Download intro to tensor networks

Introduction to Tensor Network States
Sukhwinder Singh
Macquarie University (Sydney)
Contents
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
Quantum many body system in 1-D
1
N
2
}D
dim(V )  D
Total Hilbert Space : V
N
V
N
Dimension = D
Huge !
 

i1i2
iN
 i1i2
iN
i1  i2 
 iN
N
How many qubits can we represent with 1 GB of memory?
Here, D = 2.
2 8  2
N
30
 N  27
To add one more qubit
 double the memory.
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,
H  h12  h23  h34  ...  hN 1, N
Ground states of local Hamiltonians
are special
Limited Correlations and Entanglement.
C(l )   OxOxl 
S (l )   i log i
i
Properties of ground states in 1-D
1) Gapped Hamiltonian 
C (l )  e
S (l )  const
 l /
2) Critical Hamiltonian 
C (l )  l
a
a0
S (l )  log(l )
l  
We can exploit these properties to
represent ground states more
efficiently
using tensor networks.
V
N
Ground states of local Hamiltonians
Contents
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
Tensors
Multidimensional array of complex numbers
b
a
a
Ket : 
 1 
 
 2 
 
 3


*
2
a
Matrix

*
3

 M 11

 M 21
M
 31
ik
b
a
Bra : 
*
1
Ti1i2
M 12 

M 22 
M 32 
c
Rank-3 Tensor
 M 11

c  1  M 21
M
 31
 N11

c  2  N 21
N
 31
M 12 

M 22 
M 32 
N12 

N 22 
N 32 
Contraction

=
a
a

M

a
b
M
b
ab
b
Contraction
a
Q
P
R
c
=
Rac 
a
b

c
Pab Qbc
b
contraction cost  a  b  c
Contraction
b
b
Q
S
=
a
c
S abc 
g
f
P
e
c
a
P
afe
efg
R
Q fbg Regc
Trace
a
z   M aa
M
=
b
P
b
=
a
a
R
a
c
Pab   Rabcc
c
Tensor product

a
e  a b
f  cd
c d

e  a b
b
(Reshaping)
b a
Decomposition
M
M
T
1
Q
D
Q
=
=
=
U
U
S
S
V
V
Decomposing tensors can be useful
M
d
d
=
P Q
d
 d
d
Rank(M) = 
Number of components in M = d
Number of components in P and Q =
2
2 d
Contents
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
Many-body state as a tensor
 

i1i2
 i1i2
iN
i1  i2 
iN

i1
i2
iN
 iN
Expectation values

O 
O

i1i2
 i1i2
iN
Ok 
iN

contraction cost = O  D
N

*
i1i2
iN
Correlators

O1
O2
 O1O2 

contraction cost = O  D
N

Reduced density operators


  Trsblock  
contraction cost = O  D
N

Tensor network decomposition of a state


Essential features of a tensor network
1) Can efficiently store
the TN in memory 
Total number of components =
O(poly(N))
2) Can efficiently
extract expectation

values of local
observables from
TN
Computational cost =
O(poly(N))
1


Number of tensors in TN = O(poly(N))
 is independent of N

Contents
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
Matrix Product States
MPS


1

Total number of components = N  D
2
Recall!

O 
O

i1i2
 i1i2
iN
Ok 
iN

contraction cost = O  D
N

*
i1i2
iN
Expectation values
Expectation values
Expectation values
Expectation values
Expectation values
contraction cost = O  N  D 
4
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 
C (l )  e
S (l )  const
 l /
2) Critical Hamiltonian 
C (l )  l
a
a0
S (l )  log(l )
l  
In any MPS
Correlations decay exponentially
Entropy saturates to a constant
MPS
Recall!

O1
O2
 O1O2 

contraction cost = O  D
N

Correlations in a MPS
l

l
0   1
Correlations in a MPS
l
Correlations in a MPS
l
Correlations in a MPS
l
Correlations in a MPS
M MM
l
Correlations in a MPS
M
l

L M R  L QD Q
l
l
1
l
0   1
R  L D R 
l
l
Entanglement entropy in a MPS
l
S  const
 rank (  )  const
Entanglement entropy in a MPS
Entanglement entropy in a MPS
Entanglement entropy in a MPS
Entanglement entropy in a MPS
Entanglement entropy in a MPS
dl
2
S   i log i
i
dl
rank (  )  
 S  2 log(  )
2
MPS as an ansatz for ground states
1. Variational optimization by minimizing energy
min  MPS  MPS H  MPS
0
2. Imaginary time evolution
 ground state  lim e Ht  random
 gs
t 
MPS
Contents
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA


Summary
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
Thanks !