It from Qubit Summer School

Transcription

1 It from Qubit Summer School July 27 th, 2016 Tensor Networks Guifre Vidal NSERC

2 Wednesday 27 th 9AM ecture Tensor Networks 2:30PM Problem session 5PM Focus ecture MARKUS HAURU MERA: a tensor network for scale invariant systems

3 Outline: Generalities Area law and tensor networks Definition Useful tensor networks Examples D=1 MPS vs MERA D>1 PEPS, branching MERA MERA quantum circuit RG transformation AdS/CFT

4 Outline: Generalities Area law and tensor networks Definition Useful tensor networks Examples D=1 MPS vs MERA D>1 PEPS, branching MERA MERA quantum circuit RG transformation AdS/CFT

5 Area law ρ A = Tr B Ψ Ψ ground states of local Hamiltonians S ~ A ~ D 1 generic states S = Tr ρ A log ρ A A A S ~ A ~ D ground states of local Hamiltonians H generic states

6 Many-body wave-function of N spins Ψ = Ψ i1 i 2 i N i 1 i 2 i N i 1,i 2,,i N Ψ i 1 i 2 i N = 2 N parameters tensor network i 1 i 2 i N graphical notation i i j α 1 α 2 α N a b i c ij d α1 α 2 α N i = j i k j = T ij = R ik S kj a = y M x tr(abcd) k why bother? ijklmnop A ijk B jlm C nko D kmr x i y l z n v r

7 Many-body wave-function of N spins Ψ = i 1,i 2,,i N Ψ i1 i 2 i N i 1 i 2 i N i 1 i 2 Ψ 2 N parameters i N inefficient = 2 N parameters tensor network i 1 i 2 i N O(N2χ 2 ) parameters efficient α = 1,2,, χ generic state tensor network states ground states of local Hamiltonians H (N) χ = 1 χ = 2 χ = 3 H (N)

8 Why bond indices? Product (unentangled) state Ψ = ψ 1 ψ 2 ψ N = i 1 (ψ 1 ) i1 i 1 i 2 (ψ 2 ) i2 i 2 i N (ψ N ) in i N Ψ i1 i 2 i N = ψ 1 ψ 2 ψ N i 1 i 2 i N i 1 i 2 i N Entangled state Ψ ψ 1 ψ 2 ψ N Ψ i1 i 2 i N = i 1 i 2 i N i 1 i 2 i N

9 poly(n) quantum circuit Computational efficiency matrix product state (MPS) i 1 i 2 i N i 1 i 2 i N A: Efficient representation Ψ poly(n) coefficients B: Efficient manipulation Ψ Ψ Ψ σ x i Ψ computational time and memory poly(n)

10 Matrix product state (MPS) Ψ = i 1 i 2 i N Ψ i1 i 2 i N i 1 i 2 i N Ψ i1 i 2 i N i 1 i 2 i N i 1 i 2 i N 2 N α β A: Efficient representation? O(Nχ 2 ) B: Efficient computation? parameters i 2χ 2 parameters χ 2 χ 3 χ 3 α = β = χ i = 2 Ψ Ψ = norm O Nχ 3!!! computational time Ψ σ i z Ψ = local expectation value χ 2

11 T = poly(n) Quantum circuit Ψ i1 i 2 i N i 1 i 2 i N 2 N i 1 i 2 i N A: Efficient representation? ~2 4 N T = poly(n) i k l j i = j = k = l = 2 parameters 2 4 parameters B: Efficient manipulation? Ψ Ψ norm Ψ σ i x Ψ local expectation value

12 T = poly(n) B: Efficient manipulation? u u = i 1 i 2 i N T u Ψ Ψ norm = u = =

13 T = poly(n) B: Efficient manipulation? u u = local expectation value i 1 i 2 i N Ψ σ i x Ψ = T T computational time and memory ~ exp T = exp(n q )

14 poly(n) quantum circuit Computational efficiency matrix product state (MPS) i 1 i 2 i N i 1 i 2 i N A: Efficient representation Ψ poly(n) coefficients B: Efficient manipulation Ψ Ψ Ψ σ x i Ψ computational time and memory poly(n)

15 MERA: definition Multi-scale entanglement renormalization ansatz (MERA) Matrix product state (MPS)

16 MERA also MERA!

17 A: Efficient representation? N + N 2 + N 4 + = N N Multi-scale entanglement renormalization ansatz (MERA) Matrix product state (MPS) log(n) N sites N tensors O(N) parameters N sites N N log(n) tensors? 2N tensors O(N) parameters B: Efficient manipulation? Ψ Ψ norm Ψ σ i x Ψ local expectation values Focus lecture by Markus Hauru MERA: A tensor network for scale invariant systems Wed 5pm Bob room

18 Outline: Generalities Area law and tensor networks Definition Useful tensor networks Examples D=1 MPS vs MERA D>1 PEPS, branching MERA MERA quantum circuit RG transformation AdS/CFT

19 H (N) ground states of local Hamiltonians Area law S ~ A ~ D 1 D=1 spatial dimensions Area law: S ~ A ~ 0 gapped Hamiltonians exponential decay of correlations c ~ e /ξ ogarithmic correction: S ~ log() critical systems power-law decay of correlations c ~ 1 2Δ

