TEBD and Its Generalizations

8/3/2019 TEBD and Its Generalizations

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Key laboratory of Quantum Information, CAS

TEBD and its generalizations

2010.09.21

Zhiyuan Yao


2/29

The time-evolving block decimation (TEBD) algorithm is a numericalscheme used to simulate one-dimensional quantum many-body systems,characterized by at most nearest-neighbour interactions.

What is TEBD :

1 2

1

1 2

, 1

[1] [2] [ ]( ) , , ,N

N

dss s

mps N

s s

NTr A A A s s s=

=

MPS( Matrix-product state) is a kind of pure state which has the form:

In fact the well-known numerical method for 1D quantum many-bodyproblems DMRG(Density matrix renormalization group) approximates theground state by minimizing the energy with respect to all MPSof the form.

e H =

What is MPS:


3/29

TEBD and its background:

Exponential growth of Hilbert space

1

1

... 1 2

1 1| ... ...n

n

d d

i i n

i ic i i i

= = =

Classical representation requires dn complex coefficients

n

The failure of classical description of the dynamics of many-body quantum physics :

Result: intractable to simulate.

TEBD offers an efficient description of many-body if the system

is slightly entangled.

eg: 2^50=1.110^15


4/29

TEBD has a close relationship with Schmidt decomposition .To

introduce Schmidt decomposition, we provide two proofs . The first oneis the Singular Value Decomposition(SVD) of any mn matrix A. Theanother one is what is usually seen on the physics textbooks , whichelucidates the physical motivation for TEBD. Both methods have theiradvantages in discuss different problems.

Schmidt decomposition

d d

1 1

| | |A B

A Bjn

j n

c j n= =

=

1

| | |A Bi i i

i

=

= 2

1( ) 1 min( , )i A Bi

d d = | | A A B B

j i ij j i ij = =

Schmidt decomposition

A BBA HHH =

The state vector of the system can be written as :

Schmidt decomposition:

is referred to as the Schmidt Rank

i is referred to as the Schmidt coefficents


5/29

Vidals decomposition

1 2 1

31 2

1 1 1 1 2 2 2 3 1

1 2 1

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

...

, ... 1

...n

n

n n

n

i n ii i

i ic

=

=

1

1

... 1 2

1 1| ... ...n

n

d d

i i n

i ic i i i

= = =

Representation is efficient Single qubit gate involves only local update Two-qubit gates also involve local updating

2# n parameters nd d


6/29

Vidals decomposition procedure

31 2

1 1 1 1 2 2 2 3 1

1 1

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

......

.... nn n

n

i n ii i

i ic

=

A B.....[2...,n][1]

1

1 1 1 1 1 1

1 1 1

[1][1] [1] [2 ] [1] [2 ]

1

,

| | | | |in n

i

i

= =

Schmidt decomposition of the bipartite [1]:[2...,n]

A B.....[3...,n][1,2]

1 1 2

2

[2 ] [3 ]

2 ,| | |n n

i

i

i = (i)

(ii)

(iii)

We proceed according to three steps

{ }2

1 2 1 2 2 2 22

2

[2 ][3 ] [2] [3 ] [3 ] [3 ]

, 1| | | the eigenvectors of in n n n

i is

= =

1 2

1 1 1 2 2 2

1 2 1 2

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

1 2

, , ,

| = | | |i i n

i i

i i

Iterate this process sequentially for the rest bipartite.


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31 2

1 1 1 2 2 2 3 1

1 1 1

1 2

1 1 1 2 2 2 3

1

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

1 2

1 1 ...

[ 1] [3[2 ][( 1) ] [ 1] [2]

1 1

... .... ...

...

l

l l l

l l

l

l l l l

l n

d di l ii il

l

i i

d dl i il n l

i i

where

i i i

+

+ +

+

= =

++ +

= =

=

=

3

1

1 1

] [ ]

1 2

...

