 22/ c

S NL

At Quantum Phase Transition Away from Quantum Phase Transition

Physics vs. simulationPhysics vs. simulation

Maximum entropy support for MPS

2

1

2 log

S

Maximum supported entanglement

12 ct

logmax, MPSSS

Physics vs. simulationPhysics vs. simulation

Faithfullness = Entanglement support

Lc

SLL 2log

3

Spin chainsMPS

log1

max S

Spin networks

LSLLxL

Area law

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

PEPS

Physics vs. simulationPhysics vs. simulation

Exact Cover

A clause is accepted if 001 or 010 or 100

Exact Cover is NP-complete

0 1 1 0 0 1 1 0

For every clause, one out of eight options is rejected

instance

NP-complete

Entanglement for NP-complete problems

3-SAT is NP-completek-SAT is hard for k > 2.413-SAT with m clauses: easy-hard-easy around m=4.2n

Adiabatic quantum evolution (Farhi,Goldstone,Gutmann)

H(s(t)) = (1-s(t)) H0 + s(t) Hp

Inicial hamiltonian Problem hamiltonian

s(0)=0 s(T)=1t

Adiabatic theorem:

if

E1

E0

E

t

gmin

NP-complete

Adiabatic quantum evolution for exact cover

2)1( kjiC zzzH

|0> |0> |0>|0>|1> |1>|1> |1>

(|0>+|1>) (|0>+|1>) (|0>+|1>)….(|0>+|1>)

NP-complete

2

1 zi

iz

NP problem as a non-local two-body hamiltonian!

n=100 right solution found with MPS among 1030 states

Non-critical spin chains S ~ ct

Critical spin chains S ~ log2 n

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

Fermionic systems? S ~ n log2 n

NP-complete problems3-SAT Exact Cover

S ~ .1 n

Shor Factorization S ~ r ~ n

Physics vs. simulationPhysics vs. simulation

New ideas

MPS using Schmidt decompositions (iTEBD)

Arbitrary manipulations of 1D systems

PEPS

2D, 3D systems

MERA

Scale invariant 1D, 2D, 3D systems

New ideas

Recent progress on the simulation side

2. Euclidean evolution

0 He

H

H

e

e

'

Non-unitary evolution entails loss of norm

oddeveni

ii HHhH 1,

evenoddevenoddeven HHHHH eeee 2/2/)(

oddeven HH , are sums of commuting pieces

Trotter expansion

MPS

Ex: iTEBD (infinite time-evolving block decimation)

even

odd

A A AB B

B A A

A B

B

A B

Translational invariance is momentarily broken

MPS

B A BA B BBjAAiBij

ijklij

kl U ~

la

Aa

ka

kl WV ~~

iB

Ai V

1~ B

iBi W

1~

B A~ BA~ B~

i)

ii)

iii)

iv)

MPS

B A BA BL R

RLRL ,,

Schmidt decomposition produces orthonormal L,R states

MPS

Moreover, sequential Schmidt decompositions produce isometries

B A BA BL

i

LL '

= AiAiB *2

are isometries

MPS

),(),( H

Energy

Read out

Entropy for half chain

1

22 logS

MPS

Heisenberg model

060.443147182ln4

1,0 exactE

=2 -.42790793 S=.486

=4 -.44105813 S=.764

=6 -.44249501 S=.919

=8 -.44276223 S=.994

=16 -.443094 S=1.26

Trotter 2 order, =.001

New ideasNew ideas

entropy

energy

Convergence

MPS

Local observables are much easier to get than global entanglement properties

S

M

Perfect alignment

1)( **

MPS

New ideas

PEPS: Projected Entangled Pairs

iA

physical index

ancillae

Good: PEPS support an area law!!

Bad: Contraction of PEPS is #P

New results beat Monte Carlo simulations

New ideas

A B

Entropy is proportional to the boundary

LSS BA

Contour A = L

“Area law”

Some violations of the area law have been identified

PEPS

i

i

i AAE *

''

'' '

'

''

As the contraction proceeds, the number of open indices grows as the area law

PEPS

2D seemed out of reach to any efficient representation

Contraction of PEPS is #P

Building physical PEPS would solve NP-complete problems

Yet, for translational invariant systems, it comes down to iTEBD !!

E E

Comparable to quantum Monte Carlo?

E

PEPSPEPS

E becomes a non-unitary gate

PEPS

PEPS

r

rz

rx

rr

rx hH

][]'[

)',(

][

Results for 2D Quantum Ising model (JOVVC07)

)( hhAmz

)1(332.)1(06.33

)3(346.)1(10.32

5.41

h

h

h

MC 327.044.3 h

PEPS

MERA

MERA: Multiscale Entanglement Renormalization Ansatz

Intrinsic support for scale invariance!!

),( U

MERA

MERA All entanglemnent on one line

All entanglemnent distributed on scales

MERA

Contraction = Identity

MERA

U

)(max UtrU

WV

WUU

Update

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

• Approach hard problems

• PrecisionCan we simulate better than Monte Carlo?

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

• Spin-off?

Physics:

• Scaling of entropy: Area law << Volume law• Translational symmetry and locality reduce dramatically the amountof entanglement• Worst case (max entropy) remains at phase transition points

Physics vs. simulation

Quantum Complexity Classes

QMA

11),(Pr/ xVLx yes

L is in QMA if there exists a fixed and a polynomial time verifier (V) such that

1),(Pr/ xVLx no

What is the QMA-complete problem?

Feynman idea (shaped by Kitaev)

..1 chttVH t

QMA

QMA

QMA-complete problem

• Log-local hamiltonian• 5-body• 3-body• 2-body (non-local interactions) • 2-body (nearest neighbor 12 levels interaction)!

Given H on n-party decide ifbE

aE

0

0

)(

1

npolyba

Open problems

• Separability problem (classification of completely positive maps)• Classification of entanglement (canonical form of arbitrary tensors)• Better descriptions of quantum many-body systems• Spin-off of MPS??• Rigorous results for PEPS, MERA• Need for theorems for gaps/correlation length/size of approximation• Exact diagonalisation of dilute quantum gases (BEC)• Classification of Quantum Computational Complexity classes• ….