introduction to mera sukhwinder singh macquarie university
TRANSCRIPT
![Page 1: Introduction to MERA Sukhwinder Singh Macquarie University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649f505503460f94c73601/html5/thumbnails/1.jpg)
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Introduction to MERASukhwinder Singh
Macquarie University
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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
Matrix
M M
M M
M 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 M
M M
N N
c N N
N N
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Cost of Contraction
=P
Q
R
b c
a
e f
b c
a
abc ebcf aefef
R P Q
cost a b c e f
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1 2 Ni i i
1i 2i Ni
1i 2i Ni
1
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4Total number of components = ( )O N
Made of layers
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Disentanglers & Isometries
U
†U
W†W
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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.
![Page 9: Introduction to MERA Sukhwinder Singh Macquarie University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649f505503460f94c73601/html5/thumbnails/9.jpg)
Coarse-graining transformation
Length Scale
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V
W
Coarse-graining transformation
dim( ) dim( )V W
: IsometryExample
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Layer is a coarse-graining transformation
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Coarse graining of operators
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Coarse graining of operators
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Coarse graining of operators
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Coarse graining of operators
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Coarse graining of operators
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Coarse graining of operators
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Coarse graining of operators
Cost of contraction = ( )
Local operators coarse-grained to local operators.
pO
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Scaling Superoperator
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Scaling Superoperator
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MERA defines an RG flow
0L
1L
2L
3L
Scale Wavefunction on coarse-grained lattice with two sites
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Types of MERA
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Types of MERA
Binary MERA Ternary MERA
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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.
![Page 25: Introduction to MERA Sukhwinder Singh Macquarie University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649f505503460f94c73601/html5/thumbnails/25.jpg)
Expectation values from the MERA
2
Perform contraction layer by layer
Cost = O( log )
Efficient!
p N
MERA MERAO MERA
MERA
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“Causal Cone” of the MERA
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But is the MERA good for representing ground states?
Claim: Yes!Naturally suited for critical systems.
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Recall!
1) Gapped Hamiltonian
2) Critical Hamiltonian
( ) log( )S l l
( )S l const l /( ) lC l e
( ) 0aC l l a
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In any MERA
Correlations decay polynomially
Entropy grows logarithmically
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Correlations in the MERA
log1 2
log log
( )
0 1; 0
COARSE
l
l q
Tr O
Tr S OO
l l
q
log stepsl
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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 l
q
log stepsl
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Entanglement entropy in the MERA
sitesl
loglog rank( ) ( ) lS l const
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Entanglement entropy in the MERA
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Entanglement entropy in the MERA
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Entanglement entropy in the MERA
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Entanglement entropy in the MERA
sitesl
log stepsl
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Entanglement entropy in the MERA
sitesl
log stepsl
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Entanglement entropy in the MERA
sitesl
log stepsl
logS l
ld
log l
ld
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Therefore MERA can be used a variational ansatz for ground states
of critical Hamiltonians
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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.
![Page 41: Introduction to MERA Sukhwinder Singh Macquarie University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649f505503460f94c73601/html5/thumbnails/41.jpg)
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00 0 0
0 0 0 0 0 0 0 0 0 0
0
0
0
0
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Time
Space
00 0 0
0 0 0 0 0 0 0 0 0 0
0
0
0
0
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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.
![Page 45: Introduction to MERA Sukhwinder Singh Macquarie University](https://reader036.vdocuments.us/reader036/viewer/2022062314/56649f505503460f94c73601/html5/thumbnails/45.jpg)
Figure Source: Evenbly, Vidal 2011
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g g
†g
g g
†g†g
SU(2)g
MERA and spin networks
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MERA and spin networks
a b
c ( , , )
( , , )
( , , )
a a a
b b b
c c c
a j m t
b j m t
c j m t
0 1 0 1
0 0 1 1 2
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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)
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MERA and spin networks
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MERA and spin networks
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1 2 Rj j j
MERA and spin networks
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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.