.... ...nn

l

i n i

l l ni i i

+ +

1

1

31 2

1 1 1 2 2 2 3 1

1 1 1

... 1 2

1 1

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

1 2

1 1 ...

| ... ...

... .... ...

n

n

n

n

n n

d d

i i n

i i

d di n ii i

n

i i

c i i i

i i i

= =

= =

=

=

Vidals decomposition

Finally we arrive at Vidals decomposition :

From which the Schmidt decomposition of the bipartite [1,2...,l]:[l+1,l+2...,n]is readily seen:

[ ] [1 ] [( 1) ]|l l l

l

l l l n

+ =


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Lemma 1: Updating the s ands of state |> after a unitaryoperation U acts on qubit l involves transforming only [l].

Proof :In the SD according to the splitting [1] :[2n], a unitary operation U on qubitl does not affect the Schmidt vectors for part [1] and therefore andremain the same. Then, by considering the SD for the splitting [2] :[3n], weconclude that also and remain unaffected. By the same token , weconclude that

remain unchanged.

Similarly, remain unaffected.

Write the state vector in Vidals decomposition form,we can readily find out that the only changed changes as

[1]

1

1

[1]i

2

[2 ]

2

1 2

[2 ]i

1

[ ][ ]

,and (1 j 1)

j

j j j

j ij

l

1

[ ][ 1]

,and (l+1 j )j

j j j

j ijn

[ ]l i

[ ] [ ]

, ,

1

dl i l j

ij

j

U =

= Unlike classical representation, in Vidals decomposition formalism updating

qubit l involves only the local tensor, thus it can save us a lot of calculation efforts.

U

[ 1]l [ ]l[ ]l

[ 1]l [ ]l[ ]l


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Details of the proof:

11 1 12 2 1

21 1 22 2 2

1 1 2 2

0

0

0

n n

n n

d d dn n

a x a x a x

a x a x a x

a x a x a x

+ + + = + + + =

+ + + =

2 2

1 2 1 2 1

[2] [2]i ix =

1

1 1 1 1

[1] [1]=i

ia

1

1 2 2

1 1 1 2 1 2

1

[1] [2] [2][1]

1 2( )=0 (1 , )i i i

i i d

2 2

1 2 1 2

[2] [2]

i iand The problem lies in that whether or not we can prove are indentical.

Suppose that after that one-qubit gate, the coefficient changes into ,

From what reasoned in the previous slide, we have

Denote

1 1

1 1 1 1 1 1 2

1 1 1 2 1

[1] [1][1] [2 ] [1]

1 1 , ... 2

, , ... ,

...n

n

i in

i i n

i i i i

i i U i i

= =

rank A rank AU = =

Noticing that :

So we can conclude that :2 2

1 1 2 1 2

[2] [2]

0, .

i i

x = =

2

1 2

[2 ]i

21 2

[2 ]i


10/29

Lemma 2: Updating the s ands of state |> after a unitary operationV acts on qubit l and l+1 involves transforming only [l], [l+1] and[l] .

Proof (non-degenerate case) :

1l l A BJ += H H H K = H H

Due to what is reasoned in the proof of Lemma 1, the conclusion ofLemma 2 is obvious. Now we derive the transformation of the local tensorsafter a unitary operation V.

[1 ] [( 1) ]

l

l l

l l n

+ =

1

1 1 1

1

[ ][1 ] [ 1] [1 1]

1

l

l

l l l l l

l l

dl il l l

l

i

i

=

=

1

1

1 1

1 1

[ 1][( 1) ] [ 1] [ 2 ]

1

1

l

l

l l l l

l l

dl il n l l n

l

i

i

+

+

+ +

+ +

++ + +

+

=

=

Based on Vials decomposition, the SD ofpartition[1,2...,l]:[l+1,l+2...,n] is:

