Introduction to Tensor Network States Sukhwinder Singh Macquarie University (Sydney)

Introduction to Tensor Network StatesSukhwinder Singh

Macquarie University (Sydney)

Contents

• The quantum many body problem.

• Diagrammatic Notation

• What is a tensor network?

• Example 1 : MPS

• Example 2 : MERA

}D

1 2 N

Total Hilbert Space : NV

Quantum many body system in 1-D

dim( )V D

1 2

1 2

1 2N

N

i i i Ni i i

i i i

NV

!

Dimension = ND

Huge

How many qubits can we represent with 1 GB of memory?

Here, D = 2.

To add one more qubit double the memory.

302 8 2

27

N

N

But usually, we are not interested in arbitrary states in the Hilbert space.

Typical problem : To find the ground state of a local

Hamiltonian H,

12 23 34 1,... N NH h h h h

Ground states of local Hamiltonians are special

( ) logi ii

S l

Limited Correlations and Entanglement.

( ) x x lC l O O

1) Gapped Hamiltonian

2) Critical Hamiltonian

( ) log( )S l l

( )S l const l /( ) lC l e

( ) 0aC l l a

Properties of ground states in 1-D

We can exploit these properties to represent ground states more

efficiently using tensor networks.

Ground states of local Hamiltonians

NV

Contents




• Example 1 : MPS


Multidimensional array of complex numbers

Tensors

1 2 ki i iT

1

2

3

:Ket

* * *1 2 3

: Bra

11 12

21 22

31 32

Matrix

M M

M M

M M

a

a

a

b

a

b

c

11 12

21 22

31 32

11 12

21 22

31 32

1

2

Rank-3 TensorM M

c M M

M M

N N

c N N

N N

Contraction

=

a ab bb

M

M

a ba

Contraction

=P QR

ac ab bcb

R P Q

contraction cost a b c

b caa c

Contraction

= P

Q

R

S

b

ca

b

cae

f g

abc afe fbg egcefg

S P Q R

Trace

=

=

Maa

a

z M

P Rab abcc

c

P R

a

b

a

b

a

c

Tensor product

a be a b

ab

dcf c d

e a b

(Reshaping)

Decomposition

=M Q D

1Q

=M U S V

=TU S V

Decomposing tensors can be useful

=M QP

d d d d

d

Number of components in M = 2d

Number of components in P and Q = 2 d

Rank(M) =

Contents




• Example 1 : MPS


1 2

1 2

1 2N

N

i i i Ni i i

i i i

Many-body state as a tensor

1i 2i Ni

Expectation values

O1 2 1 2

1 2

*

N N

N

i i i k i i ii i i

O

O

contraction cost = NO D

Correlators

1 2OO

1O 2O


Reduced density operators


Trs block

Tensor network decomposition of a state

Essential features of a tensor network

1) Can efficiently store the TN in memory

2) Can efficiently extract expectation values of local observables from TN

Total number of components = O(poly(N))

Computational cost = O(poly(N))

Number of tensors in TN = O(poly(N)) is independent of N

1

Contents




• Example 1 : MPS


Matrix Product States

MPS

1

2Total number of components = N D

Recall!

O1 2 1 2

1 2

*

N N

N

i i i k i i ii i i

O

O


Expectation values

Expectation values

4contraction cost = O N D

But is the MPS good for representing ground states?

But is the MPS good for representing ground states?

Claim: Yes!Naturally suited for gapped systems.

Recall!

1) Gapped Hamiltonian

2) Critical Hamiltonian

( ) log( )S l l

( )S l const l /( ) lC l e

( ) 0aC l l a

In any MPS

Correlations decay exponentially

Entropy saturates to a constant

MPS

Recall!

1 2OO

1O 2O


Correlations in a MPS

l

0 1l


l


M M M

l


lM

0 1l

1l l l lL M R L QD Q R L D R

Entanglement entropy in a MPS

l

( )

S const

const

rank


2

ld

ld

2( ) rank

2log( )S

logi ii

S

1. Variational optimization by minimizing energy

2. Imaginary time evolution

MPS as an ansatz for ground states

MPS

lim Htground state random

te

minMPS MPS MPSH

gs

0

Contents




• Example 1 : MPS


Summary




• Example 1 : MPS


Thanks !

Introduction to Tensor Network States Sukhwinder Singh Macquarie University (Sydney)

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