Introduction to MERA Sukhwinder Singh Macquarie University.

53

Transcript of Introduction to MERA Sukhwinder Singh Macquarie University.

Page 1: Introduction to MERA Sukhwinder Singh Macquarie University.
Page 2: Introduction to MERA Sukhwinder Singh Macquarie University.

Introduction to MERASukhwinder Singh

Macquarie University

Page 3: Introduction to MERA Sukhwinder 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

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

Page 4: Introduction to MERA Sukhwinder Singh Macquarie University.

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

Page 8: Introduction to MERA Sukhwinder Singh Macquarie University.

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.

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

Page 17: Introduction to MERA Sukhwinder Singh Macquarie University.

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

Page 24: Introduction to MERA Sukhwinder Singh Macquarie University.

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.

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.

Page 28: Introduction to MERA Sukhwinder Singh Macquarie University.

Recall!

1) Gapped Hamiltonian

2) Critical Hamiltonian

( ) log( )S l l

( )S l const l /( ) lC l e

( ) 0aC l l a

Page 29: Introduction to MERA Sukhwinder Singh Macquarie University.

In any MERA

Correlations decay polynomially

Entropy grows logarithmically

Page 30: Introduction to MERA Sukhwinder Singh Macquarie University.

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

Page 31: Introduction to MERA Sukhwinder Singh Macquarie University.

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

Page 40: Introduction to MERA Sukhwinder Singh Macquarie University.

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.
Page 42: Introduction to MERA Sukhwinder Singh Macquarie University.

00 0 0

0 0 0 0 0 0 0 0 0 0

0

0

0

0

Page 43: Introduction to MERA Sukhwinder Singh Macquarie University.

Time

Space

00 0 0

0 0 0 0 0 0 0 0 0 0

0

0

0

0

Page 44: Introduction to MERA Sukhwinder Singh Macquarie University.

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.

Figure Source: Evenbly, Vidal 2011

Page 46: Introduction to MERA Sukhwinder Singh Macquarie University.
Page 47: Introduction to MERA Sukhwinder Singh Macquarie University.

g g

†g

g g

†g†g

SU(2)g

MERA and spin networks

Page 48: Introduction to MERA Sukhwinder Singh Macquarie University.

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

Page 49: Introduction to MERA Sukhwinder Singh Macquarie University.

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.