Where the Schmidt vectors are

V

[ 1]l

[ ]l

[ 1]l

+[ ]l

[ 1]l +

[ 1]l [ ]l

[ 1]l +[ ]l [ 1]l +


11/29

[ 1]l +

[ 1]l

[ ]l

[ 1]l

+[ ]l

|

Aj |

Bn

BA jnC

1

[1 1]|l

l

A lj i

= 1

[ 2 ]

1| ll n

B ln i

+

++ =

Now respectively choose a set of basis

for subsystem A and B:

d d

1 1

| | |A B

A Bjn

j n

c j n= =

=

[ ]

, ,l l l

l

l H

jn j nC U V

=

1 1

1 1 1 1 1[ ] [ 1][ 1] [ ] [ 1]l l l l

l l l l l l l l l

l

i i l i l il l ljnC

+ + ++ += = 1 1 1( 1) , ( 1)l l l l j d i n d i + += + = +

1 1 1 1

1 1 1 1 1 1 1

1

[ ] [ 1][ 1] [ ] [ 1]l l l l l l l l

l l l l l l l l l l l l l

l l l

i i i i i i l i l il l l

jn i i

i i

C V

+ + +

+ +

= = = After a unitary operator acts on qubit l and l+1, the coefficients transform as:

Make SVD of Matrix C:

| U | , | |l l l l

A B H

j A n B j V n = = [ ]| | |

l l ll

l A B

=

The state vector becomes :


12/29

Recall the uniqueness of Schmidt decomposition when the eigenvaluesare different, then we have :

[1 ] [( 1)[ ]]

l l

l

l

l l nl

+

=

Write the SD of the state vector in Vidals decomposition form:

1

1 1 1

1

[ ][1 ] [ 1] [1 1]

1

l

l

l l l l l

l l

dl il l l

l

i

i

=

=

1

1

1 1

1 1

[ 1][( 1) ] [ 1] [ 2 ]

1

1

l

l

l l l l

l l

dl il n l l n

l

i

i

+

+

+ +

+ +

++ + +

+=

=

1

1 1

1

[1 ] [1 1]

( 1) ,

1

| U | = Ul

l l l l l l l

l l

dl A l

j A d i l

i

j i

+

=

= =

1 1 1

[ ][ 1]

( 1) ,Ul

l l l l l l

l il

d i

+ =

1 1 1

[ ] [ 1]

( 1) ,Ul

l l l l l l

l i l

d i

+ =

1

1 1 1

1 1

[( 1) ] [ 2 ]

,( 1) 1

1

| |l

l l l l l l l

l l

dl n B H H l n

n B d i l

i

V n V i

+

+ + +

+ +

+ + + +

=

= = =

1

1 1 1

[ 1] [ 1]

,( 1)l

l l l l l

l i l H

d iV

+

+ + +

+ +

+ = 1

1 1 1

[ 1] [ 1]

,( 1)l

l l l l l

l i H l

d iV

+

+ + +

+ + + =


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Swapping technique

In dealing with periodic boundary problems and systems with long-range

interactions, swapping technique plays a crucial role.

The state vector is :1 1

1

1 1 1

1 1 1

[1 1] [ 2 ]

1

l l

l l

l l l l

l l l l

di i l l n

l l

i i

i i

+

+

+ +

+

+ =

1 1

1 1 1 1 1

[ ] [ 1][ 1] [ ] [ 1]l l l l

l l l l l l l l l

l

i i l i l il l l

+ + +

+ + =

1

1

l l

l l

i i

jnC

=

1 1 1( 1) , ( 1)l l l l j d i n d i + += + = +

[ ]

, ,l l l

l

l H

jn j nC U V

=

1 1 1 1

[ ] [ 1]

( 1) ,Ul

l l l l l l

l i l

d i +

+ = 1

1 1

[ 1] [ 1]

,( 1)

l

l l l l l

l i H l

d iV +

+ +

+ +

+ =

[ 1]l +[ 1]l [ ]l [ 1]l +[ ]l|

Aj |

Bn

BA jnC

1

[1 1]

| A ll

lj i

=

1

[ 2 ]

1| B ll n

ln i ++

+ =

[ 1]l +[ 1]l [ ]l [ 1]l +[ ]l

|A

j |B

n

BA jnC

1

[1 1]

1| A ll

lj i

+ =

1

[ 2 ]|B l

l n

ln i +

+ =

swapRecall the procedure of Vidals decomposition, wecan conclude that all the tensors exceptremain the same.

