Entanglement Branching Operator Kenji Harada Graduate School of - - PowerPoint PPT Presentation

Entanglement Branching Operator

Kenji Harada

Graduate School of Informatics, Kyoto University

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NQS2017@YITP November 6, 2017

Entanglement and Singular Value Decomposition

Quantum state of a two-body system Singular Value Decomposition (SVD) = Schmidt decomp.

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∀i, ∀j 2 {1, · · · , D} : λi > 0, hui|uji = δij, hvi|vji = δij

• |ψi =

X

mn

(Tmn) |mi ⌦ |ni =

D

X

l=1

(λl) |uli ⌦ |vli

Schmidt decomposition

Entangled ⇔ D > 1

T = UΛV †,

• Λ = diag(λ1, · · · , λD), Unitary : U, V
• m

n V U m n

T

m n V' U'

= ≈

Λ Λ0 D D0

SVD

X

l>D0

l < ✏

Select effective entanglements

Diagrammatic representation of entanglements

(Matrix, Matrix Product) ⇒ (Tensor, Tensor contraction) 1D quantum state ⇒ Matrix Product States (MPS)

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m n V U

m n

T

= = = Λ

√ Λ √ Λ

An entanglement flows on a link

m (m,n) n

⇔

⇒ ⇒

s1 s2 s3 s4

SVD

Recursively

combine Rank 3 Rank 5

U V

Rank 6 Rank 7

Tensor network

Tensor Network States (TNS)

Entanglement entropy of a TNS

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× ×

× × × × × × × × × × × × × ×

1D 2D E.E. = const E.E. = O(L)

The area law is satisfied

MPS

Projected Entangled Pair States

(PEPS) (Verstrate&Cirac ’04, Nishino, et al. ’01)

E.E. = Tr(−ρA log ρA) = X

l

−λ2

l log λ2 l ∝ log(Num. of cut links)

Tensor network algorithms

For a ground state calculation, Variational method on TNS Imaginary time evolution For a partition function calculation, Renormalization group method on TN

5 Imaginary time evolution op. SVD Tensor contraction

iTEBD(Vidal ’07) (1+1)-d

MPS

e−∆tHi,i+1

TRG (Levin & Nave ’07) TNR (Evenbly & Vidal ’15)

Current status for 2D quantum systems

Variational method Full update of PEPS Corner Transfer Matrix (CTM) (Nishino & Okunishi ’96) Multi-scale Entanglement Renormalization Ansatz (MERA)

(Vidal ’07)

Renormalization group method 3D = (2+1)-d tensor network Higher-Order Tensor Renormalization Group (HOTRG)

(Xie, et al. ’12)

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However, not enough!

Tensor operations

Tensor contraction Tensor decomposition

Based on a matrix decomposition : SVD, QR, …

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A D B C

a b c d i j k l

T

i l j k

two-body TN

T

i l j k

L

i j

R

l k m

A D B C

a b c d i j k l

four-body TN

L

i j

R

l k m

• r …

Accumulation Extraction

Entanglement branching (EB) EB operator

Splitting a composite entanglement flow in a link

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The pair of branching operators can
 be freely inserted on a link

T j k i l B m

= B B+

• K. Harada, arXiv: 1710.01830

Optimize an EB operator

Bond dimensions on a link a and b are squeezable, when B, W, and V are optimized. Minimization of a distance between two tensor networks

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i l a b

W W V V

T j’ k’ m j k B

T j k i l B i l

W W V V

T j k B

|| ||

• 2

= const -

W V

T B

|| ||

2

T j k i l B m

≈

Applications of EB

Example 1. Detecting a specific entanglement flow New HOTRG algorithm to catch a proper RG flow Example 2. Many-body decomposition of a tensor Perfect disentangling among tensors Deviation of PEPS from a wave function

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Tensor network

EB operation

+ = ?

T j k i l B m

Renormalization group on tensor network

Levin & Nave (2007) Tensor Renormalization Group (TRG) Xie, et al. (2012) Higher-Order Tensor Renormalization group (HOTRG) Evenbly & Vidal (2016) Tensor Network Renormalization (TNR) TRG with disentanglers Proper RG flow!

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Not a proper RG flow!

Gu & Wen (2009) : tensor entanglement filtering, Yang, el al. (2016) : loop optimization

Entanglement structure and RG procedure

Necessary condition of a proper RG = To erase entanglements under a renormalized scale

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T T T T T T T T T T T T

T T T T T T T T P P P P P P P P

T’ T’ T’ T’ T’ T’

In the HOTRG algorithm,

(Red) loop entanglement structures remain

To split the shortest entanglement flow

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B

i T T j k m

T T m i j k a

B w w

Squeezing operators Entanglement branching

√ D D D

• const. -

B

i T T j k T T i j k

B w w

T T

B w

||

2

• ||

=

2

|| ||

b

Optimization problem for B and w

HOTRG with entanglement branching operators

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B

T T

B

i j k i’ j’ k’

L R i j k i’ j’ k’

L R P P L R P P L R P P L R P P

T’ T’ T’ T’

SVD

HOSVD

There is no entanglement between L and R.

Gather loop entanglement structures in the combination of R and L.

Accuracy of new HOTRG algorithm

Free energy of the 2D Ising model

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Better precision than a HOTRG algorithm

10-10 10-9 10-8 10-7 10-6 10-5 0.9 0.95 1 1.05 1.1 Precision of free energy T/Tc

D=8 D=12 D=16 D=20 D=24 D=8 (HOTRG) D=12 (HOTRG) D=16 (HOTRG) D=20 (HOTRG) D=24 (HOTRG)

Solid : new alg. Dash : HOTRG

Effect of an EB operator

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At the critical point,

T T

B

T T

B

0.2 0.4 0.6 0.8 1 1.2 2 4 6 8 10 12 14 16 Entropy Renormalization step Before After

reduced

Entropy = −Tr˜ Λ log ˜ Λ, ˜ Λ = Λ/TrΛ

D=24

Entropy of nearest neighbor tensors

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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 6 8 10 12 14 16 Entropy Renormalization step Tc(HOTRG) Tc 0.9Tc 1.1Tc ln(2)

Entropy = −Tr˜ Λ log ˜ Λ, ˜ Λ = Λ/TrΛ

const

New tensor network state as like MERA

Repeating a new HOTRG procedure to a tensor network representation of a density operator

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T T T T T T T T T T T T

Open boundary Open boundary

Imaginary time direction

New tensor network

Log correction of E.E. : ok!

Many-body decomposition

Perfect disentangling for a loop entanglement

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Contracting

Many-body
 decomposition

No loop entanglement Loop entanglement

Complexity O(!8)

T SVD Branching Contracting SVD

Deviation of PEPS from a wave function

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If the area law of entanglement entropy holds,
 bond dimensions of a derived PEPS are finite

The metric of space is related
 to entanglement structures

Branching Many-body decomposing Repeating

Summary

Entanglement branching (EB) operation Isometric EB operator Optimization problem by squeezing operators Application of EB New HOTRG algorithm which catches a proper RG Better accuracy than the original HOTRG algorithm Many-body decomposition Perfect disentangler, and a derived PEPS

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T j k i l B m

i l a b

W W V V

T j’ k’ m j k B

Minimization arXiv: 1710.01830