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.