Make the SVD of Matrix C:

[ ] [ ] [ 1], andl l l

+


14/29

Swapping techniques for ( periodic boundary condition):

1,nH

1,nH

1 2 1n ni i i i

1 2 2 1n n ni i i i i

1 2 3 2 1n n n ni i i i i i

1 2 2 1n n ni i i i i

PBC


15/29

Imaginary time evolution for ground state

lim

H

gr H

e

e

+ =

The ground state of a Hamiltonian system can be found out bymeans of Imaginary time evolution so long as 0.gr

i i

i

a = is the egienvector of corresponding the egienvalue of

i iH E

i

EH

i i

ie a e

=

Proof:

The proof is almost the same with that of using power method to findout the egienvector with the biggest eigenvalue in computational methods.

0

0

00

0 0

lim = lim

EH

grH E

e e

ae

a

e

+ += It is easy to see


16/29

Generalizations for excited state

In the orthogonal complement space of the subspace spanned by ,the lowest eigenvalue is E1 .

0

0 0 00 = =

10 , for the same reason:if

1lim

H

H

e

e

+

=

This can be done sequentially to find out the other excited states.

0 0 0 1 0 = = =

2lim

H

H

e

e

+

=


17/29

Generalizations for excited state

Apart from the first scheme in which we find the first excited state in theorthogonal complement space , we can construct a new Hamiltonian of whichthe excited state of the original Hamiltonian turns out to be the ground state ofthe new Hamiltonian.

0 0 i 0 0

i 0

0 0 0 1 1 1 2 2 2( )

i i H H E

E E E

=

= + = +

= + + + +

1

( is the energy gap between the ground state and the first excited

state), then turns out to be the ground state of . So what is designed to find

the ground state can be used to

Take

H

>

find the excited states.

A combination of the previous two schemes can also be used to findthe excited states.

Note:


18/29

Efficient simulation by means of Vidals decomposition formalism

In the open boundary problem(OBC), the Hamilton of a systemcounting in adjacent interacting effect can be written as :

, 1

, 11, 1( )

n n

j i i even odd j i H H H H H

+= == + = +

1 12 2 23 1,n n n H H H H H H H = + + + + + +

Make a division of the Hamilton as :

2 1 2 1,2 2 1

1 2 1 2

( )odd m m m m

m n m n

H H H

= + = H

2 2 ,2 1 21 2 1 2

( )even m m m mm n m n

H H H +

= + =

H

1 12 3 34odd H H H H H = + + + +

2 23 4 45even H H H H H = + + + +

where

One advantage of such division is :

2 2[ , ] 0i j =H H 2 1 2 1[ , ] 0i j =H H

n


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Division of Hamilton in 1D with next-nearest-neighbor interaction in consideration :

1 1,2 2 2,3 3 3,4 4 4,5

1,3 2,4 3,5 4,6

1 2

1 1,2 2 2,3 3 3,4 4 4,5

1,3 2,4 3,5 4,6 5,7 6,8 7,

( )

( )

NB NNB NB NNB NNB

NB

NNB

H H H H H H H H H

H H H H

where

H H H H H H H H

H H H H H H H

= + + + + + + + +

+ + + + +

= + = + +

= + + + + + + + +

= + + + + + +

H H H H H

H

H

9 8,10 9,11

1 1,3 2,4 5,7 6,8

2 3,5 4,6 7,9 8,10

NNB

NNB

H H

H H H H

H H H H

+ + +

= + + + += + + + +

H

H

Advantages:

1. Generally speaking, hence the error produced by Suzuki-Trotter

expansion will be reduced. NNB NBH H

2. Less swapping times will be needed, thus the calculation effort is saved.

After we swap site 2 and site 3, two terms ( ) can be handled, the

same with site 4 and site 5, etc. Therefore, the swapping time is ashalf as theprevious method.

