Quantum Information and the simulation of quantum systems

José Ignacio Latorre Universitat de Barcelona Perugia, July 2007 In collaboration with: Sofyan Iblisdir, Luca Tagliacozzo Arnau Riera, Thiago Rodrigues de Oliveira, José María Escartín, Vicent Picó Román Orús, Artur García-Sáez, Frank Verstraete, Miguel Aguado, Ignacio Cirac

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 computer Quantum computer Classical simulation of Quantum Mechanics is related to our ability to support large 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)

?

Is it possible to classically simulate faithfully a quantum system?

Heisenberg model

H

 

i S



i



S



i

 1

H

 0 

E

0  0

represent

 0

U

(

t

)  0  0

O

1

O

2  0

evolve read

Misconception: NO Exponential growth of Hilbert space

n

|   

i

1 1 ...

i n d d

    1

c i

1 ...

i n

|

i

1 ...

i n

 Classical representation requires

d n

complex coefficients

A random state carries maximum entropy



L



Tr

(

n



L

)  

S

( 

L

)  

Tr

 

L

log 

L

 

l

log

d

Refutation Realistic quantum systems are not random

• •

symmetries (translational invariance, scale invariance) local interactions We do not have to work on the computational basis

•

use an entangled basis

e.g: efficient description for slightly entangled states

Schmidt decomposition

H



H A



H B

A B |  

AB

|  

AB

  dim

i

1 

H

 1

A



k

  1 dim

i

2 

H

 1

B

c

i

1

i

2

p k

|

i

1 

A

|

i

2 

B

| 

k



A

| 

k



B c i

1

i

2  

= min(dim H A , dim H B )

Schmidt number

k

   1 

k

[ 1 ]

i

1 

k



k

[ 2 ]

i

2

c

i

1

i

2 

U

i

1

k

p

k

V

ki

2  A product state will have   1

Vidal 03

: Iterate this process |   

i

1 1 ...

i n d d

    1

c i

1 ...

i n

|

i

1 ...

i n



c i

1 ...

i n

  1 ...

 

n

 1  [ 1 ]

i

1  1  [ 1 ]  1  [  2 1 ]

i

2  2  [  2 2 ]  [  3 ] 2 

i

3 3 ....

 [ 

n n

]

i n

 1 A product state iff 

i

 1 #

parameters



nd

 2 

d

n

Slight entanglement iff 

poly(n)<< d n

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

Matrix Product States

|   

i

1 1 ...

i n d d

    1

c i

1 ...

i n

|

i

1 ...

i n



i

A

  

α

A

[  1 ] 1 

i

1 2

c i

1 ...

i n A

[  2 ]

i

2 2  3 

A

[  3 3 ] 

i

3 4

A

[  4 4 ] 

i

4 5  2 ...

 

n

 1

A

[ 1 ]

i

1 1  2

A

[  5 5 ] 

i

5 6

A

[  2 2 ]

i

2  3

A

[  6 6 ] 

i

6 7

A

[  7 ]

i

7 7  8

A

[  3 ]

i

3 3  4 ....

A

[ 

n n

1 ]

i n

canonical form PVWC06 

i A

[

i

]

A

[

i

]  

I



i A

[

i

]   [

i

 1 ]

A

[

i

]   [

i

]

Approximate physical states with a finite



MPS

Graphic representation of a MPS 

A

[ 

j

]

i j



j j

 1 Efficient computation of scalar products   1 ,  , 

i

 1 ,  ,

d

operations

d

 2

nd

 3

Local action on MPS

U

U

ij kl



i

   

j

 

k

   ~

l



Intelligent way to represent and manipulate

entanglement

c i

1 ...

i n

  2 ...

 

n

 1

A

[ 1 ]

i

1 1  2

A

[  2 2 ]

i

2  3

A

[  3 ]

i

3 3  4 ....

A

[ 

n n

1 ]

i n c i

1 ...

i n

  1  ...



n

 1  [ 1 ]

i

1  1  [ 1 ]  1  [  1 2 ] 

i

2 2  [  2 2 ]  [  3 2 ]

i

3  3 ....

