Computations in Quantum Tensor Networks

Transcription

1 Coputtons n Quntu Tensor etwors Thos ucle Thos Schulte-erbrüggen Konrd Wldherr Bonn 4.8.

2 Overvew () Proble settng: Coputton of Ground Sttes Physcl odel nd th. descrpton () Effcent Representton of Vectors s Tensors: () Mtr Product Sttes (MPS) nd Tensor Trns (TT): escrpton Coputtons/Contrctons orlztons (SV MRG) (b) Mtr Product Opertors (MPO) (c) tlzng syetres n the vector representton (d) Krylov ethods for egenvectors n MPS representton

3 . Coputton of ground sttes Physcl syste wth prtcles (here spn chn) 4 5 ntercton wthn the syste (e.g. nerest-neghbor ntercton) 4 5 Eternl ntercton (e.g. eteror gnetc feld) 4 5 Gol: Fnd ground stte sllest energy level of the syste nu egenvlue/vector

4 Quntu syste descrbed by ltonn opertor. The rel egenvlues of re the energy levels of sttonry sttes descrbed by the relted egenvectors: ψ > E ψ > ny stte s represented by vector ltonn tr C C 4

5 5 ltonn y be forulted s weghted su of Kronecer products of Pul trces ( ertn untry trces). Pul trces s spn opertors: σ y σ z σ Typcl spn vectors for spn up: > > Kronecer product Tensor product

6 6 Eple: sng-type ltonn z z z z z z z z σ σ σ σ σ σ σ σ σ σ σ σ σ 4 5

7 Typcl Pttern spn chn control ltonn Sprsty: O(n log(n)) Structured: constnt long dgonls 7

8 Q Generl ltonn M α ( ) ( ) Q Q Q ( ) { σ σ σ } Pul trces ( ) y z : Proble: For 5 egenvector hs 5 coponents! Soluton: Fnd sutble vectors wth sprse representton tht llow - good pprotons of the egenvector we re loong for - esy coputtons y for nuercl egenvlue coputtons (Rylegh Quotent Vector terton. n sutble subset solve n T T eff eff 8

9 9 Typcl ltonns Open Boundry Condtons OBC otton: z y P S S S S P S P σ t ste ( ) ( ) P P P P ( ) ( ) ( ) P P P P P P z z zz y y yy P P P P OBC:

10 Typcl ltonns Perodc Boundry Condtons PBC: sotropc esenberg- J J ' yy esenberg odel P P od (generlzed) nsotropc esenberg-y J J Y yy esenberg-z J J Z zz sotropc esenberg- J J yy J zz λ esenberg-z J J yy J Z zz esenberg-yz J J Y yy J Z zz

11 Typcl ltonns Blner bqudrtc - KLT KLT odel: ( S S ) S S ( ) sn Blner bqudrtc: cos( θ ) S S ( θ )( S S ) 8 : Suton over neghbors wth nde : 4 5 SS : neghbors ( S S ) 6

12 . Sprse nd effcent representton/pproton of vector : Consder vector s bnry tensor: ( ) ( ) v bnry representton of nde. Reshpe! >... > >...

13 Grphcl otton Vector ( leg): Mtr ( legs): ( ) Generl tensor wth legs... Mtr-vector product contrcton over nde : ( ) ( ) ( y )

14 4 Frst Subset of tensor ppro: Consder ll vectors of the for R b b b b nner product: y y O() Mtr-vector product: ( ) ( ) ( ) M M Q Q Q Q α α Costs: (M) but unstsfctory pproton property. CP bd ppro. Tucer cnnot be ppled.

15 5 () Mtr Product Sttes Tensor Trn - pprotons se rn- ters le CP but n lned for tht - reflects the underlyng Physcs - llows fst coputtons Quntu Physcs: Verstrete Schollwöc Mthetcs: Tyrtyshnov Oseledets

16 6 Mtr Product Sttes cont Suton overlp reflects neghborhood relton n spn chn

17 7 Mtr Product Sttes cont. Perodc boundry condtons (Tensor chns): ( ) trce For ect representton of one needs lrger! We re only nterested n sll tr szes nd pprotons!

18 - [ ]

19 ' ' ' ' ' ' nd so on. For notton see Khoros/Kzeev: rn core product :

20 Core Tensors : trce...

21 For the coputton of y we need the nner product of two MPS vectors: ( ) ( ) p p p p p p b b b... ;... ;... Therefore we represent the sngle fctors n the bove su s sll tensors wth three legs (ndces) structured by ndces tht pper n two dfferent tensor fctors. ;... ; ) ( ; ; ere we hve to decde bout the order of the suton (contrctons).