20 Structural properties Matrix product state (MPS) Multi-scale entanglement renormalization ansatz (MERA) Decay of correlations Scaling of entanglement

21 Ψ o 0 o() Ψ MPS Problem session = = = 1 aλ = ae /ξ ξ 1 log λ exponential decay of correlations

22 Ψ o 0 o Ψ = MERA Focus lecture by Markus Hauru

23 Ψ o 0 o Ψ MERA Focus lecture by Markus Hauru O(log()) = = λ log 3 = λ 2 log 3() λ log 3() = 2 log 3(λ) = p x log 3(y) = y log 3(x) p 2 log 3 (λ) polynomial decay of correlations

24 Correlations: geometric interpretation multi-scale entanglement renormalization ansatz (MERA) matrix product state (MPS) log() structure of geodesics: structure of geodesics: o 0 o e /ξ o 0 o 1 2Δ exponential power-law

25 Entanglement entropy Problem session Focus lecture by Markus Hauru multi-scale entanglement renormalization ansatz (MERA) matrix product state (MPS) log() A A connectivity: S A const area law! connectivity: S A log logarithmic correction!

26 H (N) ground states of local Hamiltonians Area law S ~ A ~ D 1 energy spectrum space dimension D=1 D=2 D=3 gapped Δ > 0 S const MPS S S 2 small surface of zero modes N/A S S 2 gapless Δ = 0 large [(D-1)- dimensional ] surface of zero modes S log() MERA S log() S 2 log() area law S D 1 S D 1 log() logarithmic correction

27 Projected entangled pair states (PEPS) A: Efficient representation? α δ i β γ 2χ 4 parameters α = β = γ = δ = χ i = 2 O(Nχ 4 ) parameters

28 Projected entangled pair states (PEPS) Norm Ψ Ψ Ψ B: Efficient manipulation? Ψ

29 Projected entangled pair states (PEPS) Norm? Ψ Ψ B: Efficient manipulation? N y A A N x Cost exact N contraction x exp(n y ) (if N y < N x ) Cost approximate contraction O(N x N y )

30 Projected entangled pair states (PEPS) Structural properties Entanglement entropy: area law S 4 log (χ) (= D 1 )

31 log() log() scale scale D=1 dimensions S ~ log() Entanglement entropy in MERA logarithmic correction D=2 dimensions S ~ area law A A ~ /4 ~ / 2 ~ n A ~log() n A ~( ) 2

32 H (N) ground states of local Hamiltonians Area law S ~ A ~ D 1 energy spectrum space dimension D=1 D=2 D=3 gapped Δ > 0 S const MPS S PEPS S 2 PEPS small surface of zero modes N/A S MERA S 2 MERA gapless Δ = 0 large [(D-1)- dimensional ] surface of zero modes S log() MERA S log() S 2 log() area law S D 1 S D 1 log() logarithmic correction

33 log(n) MERA branching MERA N N

34 log(n) MERA branching MERA N N N S log() S (log()) 2 S logarithmic correction volume law!

35 branching MERA D=1 spatial dimensions S log() S D>1 spatial dimensions S D 1 S D 1 log S D

36 H (N) ground states of local Hamiltonians Area law S ~ A ~ D 1 energy spectrum space dimension D=1 D=2 D=3 gapped Δ > 0 S const MPS S PEPS S 2 PEPS small surface of zero modes N/A S MERA S 2 MERA gapless Δ = 0 large [(D-1)- dimensional ] surface of zero modes S log() MERA S log() branching MERA S 2 log() branching MERA area law S D 1 S D 1 log() logarithmic correction

37 Outline: Generalities Area law and tensor networks Definition Useful tensor networks Examples D=1 MPS vs MERA D>1 PEPS, branching MERA MERA quantum circuit RG transformation AdS/CFT

38 MERA as a quantum circuit: 0 disentangler two-body unitary gate isometry = also a two-body unitary gate

39 MERA as a quantum circuit: disentangler two-body unitary gate isometry = also a two-body unitary gate

40 MERA as a quantum circuit: disentangler two-body unitary gate isometry = also a two-body unitary gate

41 The success of MERA informs us about the structure of ground states MERA as a quantum circuit: quantum circuit V auxiliary time = scale ground state Ψ = V 0 N entanglement introduced sequentially at different length scales

42 MERA as a coarse-graining transformation: Ψ Ψ Ψ Ψ

43 MERA implements a scale transformation (Wilson s RG) both in the space of wave-functions Ψ Ψ Ψ Ψ Phase A Phase B stable fixed point A stable fixed point B critical fixed point and in the space of Hamiltonians H H H H = Focus lecture by Markus Hauru scale invariance CFT data

44 MERA has been conjectured to be a lattice realization of the holographic principle AdS 3 /CFT 2 Swingle 2009, 2012 MERA = time slice of AdS 3 (hyperbolic plane H 2 ) Czech, amprou, McCandlish, Sully, MERA = kinematic space (integral transform of H 2 ) ds 2 orentzian signature/ causal structure boundary intervals under RG

45 Summary: Generalities Area law and tensor networks Definition Useful tensor networks Examples D=1 MPS vs MERA D>1 PEPS, branching MERA MERA quantum circuit RG transformation AdS/CFT