1,3 2,4andH H

1 3 42 65

3. The program can be easily written based on the original one.

1 3 42 65


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Here we provide two alternative partitions of the 2D Hamilton. The first one

is from the perspective of effective interaction range . The anther one, however,utilize the concept of solid physics.

Division of Hamilton in 2D:

( , )x yn n ( , )x yn n ,

,x y

x y

n n

n nH We denote by the interaction Hamilton between site and .

Then the Hamilton can be divided as (with only the on-site and nearest-neighborinteraction taking into consideration) :

2 2 ,2 2 1,2 1

2 1,2 2 1,2 2 1,2 2 1,2

,

2 1,2 2 ,2 1 2 1,2 1 2 ,2 2

2 ,2 2 ,2 2 ,2 2 ,2 1 2 ,2 1 2 ,2 1

, ,

1, , 1, , ,

,

+ ( )+ (

( )+ ( )

( )

i j i j

i j i j i j i

i j

x y x y

x y x y x y

x y

i j i j i j i j

i j i j i j i j i j i j

i j i j

n n n nn n n n n n

n n

H H H H

H H H H H H

H H H H

+ + +

+ + + +

+ + + + +

+ + +

+ +

+ +

+ + + +

= + +

=

2 2,2 1 2 1,2 2

1 2 1,2 1 2 1,2 1

,

2 ,2 2 ,2 1 2 1,2 2 1,2 1

,

0,0 0,1 1,0 1,1

)

= ( )

=

i j i j

j i j i j

i j

i j i j i j i j

i j

H H+ + + +

+ + + + +

+ + + +

+ +

+ + +

+ + +

H H H H

H H H H

2 ,2 2 ,2[ , ] 0 , , k 0,1

i k j i k ji j i j

+ + + + = =H H Any two terms belong to the same commute!H H


21/29

1, , 1

, , ,

,

1, , 1

, , ,

,

, ,

( )

=

=

x y x y

x y x y x y

x y

x y x y

x y x y x y

x y y x x y

n n n n

n n n n n n

n n

n n n n

n n n n n n

n n n n n n

s r i l i

i i

s r l

H H H H

H H H

+ +

+ +

= + +

= + +

+ +

+ +

H H H

H H H

The other partition is :

, , , ,

Also [ , ] [ , ] 0 (i j)r i r j l i l j

= = H H H H

After making a reasonable division of the Hamilton, then the problemscome down to partite the 2D system.

An alternative division of Hamilton in 2D:


22/29

The corresponding path topartite the 2D system

In such decomposition, we can reduce the 2D problems into 1D by the means of.swapping techniques

According to the way we partite the Hamilton of the system, we can present areasonable way to partite the system .

Swapping needed


23/29

Problems with generalizations of TEBD in 2D and 3D cases:

In previous discussion, we have generalized the TEBD algorithm to handle 2D,

3D problems . Then one might be tempted to claim that many 2D, 3D problems can becalculated by means of TEBD algorithm. However, we are not so fortunate. An

investigation of Area Laws will reveal the very reason.

To introduce Area Laws, it necessary to introduce the concept of entanglement

entropy.

In classical physics concepts of entropy quantify the extent to which we areuncertain about the exact state of a physical system at hand or, in other words, the

amount of information that is lacking to identify the microstate of a system from all

possibilities compatible with the macrostate of the system. If we are not quite sure

what microstate of a system to expect, notions of entropy will reflect this lack of

knowledge.