22 MPS grphcl trce ( )

23 MPS orlzton MPS representton s not unque. Between tr products we cn nsert - wthout chngng the vector coponents. Trnsforng the trces nto untry trces v SV. Cobne the two trces t poston nto rectngulr bloc tr: trce ( ) or ( ) nfoldng or trcston of -leg tensor

24 Replce the two trces by prts of untry tr: Copute SV: Λ V Replce trces by. Multply the Λ V prt on the rght neghborng pr. n the se wy we cn consder the SV ( ) V Λ ( ) Then we cn ove the V Λ to the left neghbour pr -. So we cn replce ll by upto (the lst renng one). 4

25 For open boundry condtons ths cn be used to orthogonlze every tr pr e.g. fro the left upto the lst vector pr t the rght end u u u u ru 5

26 OBC: n the open boundry cse the fctor r n the left ost pr cn be etrcted s fctor for the whole MPS-vector nd cn be gnored. dvntge n Rylegh Quotent nzton: eff nd fster convergence. PBC: n the perodc cse we cnnot get rd of the lst tr pr. Therefore untry trces cn be cheved upto one tr pr. dvntge n Rylegh Quotent nzton: uercl stblty n eff nd fster convergence. 6

27 7 The norlzton v SV leds to the condtons or n the nonperodc cse the frst nd lst eleents re vectors wth p p p p p p p p p b b or δ δ Wrtten coponentwse: or δ δ s then clled Krus opertor.

28 Orthogonlzton v MRG Cobne two neghbourng tr prs nd pply SV: trce ( ) ( ) ( ΛV ΛV ) ( V V ) Λ Λ ΛV Strt e.g. on the left nd orthogonlze ech tr pr nd then go the rght neghbour. 8

29 9 Further Trnsforton Σ W V W V Σ se SV nonnegtve dgonl wth W B W B B B B B u u wth Σ W W W V

30 MPS MPS gves lrger MPS: ( ) ( ) B B B trce B B B trce trce b b b y So for orgnl MPS vectors wth tr sze the su s MPS vector wth tr sze. MPS MPS MPS

31 MPS wth dgonl trces CP trce

32 MPS Mnfold The spce M of MPS vectors s no lner subspce but hs certn propertes: The unt vectors re n M wth : ( e ) ( ) e δ δ δ Sprse vectors wth nnz re ebers of M for tr sze. M s so clled tr nfold Tngent spce R.Schneder e.. bsl e..

33 PEPS nner product between two PEPS tensors ( nd ): Contrcton begnnng fro down left wth the frst left colun: ~ ~ ~ ~ ~ 4 ~ 4 ~ 4 ~ ~ ~ 4 ~ ~ 4 4 ~ ~ ~ ~ ~ ~ ~ 4 ~ 4 4 ~ ~ ~ ~

34 nner product of two tensors (relted to nde resp. ). Frst step: Contrcton n ll ndces

35 eltng nd to longer of length r. Contrctons prwse n frst nd second colun:

36 Meltng nd reducton to short ndces:.. { 5 5 } 4 { 4 4 } Reduce ndces to hlf length gn! { } { 4 4 } 4 4 { } { } 4 { } 4 6

37 ow ll ndces re of short length r gn Repet untl one colun left 4 7

38 MER Lyers wth untry tensors nd soetres: B 8

39 9 B ' ' ' ' ' δ δ ' ' ' B B δ untry core tensor: soetry:

40 C ; ;; ; ; ; B ;; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; B ; ; ; C ; ; ; 4

41 (b) MPO - Mtr Product Opertor O... trce ( )... σ σ wth e.g. σ σ σ z or O (... ) trce ><... >< Quntu Physcs: Verstrete Schollwöc.. Mthetcs: Oseledets Khoros R. Schneder 4

42 ;; ;; ;; ;; ;; ;; ;; ;; O MPS: MPO:

43 4 [ ] ' ' ' ' ' ' ' ' : : : :...

