introduction to tensor network states
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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 = NDHuge
How many qubits can we represent with 1 GB of memory?
Here, D = 2.
To add one more qubit double the memory.
302 8 227
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
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
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
MatrixM MM MM 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 MM M
N Nc 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
RS
b
ca
b
cae
f g
abc afe fbg egcefg
S P Q R
Trace
=
=
Maa
a
z M
P R ab abccc
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
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
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
contraction cost = NO D
Reduced density operators
contraction cost = NO D
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
• The quantum many body problem.
• Diagrammatic Notation
• What is a tensor network?
• Example 1 : MPS
• Example 2 : MERA
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
contraction cost = NO D
Expectation values
Expectation values
Expectation values
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
contraction cost = NO D
Correlations in a MPS
l
0 1l
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
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
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
2
ld
ld
2( ) rank2log( )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 randomt
e
minMPS MPS MPSH
gs
0
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 !