In quantum mechanics, however, positive entropies may occur even without a lackof information. Take two electrons which are in EPR state for example, the whole system

has the vanishing Von Neumann entropy :

2( ) [ log ]S tr =


24/29

Area Laws:

( )12

A B A B =

A B

( ) ( )12Are B A A A ATr = = +

EPR

2( ) [ log ] 1 A A A

re re reS tr = =

2( ) [ log ] 0S tr = =

Entanglement entropy and Area Laws:

But the entropy of the subsystem A or B has nonvanishing Von Neumann

entropy.

The mentioned quantity, the entropy of a subsystem, is calledentanglement

entropy orgeometric entropy and in quantum information

entropy of entanglement.

For ground states, the entanglement entropy

of the reduced state of a subregion often merelygrows like the boundary area of the subregion,

and not like its volume, in sharp contrast with an

expected extensive behavior.


25/29

The entanglement entropy of ground states of local gapped models of the type of equation:

for bosons and of equation:

for fermions for arbitrary lattices G=(L,E) andarbitrary regionsI satisfies

( )

, , , ,

,

1

2i i j j i i j j i i j j i i j j

i jH b A b b A b b B b b B b= + + +

( ), , , ,,

1

2i i j j i i j j i i j j i i j j

i j

H f A f f A f f B f f B f = + +

Area Laws in one dimension:

Area Laws in higher dimension(Area law for gapped quasifree models):

Area Laws

In fact, MPS satisfies this Area Law.

1 2

1

1 2

, 1

[1] [2] [ ]( ) , , ,N

N

dss s

mps N

s s

NTr A A A s s s=

=

1 m2 m+L N

, 1

, ,m m L

D

L m m L m m L

+

+ +

=

=

1

1 111

11 1

,

, , 1[ ] [ 1],

S Sm m L

m m m L m LS Sm m L

m m L

d D

m m L m m Lm m L A A s s

+

+ + +=+

=+ +

+ + + =

( ) ( )2( ) log 1LS n O = +

For a gapped system, away from critical point,the entanglement entropy of a contiguous set of

quantum system of a chain, a block I={1,,n},

always scales as

2( ) 2 log LLS D

( ) ( )LS s I


26/29

Implications from Area Laws:

9 101

1 1

i i

z z z

i i

H +

= =

= +

For one dimension problems, according to Area law the entanglment entropy is

bounded by

( ) ( )2( ) log 1LS n O = +A typical value of is 1/3, for MPS: , we now calculate

the dimension D needed to truly represent the genuine ground state.2( ) 2logLS D

2 2

12log log 32

3 D n n D= =

This is conducive to our understanding of the great success of DMRG in one dimension.

e.g

In two and higher dimension, however, the

dimension of the matrix needed will be very big.

Therefore, the calculation will become intractable.

2

22log 4 2

n

D n D

= =


27/29

G

A global perspective and Conclusions:

G

Area LawSymmetry..

Hilbert space

G

G

G

The Hilbert space is large!

The space the ground statebelongs to gets smaller!

In MPS framework, we can naturally generalize TEBD method to deal with 2D

and 3D problems, but the matrix dimension D needs to be chosen so big that the

calculation is intractable. The failure of DMRG in two dimension also suggests this

point.


28/29

Development and Prospects:

Two methods, PEPS(projected entanglement pair state) and MERA(multiscale

entanglement renormalization ansatz) which aim at dealing with higher dimensionalquantum many-body problems have been put forward. Both of which satisfy Area

Laws.


29/29

Reference

G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); Phys. Rev. Lett.

93, 040502 (2004).

[1]

[4]

[2]

[3]

Ippei Danshita and Pascal Naidon , Phys. Rev. A. 79, 043601(2009)

Diplomarbeit von Mara Gracia Eckholt Perotti Matrix ProductFormalism Garching, September 2005

Open Source TEBD v2.0 Users Guide

[7]

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[6] F. Vestraete and J. I.Cirac, arXiv:cond-mat/0407066v1

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TEBD and Its Generalizations

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