 [ 

n

]

i n n

 1

Classical analogy:

I want to send

16,24,36,40,54,60,81,90,100,135,150,225,250,375,625

Instruction:

take all 4 products of 2,3,5 MPS= compression algorithm

Crazy ideas: Image compression

i 2 =1 i 2 =2 i 1 =1 i 1 =2 i 2 =3 i 2 =4

   RG addressing

i 1 =3 i 1 =4

|    1

image

   ,...,  

n

  1  ( 1 ) 1 

i

1 2

i

1 , ..., 4 

c

 1

i

1

i n

...

i n

|

i n

...

i

1 

level of grey

....

  (

n n

) 

i n

1 |

i n

...

i

1 

pixel address

QPEG

• • • •

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

 • • •

Find optimal

{  (

a

) }

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

....

Max



= 81



= 1 PSNR=17



= 4 PSNR=25



= 8 PSNR=31

O

(  1 ,  2 ,..., 

n

)

f

(

x

1 ,

x

2 ,...,

x n

)  0

f

(

x

1 ,

x

2 ,...,

x n

) 

tr

(

A i

1

A i

2 ...

A i n

)

h i

1 (

x

1 )

h i

2 (

x

2 )...

h i n

(

x n

) min {

A

}

Of

2 Constructed: adder, multiplier, multiplier mod(N)

Note: classical problems with a direct product structure!

Back to the central idea: entanglement support

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

Physics

S exact

Simulation

S

(  ) If

S exact

 log

n

MPS is in very bad shape

Exact entropy for a reduced block in spin chains At Quantum Phase Transition Away from Quantum Phase Transition

S L



L



c

3 log 2

L S L



N

/ 2   

c

6 log 2 | 1   |

Maximum entropy support for MPS

S

      1   log   Maximum supported entanglement   

ct

 1 

S



S MPS

, max  log 

Faithfullness = Entanglement support

S L

Spin chains 

L



c

3 log 2

L

   1  

S

max  log 

MPS

Spin networks

S

LxL



L



L

Area law

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

Physics

S L



c

3 log 2

L d dt S L

(

t

)  0 VLRK02-03 LLRV04 Simulation

S L



L

For 3-SAT OL04 Exact RG on states VCLRW05 Lipkin model

E

1  1 3

S

Area law OLRV05 100-qubit Ex-cover instance Image compression L05 BOLP05 OLEC06 RL06 Laughlin ILO06 Continuous variables ILO06

Non-critical spin chains

S ~ ct

Critical spin chains

S ~ log 2 n

Spin chains in d-dimensions

S ~ n d-1/d

Fermionic systems?

NP-complete problems Shor Factorization

S ~ n log 2 n S ~ .1 n S ~ r ~ n Local (12 levels), nearest neighbor H is QMA-complete!!

AGK07

Keep in mind: Area law << Volume law Translational symmetry and locality have reduced dramatically the amount of entanglement Worst case (max entropy) remains at phase transition points

MPS and PEPS are a good representation of QM • Approach new problems • Precision Can we do any better than DMRG?

e.g.: Faithfull numbers for entropy? Exact solutions? Smaller errors?

• Can we simulate better than lattice Monte Carlo?

• Are MPS and PEPS the best simulation solution?

Simulation of the Laughlin wave function

 (

z

1 ,  ,

z n

)  

i



j

(

z i



z j

)

e

 1 2 

i z i

2 

i

(

z

) 

z a e

 1 2

z

2 Local basis:

a=0,..,n-1

Dimension of the Hilbert space

n n

Analytic expression for the reduced entropy log   

n k

    2

n



n n S

(

n

,

k

)  log  

n k

 

ILO06

Exact MPS representation of Laughlin wave function  (

z

1 ,  ,

z n

)  

i



j

(

z i



z j

)

e

 1 2 

i z i

2  (

z

1 ,  ,

z

n

)  

i



j

(

z

i



z

j

)  

i

1

i

2 

i n

z

1

i

1

z

2

i

2 

z

n i n



i

1

i

2 

i n



Tr

 

i

1 

i

2  

i n

 5   

a

, 

b

  2 

ab I

Clifford algebra

  dim 

a

 2  

n

2   

n n

 5   0  1  

n

 1  2  2

S

  

n k

   2

n

Optimal solution!

