Fast algorithm for efficient simulation of quantum algorithm gates on classical computer

Transcription

1 Fast algort for effcet sulato of quatu algort gates o classcal coputer Sergey A PANFILOV Sergey V LYANOV Ludla V LITVINTSEVA Yaaa Motor Europe NV R&D Offce Va Braate Crea (CR) Italy Alexader V YAZENIN Dept of Iforatcs Tver State versty l Zelyabova Tver Russa Federato ABSTRACT Te geeral approac for quatu algort sulato o classcal coputer s troduced Effcet fast algort for sulato of Grover's quatu searc algort usorted database s preseted Coparso wt coo quatu algort sulato approac s deostrated Ts dvso perts to geeralze te approac of QA sulato ad to create a classcal tool to sulate ay type of kow QA Furter ore local optzato of QA copoets accordg to specfc ardware realzato akes t possble to develop approprate ardware accelerator of QA sulato usg classcal gates [3 4] Keywords: Quatu algort effcet sulato fast algorts INTRODCTION Quatu algorts (QA) deostrate great effcecy ay practcal tasks suc as factorzato of large teger ubers were classcal algorts are falg or draatcally effectve [] Practcal applcato s stll away due to lack of te pyscal ardware pleetato of quatu coputers Te dfferece betwee classcal ad QAs s followg: proble solved by QA s coded te structure of te quatu operators Iput to QA ts case s always te sae Output of QA says wc proble was coded I soe sese you gve a fucto to QA to aalyze ad QA returs ts property as a aswer Forally te probles solved by QAs could be stated as follows: Iput Proble A fucto f:{0} {0} Fd a certa property of f Repeated k tes x> x> S S F INT Iput Superposto Etagleet Iterferece Output > (a) F D INPT STEP STEP STEP 3 OTPT bt bt bt bt M E A S R E M E N T ϕ0> ϕν > ϕν> Tus QA studes qualtatve propertes of te fuctos Te core of ay QA s a set of utary quatu operators or quatu gates I practcal represetato quatu gate s a utary atrx wt partcular structure Te sze of ts atrx grows expoetally wt te uber of puts akg t possble to sulate QAs wt ore ta puts [] o classcal coputer wt vo Neua arctecture I ts report we preset a practcal approac to sulate ost of kow QAs o classcal coputers We preset te results of te classcal effcet sulato of te Grover s quatu searc algort (QSA) as a becark of ts approac STRCTRE OF QA GATE SYSTEM DESIGN Te backgroud of QA sulato s a geeralzed represetato of QA as a set of sequetally appled saller quatu gates as t s preseted o te Fgure a Fro te structural pot of vew eac QA requres a partcular set of quatu gates but geerally eac partcular set ca be dvded to tree a subsets wt sae fucto for all QAs: Superposto operators Etagleet operators ad Iterferece operators (b) Fgure : a) Crcut represetato of QA; b) Quatu crcut of Grover s QSA Geeralzed approac QA sulato I geeral ay QA ca be represeted as a crcut of saller quatu gates as t s deostrated o te Fgure [3] Te crcut preseted te Fgure s dvded o fve geeral steps: Step : Iput Quatu state vector s set up to a tal value for ts cocrete algort For exaple put for Grover s QSA s a quatu state φ0 descrbed as a tesor product φ = a 00 = a 0 0 () were 0 = ; = ; deotes Kroecker tesor 0 product operato [] Suc a quatu state ca be preseted as t s sow o te Fgure a SYSTEMICS CYBERNETICS AND INFORMATICS VOLME - NMBER 3 63

2 Step 5: Output O ts step perfored easureet operato (extracto of te state wt axu probablty) ad followg terpretato of te result For exaple case of Grover s QSA requred dex s coded frst bts of te easured bass vector Steps of QAs are realzed by utary quatu operators Sulato of quatu operators s a key pot geeral QA sulato I order to accelerate QAs basc quatu operators ust be studed Fgure : Dyacs of Grover s QSA probablty apltudes of state vector o eac algort step Te coeffcets te Eq () are called probablty apltudes [3] Probablty apltudes ay take egatve or eve coplex values Te oly oe costrat o te values of te probablty apltudes s a = () Te actual probablty of te arbtrary quatu state a to be easured s calculated as a square of ts probablty apltude value p = a Step : Superposto Te state of te quatu state vector s trasfored te way tat probabltes are dstrbuted uforly aog all bass states Te result of te superposto step of Grover s QSA s preseted o te Fgure b probablty apltude represetato ad te Fgure 3b probablty represetato Step 3: Etagleet Probablty apltudes of te bass vector correspodg to te curret proble are flpped wle rest bass vectors left ucaged Etagleet s doe va cotrolled NOT operato Result of etagleet operato applcato to te state vector after superposto operato s sow o te Fgure c ad te Fgure 3c Note tat a etagleet operato does ot affect te probablty of state vector to be easured Actually etagleet prepares a state wc ca ot be represeted as a tesor product of spler state vectors For exaple cosder state φ preseted o te Fgure b ad state φ preseted o te Fgure c: ( ) ( )( ) ( ) ( + + ) ( ) φ = = = φ = = = As t was sow above descrbed state φ ca be preseted as tesor product of spler states wle state φ ca ot Step 4: Iterferece Probablty apltudes are verted about te average value As a result te probablty apltude of states arked by etagleet operato wll crease Result of terferece operator applcato s preseted o te Fgure a a probablty apltude represetato ad te Fgure 3a a probablty represetato Fgure 3: Dyacs of Grover s QSA probabltes of state vector o eac algort step Ma QA operators We cosder superposto etagleet ad terferece operators fro sulato vew pot I ts case superposto ad terferece ave ore coplcated structure ad dffer fro algort to algort Ad te we cosder etagleet operators sce tey ave slar structure for all QAs ad dffer oly by fucto beg aalyzed Superposto operators of QAs I geeral te superposto operator cossts of te cobato of te tesor products adaard operators wt detty operator I : 0 = I = 0 For ost QAs te superposto operator ca be expressed as Sp = S = S = = (3) were ad are te ubers of puts ad of outputs respectvely left sde power operato eas tesor power Operator S depedg o te algort ay be or adaard operator or detty operator I Nubers of outputs as well as structures of correspodg superposto ad terferece operators are preseted te Table for dfferet QAs Note tat superposto ad terferece operators are ofte cota tesor power of adaard operator ( ) wc s called Wals-adaard operator ( W ) It s kow [3] tat eleets of te Wals-adaard operator could be obtaed as ( ) * (4) were = 0 = 0 + = / 64 SYSTEMICS CYBERNETICS AND INFORMATICS VOLME - NMBER 3

