Introduction to MERA
Sukhwinder Singh
Macquarie University
Tensors
Multidimensional array of complex numbers
Ti1i2
b
b
a
a
a
B ra : 
K et : 
 1


 2

 3






*
1

*
2
a
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 
Cost of Contraction
c
b
R
c
b
P
=
f
e
Q
a
a
R abc 
P
ebcf
Q aef
ef
co st  a b c e f
 i1i
2
iN

i1
i1 i 2
i2
iN
1

iN
Made of layers
Total number of components = O( N  4 )
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
Coarse-graining transformation
W
dim (V )  dim (W )
V
E xam ple : Isom etry



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 = O(  p )
Local operators coarse-grained to local operators.
Scaling Superoperator
Scaling Superoperator
MERA defines an RG flow
Scale
L3
L2
L1
L0
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
 MERA
 M ERA O  M ERA
Perform contraction layer by layer
Cost = O( p log 2 N )
Efficient!
 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 
C (l )  e
S ( l )  const
 l /
2) Critical Hamiltonian 
C (l )  l
a
a0
S ( l )  log( l )
l  
In any MERA
Correlations decay polynomially
Entropy grows logarithmically
Correlations in the MERA

log l steps
 T r (  O C O A R SE )

 Tr  S

log l
log l
l
O1O 2
log 
l
0    1; q  0

q
Correlations in the MERA

log l steps
M
 T r (  O C O A R SE )

 Tr  M
log l

log 
log l
l
O1O 2 M
l
0    1; q  0
q
† log l

Entanglement entropy in the MERA

l sites
S  log l  rank (  )  ( const )
log l
Entanglement entropy in the MERA

Entanglement entropy in the MERA

Entanglement entropy in the MERA

Entanglement entropy in the MERA
l sites

log l steps
Entanglement entropy in the MERA
l sites

log l steps
Entanglement entropy in the MERA
l sites

log l steps
l
d

S  log l
d
l
log l
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.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Time
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Space
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
MERA and spin networks
g
g
†

g
 g  S U (2)
g
†
g
g
†

g
MERA and spin networks
0  0 11 2
c
a  ( ja , m a , t a )
b  ( jb , m b , t b )
a
01
b
01
c  ( jc , m c , t c )
MERA and spin networks
( jc , t c )
( jc , m c , t c )
( jc , m c )

( j a , m a , t a ) ( jb , m b , t b )
( ja , t a )
( jb , t b )
( ja , m a )
(Wigner-Eckart Theorem)
( jb , m b )
MERA and spin networks

MERA and spin networks
MERA and spin networks

j1 j 2
jR
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.