(all matrices equal but the last!)

 (

z

1 ,  ,

z n

)  

i



j

(

z i



z j

)

m e

 1 2 

i z i

2

m=2



i

1

i

2 

i n



j

1

j

2 

j n

 

Tr Tr

   

i

1

a

1 

i

2    

a n



i n

 5 5  5    

j

1 

j

2  

j n

 5  

a



i

  

j a



i i j

  

i

 

j

 5  5   5   5

optimal

 dim 

a

 2

m

 

n

2  

Example: Normalization of wave function for m=2

n

 2

n

 4

n

 6

n

 8

A



6

A



2520

A



7484400

A



8172964800 0

So far, we have not managed to exploit the product structure

        

Translational invariant spin chains

Vidal05: iTEBD translationaly invariant infinite system algorithm

e

 

H

   0

H

 

i S



i



S



i

 1  

ieven S



i



S



i

 1  

iodd S



i



S



i

 1

commute commute All even gates can be performe simultaneously All odd gates can be performe simultaneously Use Trotter to combine them



B



A



A



B



B



B A



A B



B



ij

  

B

 

Ai

  

A



Bj

 

B



kl

 

U ij kl



ij



kl

 

V



k a



a A W



l a



Ai

   1

B



V i



Bi

 

W i

  1

B



  2  

Ai



are isometries

 * 

Ai

   Energy  (  ,  )

H

 (  ,  )

=

Heisenberg model

E

0 ,

exact

 1 4  ln 2   0.44314718

06  =2  =4  =6  =8  =16

Trotter 2 order,



=.001

-.42790793

-.44105813

-.44249501

-.44276223

-.443094

S=.486

S=.764

S=.919

S=.994

S=1.26

Trotter 2 nd order

.1

.01

.001

-.42474325

-.425759

-.427876149

.0001

-.427904824

.00001 -.427907692

  2

S



S

max  log 

Exponential distribution λ Poorness of DMRG

Advantage: clean results for infinite half chain entropy Problem: Poor convergence of entropy entropy energy Maximum half-chain entanglement for Heisenberg model

H



c XX XX



c YY YY



c ZZ ZZ

{

c XX

max ,

c YY

,

c ZZ

}

S



c XX



c YY



c ZZ

Consistent with central charge c=1 Attention to spontaneous symmetry breaking

To compute block entropies, use exact coarse graining of MPS

  

A i

1

A i

2 

A i n i

1

i

2 

i n A p



A q



A p

 

A

( (

pq

 ) )  min(

l d

  2 1 , 

U

2 )

l pq



l V l

,  Local basis  

A

'

l

  

l V l

, 

Optimal choice!

VCLRW

 remains the same and locks the physical index!

After

L

spins are sequentially blocked

A

( 

L

    )  2 Entropy is bounded Exact description of non-critical systems

Exact solution for



=2

min

min

 ,  

Tr

(



B

2 

Ai



A



Bj



B

2  *

Bk



A

 *

Al

)

H ijkl



=

  2 

Ai

  * 

Ai

   

A

 

B



A

1  

B

2 

A

2   

B

1    *  

T

 1    0 0

b

2  1  2 2 0

a

   2    0

b a

2  2  2  2 4 2  1

b

0   4  3  2 2  14  4 2  8  6 2  16  8 2  0

E

0  9 4  1  2 2 S= .485704202

   1 2  2  1  2  2  2 2   .

427908010

Numerics

Precision for entropy requires some extra effort

Trotter higher order Random seeds

(avoiding hysteresis cycles associated to the minimization procedure)

Boost



i

  '

i



i

  

i

 

m

(  '

i

  

i

 )

S M

Perfect alignement  * (  )      *  1

MPS support of entropy obeys scaling law!!

S S

 1 6 log 

χ

   2 ??

Physics Entanglement

So far

Simulation technique • representation • evolution • observables Entanglement support NEXT Exploit MPS, PEPS, MERA

Contraction of PEPS is #P Yet, for translational invariant systems, it comes to iTEBD JOVVC07 Beats quantum Montecarlo!!

VIDAL Beyond MPS: Entanglement RG MERA Unitary networks Building the program: detailed check vs MPS