44 44 Further MPOs [ ] r r r r r r r r ' ' ' ' ' ' ' ' r r r r r Y Y r r Z Y Z Y

45 45 Further MPOs r r Z Y Z Y sng r r r Z Y Z Y Z W V Y W V Z Y

46 46 Core tensor n dfferent fors: Z Y Z Y ( ) Z Z Y Y Z Z Y Y Z Y r r

47 47 MPO wth Z Y Z Y Z Y Z Y Z Z Y Y Z Z Y Y Z Y r r

48 48 dvntge: MPO MPS ( ) ( ) ( ) ( ) ( ) ( ) O O O O O O O MPS b b MPS MPS MPO ' ' ' ' ' ' ;; ;; ;; ;; ;; ;; ;; ;; O O O MPO MPO MPO

49 (c) Syetres The consdered ltonns often hve specl syetres: J the nt-dentty: J T nd JJ T syetrc persyetrc Then the egenvlues re syetrc: J ± Other syetres: J ± Or for generl peruttons P: P ± P ± Queston: ow to odel these syetres n the MPS nstz? 49

50 ( ) tr Eple... wth the se tr pr t ech poston Ths s relted to trnslton nvrnt spn perodc spn syste or T MPS. Leds to strong syetres e.g. tr ( ) tr( ) tr( ) Mn property: trce(b) trce(b) bt shft syetry:

51 5 Slrly blocwse: ) ( C B C B tr ) ( ) ( B C B C tr B C B C tr ) ( B B B tr ) ( ) ( ) ( B B B tr B B B tr B B B tr

52 Eple Btreversl Syetry: ( ) T MPS tr wth tr tr... ( ) ( ) tr ( T T ) tr( )... T Specl cse T MPS: tr wth T ( )... 5

53 ( )... tr Eple wth B ± B b ± b Slrly f the lst tr pr hs ths property then B ± B b ± b b ± b 5

54 Persyetry Bt flp Syetry tr wth nvolutons: (( )( ) ) tr t holds:. resp.. n generl: tr tr ~ ( ) (... ~ ~ ~~ )... 54

55 55 Bt flp orl For nvoluton S - ± S ± ± ± ± ± ± ± ± ± ± ± ± ~ ~ ~ ~ ~ ~ tr S S S S S S S S tr S S S S S S S S tr tr Ebed ll ± n nt-denty J (Jordn bloc J )

56 Qus nqueness ssue tht the MPS vector s of the for tr V V V wth untry trces V nd. ssue tht t holds J for ll possble choces of. Then t follows tht V re nvolutons for ll. 56

57 57 Full Bt Syetry Gves bt reverse/flp/shft syetry (we cn ssue sy. nvol.) J J J J J J J J tr T Wthout Persyetry: Λ Λ Λ Λ Λ Λ B B tr tr tr T T Λ B s possble norl for.

58 dvntges Reducton n degree of freedo by usng syetres: More copct norl for. bt shft / bt reversl / bt flp / Less storge fster convergence nd hgher ccurcy n vector pproton. 58

59 (d) Krylov - MPS Replced Rylegh Quotent Mnzton by Krylov Subspce nzton n MPS spce. Generte Orthonorl bss of K n ( ) spn { n... } Proble: MPS nd prwse orthogonlzton of MPS gves MPS vectors wth lrger blocsze new. Therefore we hve to pply bc proecton nto MPS -spce MPO representton of reduces the costs for drtclly! 59

60 6 Proected Krylov Subspce terton - vod orthogonlzton - use subspces of fed sze. n n n n n n B n n n n n n Solve n y λ B n y ew egenvector pproton y y y n n new...

61 Proectons Replce n subspce y MPS_ ( o MPS vector) by proecton nto MPS spce: n y y MPS _ MPS _ O Solve ths nzton pprotely by - SV copresson - lterntng Lest Squres nzton 6

62 6 Subspce terton { } { } n n spn P P P P P P spn K... )...)) (... ( ( ))... ( ( ) ( ) ( ~ n n n n n n n n n n B Solve n y λ B n y Gves new egenvector estte ( ) n n MPS y y P...

63 6

64 64

65 65

66 66

67 - Fster convergence by proecton becuse ect egenvector s very close to MPS nfold - se soluton for s strt pproton for - Cn copute ore egenvlues/vectors - Only proecton s needed Modfctons: - Sze of subspce - - use ect Krylov tr - nclude proected orthogonlzton r : / : ; ( ) ; P( ); 67

68 Conclusons: - MPS-TT llows effcent nd hgh qulty pproton of egenvectors of huge ltonns - Mtr Product Opertors re very useful n connecton wth MPS vectors - Syetres n the vector cn be epressed n the MPS nstz - Krylov ethods cn be ppled ncludng proectons Thn you 68