Introduction to MERA
description
Transcript of Introduction to MERA
Introduction to MERASukhwinder Singh
Macquarie University
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
Cost of Contraction
=P
QR
b c
a
e f
b c
a
abc ebcf aefef
R P Q
cost a b c e f
1 2 Ni i i
1i 2i Ni
1i 2i Ni
1
4Total number of components = ( )O N
Made of layers
Disentanglers & Isometries
U
†U
W†W
Different ways of looking at the MERA
1. Coarse-graining transformation.2. Efficient description of ground states on a
classical computer.3. Quantum circuit to prepare ground states on
a quantum computer.4. A specific realization of the AdS/CFT
correspondence.
Coarse-graining transformation
Length Scale
V
W
Coarse-graining transformation
dim( ) dim( )V W
: IsometryExample
Layer is a coarse-graining transformation
Coarse graining of operators
Coarse graining of operators
Coarse graining of operators
Coarse graining of operators
Coarse graining of operators
Coarse graining of operators
Coarse graining of operators
Cost of contraction = ( )Local operators coarse-grained to local operators.
pO
Scaling Superoperator
Scaling Superoperator
MERA defines an RG flow
0L
1L
2L
3L
Scale Wavefunction on coarse-grained lattice with two sites
Types of MERA
Types of MERA
Binary MERA Ternary MERA
Different ways of looking at the MERA
1. Coarse-graining transformation.2. Efficient description of ground states on a
classical computer.3. Quantum circuit to prepare ground states on
a quantum computer.4. A specific realization of the AdS/CFT
correspondence.
Expectation values from the MERA
2
Perform contraction layer by layer
Cost = O( log )Efficient!
p N
MERA MERAO MERA
MERA
“Causal Cone” of the MERA
But is the MERA good for representing ground states?
Claim: Yes!Naturally suited for critical 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 MERA
Correlations decay polynomially
Entropy grows logarithmically
Correlations in the MERA
log1 2
log log
( )
0 1; 0
COARSE
l
l q
Tr O
Tr S OO
l lq
log stepsl
Correlations in the MERA
M
log †log1 2
log log
( )
0 1; 0
COARSE
l l
l q
Tr O
Tr M OO M
l lq
log stepsl
Entanglement entropy in the MERA
sitesl
loglog rank( ) ( ) lS l const
Entanglement entropy in the MERA
Entanglement entropy in the MERA
Entanglement entropy in the MERA
Entanglement entropy in the MERA
sitesl
log stepsl
Entanglement entropy in the MERA
sitesl
log stepsl
Entanglement entropy in the MERA
sitesl
log stepsl
logS l
ld
log l
ld
Therefore MERA can be used a variational ansatz for ground states
of critical Hamiltonians
Different ways of looking at the MERA
1. Coarse-graining transformation.2. Efficient description of ground states on a
classical computer.3. Quantum circuit to prepare ground states on
a quantum computer.4. A specific realization of the AdS/CFT
correspondence.
00 0 0
0 0 0 0 0 0 0 0 0 0
0
0
0
0
Time
Space
00 0 0
0 0 0 0 0 0 0 0 0 0
0
0
0
0
Different ways of looking at the MERA
1. Coarse-graining transformation.2. Efficient description of ground states on a
classical computer.3. Quantum circuit to prepare ground states on
a quantum computer.4. A specific realization of the AdS/CFT
correspondence.
Figure Source: Evenbly, Vidal 2011
g g
†g
g g
†g†g
SU(2)g
MERA and spin networks
MERA and spin networks
a b
c ( , , )( , , )( , , )
a a a
b b b
c c c
a j m tb j m tc j m t
0 1 0 1
0 0 1 1 2
MERA and spin networks
( , , )a a aj m t ( , , )b b bj m t
( , , )c c cj m t
( , )a aj t ( , )b bj t
( , )c cj t ( , )c cj m
( , )a aj m ( , )b bj m
(Wigner-Eckart Theorem)
MERA and spin networks
MERA and spin networks
1 2 Rj j j
MERA and spin networks
Summary – MERA can be seen as ..
1. As defining a RG flow.2. Efficient description of ground states on a
classical computer.3. Quantum circuit to prepare ground states on
a quantum computer.4. Specific realization of the AdS/CFT
correspondence.