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

November 6, 2017 NQS2017@YITP Entanglement Branching Operator Kenji Harada Graduate School of Informatics, Kyoto University 1

Entanglement and Singular Value Decomposition Quantum state of a two-body system Schmidt decomposition D X X | ψ i = ( T mn ) | m i ⌦ | n i = ( λ l ) | u l i ⌦ | v l i mn l =1 � ∀ i, ∀ j 2 { 1 , · · · , D } : λ i > 0 , h u i | u j i = δ ij , h v i | v j i = δ ij � Entangled ⇔ D > 1 Singular Value Decomposition (SVD) = Schmidt decomp. T = U Λ V † , � � Λ = diag( λ 1 , · · · , λ D ) , Unitary : U, V X Λ 0 � l < ✏ Λ SVD = ≈ T l>D 0 U V U' V' D D 0 m n m n m n Select e ff ective entanglements 2

Diagrammatic representation of entanglements Λ √ √ Λ Λ = = = T U V m n m n An entanglement flows on a link (Matrix, Matrix Product) Rank 5 Rank 6 ⇒ (Tensor, Tensor contraction) U V combine m ⇔ (m,n) n Rank 7 1D quantum state ⇒ Matrix Product States (MPS) Rank 3 SVD ⇒ ⇒ Recursively Tensor network s 1 s 2 s 3 s 4 3

Tensor Network States (TNS) × × 1D E.E. = const MPS 2D × × × × E.E. = O(L) × × × × The area law is satisfied × × × × × × Projected Entangled Pair States (PEPS) (Verstrate&Cirac ’04, Nishino, et al. ’01) Entanglement entropy of a TNS X − λ 2 l log λ 2 E.E. = Tr( − ρ A log ρ A ) = l ∝ log(Num. of cut links) l 4

Tensor network algorithms For a ground state calculation, Variational method on TNS MPS Imaginary time evolution SVD Imaginary time evolution op. iTEBD(Vidal ’07) Tensor contraction e − ∆ tH i,i +1 For a partition function calculation, Renormalization group method on TN TRG (Levin & Nave ’07) (1+1)-d TNR (Evenbly & Vidal ’15) 5

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) However, not enough! 6

Tensor operations Tensor contraction i l i l a A D i l or … m T L R b d j k B C j k c j k Accumulation Tensor decomposition Based on a matrix decomposition : SVD, QR, … Extraction i l i l m two-body TN T L R j k j k i l a A D four-body TN d b B C c 7 j k

Entanglement branching (EB) EB operator Splitting a composite entanglement flow in a link i l B + B = m B T The pair of branching operators can 
 j k be freely inserted on a link K. Harada, arXiv: 1710.01830 8

Optimize an EB operator Bond dimensions on a link a and b are squeezable, when B , W , and V are optimized . i l i l B B a b m ≈ m V W W V T T j j’ k’ k j k Minimization of a distance between two tensor networks 2 2 i l || || || || i l - B B B = const - V W W V V W T T T j k j k 9

Applications of EB Tensor network EB operation = ? i l + B m T j k 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 10

Renormalization group on tensor network Levin & Nave (2007) Tensor Renormalization Group (TRG) Xie, et al. (2012) Higher-Order Tensor Renormalization group (HOTRG) Not a proper RG flow! Evenbly & Vidal (2016) Tensor Network Renormalization (TNR) TRG with disentanglers Proper RG flow! Gu & Wen (2009) : tensor entanglement filtering, Yang, el al. (2016) : loop optimization 11

Entanglement structure and RG procedure Necessary condition of a proper RG = To erase entanglements under a renormalized scale In the HOTRG algorithm, T’ T’ T T T T (Red) loop entanglement T’ T’ T T T T structures remain T’ T’ T T T T P P T T T T P P P P T T T T P P 12

To split the shortest entanglement flow Entanglement branching Squeezing operators i j k i j k w a √ D D w B m B m T T T T D Optimization problem for B and w 2 2 i j k i j k || || || || w - w b w B const. - = B B T T T T T T 13

HOTRG with entanglement branching operators i j k i j k P P R L R L B P P T T L R P P HOSVD B SVD R L R L i’ j’ k’ i’ j’ k’ P P There is no entanglement between L and R. T’ T’ Gather loop entanglement structures T’ T’ in the combination of R and L. 14

Accuracy of new HOTRG algorithm Free energy of the 2D Ising model Solid : new alg. 10 -5 Dash : HOTRG Precision of free energy 10 -6 10 -7 10 -8 D=8 D=8 (HOTRG) 10 -9 D=12 D=12 (HOTRG) D=16 D=16 (HOTRG) D=20 D=20 (HOTRG) D=24 D=24 (HOTRG) 10 -10 0.9 0.95 1 1.05 1.1 T/T c Better precision than a HOTRG algorithm 15

E ff ect of an EB operator At the critical point, Entropy = − Tr˜ Λ log ˜ ˜ Λ , Λ = Λ / Tr Λ 1.2 reduced 1 0.8 Before After Entropy 0.6 B 0.4 T T T T B D=24 0.2 0 2 4 6 8 10 12 14 16 16 Renormalization step

Entropy of nearest neighbor tensors 1.8 T c (HOTRG) 1.1T c 1.6 T c ln(2) 0.9T c 1.4 1.2 Entropy 1 const 0.8 0.6 Entropy = − Tr˜ Λ log ˜ ˜ 0.4 Λ , Λ = Λ / Tr Λ 0.2 0 2 4 6 8 10 12 14 16 Renormalization step 17

New tensor network state as like MERA Repeating a new HOTRG procedure to a tensor network representation of a density operator Open boundary Imaginary time direction B B P T T T T T T T T T T T T P B B Open boundary New tensor network Log correction of E.E. : ok! 18

Many-body decomposition Branching SVD SVD Contracting T Perfect disentangling for a loop entanglement Many-body 
 Contracting decomposition Loop entanglement No loop entanglement Complexity O( ! 8 ) 19

The metric of space is related 
 to entanglement structures Deviation of PEPS from a wave function Repeating Branching Many-body decomposing If the area law of entanglement entropy holds, 
 bond dimensions of a derived PEPS are finite 20

Summary Entanglement branching (EB) operation arXiv: 1710.01830 Isometric EB operator Optimization problem by squeezing operators i l i l B B Minimization m a b m V W W V T T j j’ k’ k j k 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 21