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 H ilbert Space : V
N
V
N
D im ension = D
H uge !
 

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  h 23  h34  ...  h N 1, N
Ground states of local Hamiltonians
are special
L im ited C orrelation s an d E n tan glem en t.
C (l )   O x O x  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
Ti1i2
b
b
a
a
 1


 2

 3
a
a
B ra : 
K et : 






*
1

*
2
M a trix

*
3

 M 11

M
 21
M
 31
ik
M 12 

M 22

M 3 2 
c
R a n k-3 T en so r
 M 11

c  1 M 21

M
31

M 12 

M 22

M 3 2 
 N 11

c  2
N
 21
N
 31
N 12 

N 22

N 3 2 
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
P
e
P
afe
efg
R
c
a
c
S abc 
g
f
Q fbg R egc
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
b
d

e  ab
(R esh ap in g)
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
2
Number of components in P and Q = 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

i1 i2
 i1i2
iN
Ok 
iN

con traction cost = O  D
N

*
i1i 2
iN
Correlators

O1
O2
 O1O 2 

con traction cost = O  D
N

Reduced density operators


  T rs b lo ck 
con traction 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
M PS


1

Total num ber of com ponents = N  D
2
Recall!

 O  
O

i1 i2
 i1i2
iN
Ok 
iN

con traction cost = O  D
N

*
i1i 2
iN
Expectation values
Expectation values
Expectation values
Expectation values
Expectation values
con traction 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
M PS
Recall!

O1
O2
 O1O 2 

con traction 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 M M
l
Correlations in a MPS
M
l
 
L M
l
R  L QD Q
l
1
l
0   1
R  L D
l
R 
l
Entanglement entropy in a MPS
l
S  co n st
 ra n k (  )  co n st
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
d
l
2
S     i log  i
i
d
l
ra n k (  )  
2
 S  2 lo g (  )
MPS as an ansatz for ground states
1. Variational optimization by minimizing energy
m in
 MPS
 MPS H  MPS
0
2. Imaginary time evolution
 ground state  lim e
t
 Ht
 random
 gs
M PS
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 !