3 Table : Paraeters of superposto ad terferece operators of a quatu algorts Algort Superposto Iterferece Deutsc s I Deutsc- Jozsa s I Grover s D I So s I I Sor s I QFT I Ts approac proves greatly speedup of classcal sulato of te Wals adaard operators sce ts eleets could be obtaed by te sple replcato accordg to te rule preseted Eq (4) Exaple : Cosder superposto operator of Deutsc s algort = = S = I : ( ) * Deutsc [ Sp] = I / (5) 0*0 0* ( ) I ( ) I I I = *0 * = ( ) I ( ) I I I Exaple : Cosder superposto operator of Deutsc-Jozsa s ad of Grover s algort for te case = = S = : ( ) * Deutsc Jozsa ' s Grover ' s [ Sp] = / 0*0 0*0 0* 0* ( ) ( ) ( ) ( ) 0*0 0*0 0* 0* ( ) ( ) ( ) ( ) = (6) *0 * * * ( ) ( ) ( ) ( ) *0 * * * ( ) ( ) ( ) ( ) = Exaple 3: Superposto operator of So s ad of Sor s algorts = = S = I : ( ) * So Sor [ Sp] = I = / I I I I I I I I = I I I I I I I I Iterferece operators of a QAs Iterferece operators ust be selected for eac algort dvdually accordg to te paraeters preseted te Table Cosder soe partcular parts of terferece operators Iterferece operator cossts of terferece part wc s dfferet for all algorts ad fro easureet part wc s te sae for ost of algorts ad cossts of tesor power of detty operator Cosder terferece operator of eac algort Iterferece operator of Deutsc algort Iterferece operator of Deutsc s algort cossts of tesor product of two adaard trasforatos ad ca be calculated usg Eq (4) wt = : * Deutsc ( ) It = = = (7) / Note tat Deutsc s algort Wals-adaard trasforato terferece operator s used also for te easureet bass Iterferece operator of Deutsc-Jozsa s algort Iterferece operator of Deutsc-Jozsa s algort cossts of tesor product of power of Wals-adaard operator wt a detty operator I geeral for te block atrx of te terferece operator of Deutsc-Jozsa s algort ca be wrtte as: * Deutsc Jozsa ' s ( ) It = I (8) were = 0 = 0 Exaple 4: Iterferece operator of Deutsc-Jozsa s algort = = : * Deutsc Jozsa ' s ( ) It = I I I I I (9) I I I I = I I I I I I I I Iterferece operator of Grover s algort Iterferece operator of Grover s algort ca be wrtte as a block atrx of te followg for: Grover It = D I = I I / (0) I = = + I I = I / / / = were = 0 = 0 D refers to dffuso operator: AND( = ) ( ) [ D ] = / Exaple 5: Iterferece operator of Grover s QSA = = : SYSTEMICS CYBERNETICS AND INFORMATICS VOLME - NMBER 3 65

4 = = Grover It D I I I / = + I I = I I I I I I I I = I I I I I I I I () Wt = we ca observe te followg relato: QFT e e = = = J*(0*0) π / J*(0*) π / k k = J*(*0) π / J*(*) π / e e (4) Eq (3) ca be also preseted aroc for usg Euler forula: Note tat wt growg uber of qubts ga coeffcet wll becoe saller Deso of te atrx creases accordg to but eac eleet ca be extracted usg Eq (0) wtout allocato of etre operator atrx Iterferece operator of So s algort Iterferece operator of So s algort s prepared te sae aer as superposto (as well as superposto operators of Sor s algort) ad ca be descrbed as followg Eq () ad Eq(a): * So ( ) It = I = I / () 0*0 0* 0* ( ) ( ) I ( ) I ( ) I *0 * * ( = ) / ( ) I ( ) I ( ) I ( )*0 ( )* ( )*( ) ( ) I ( ) I ( ) I Reark I geeral terferece operator of So s algort cocdes wt terferece operator of Deutsc-Jozsa s algort Eq (8) but eac block of te operator atrx Eq () cossts of tesor products of detty operator Reark Eac odd block (we product of te dexes s a odd uber) of te So s terferece operator Eq () as a egatve sg Actually f = 0 4 or = 0 4 te block sg s postve else block sg s egatve Ts rule s applcable also for Eq (8) of Deutsc- Jozsa s algort terferece operator Te t s coveet to ceck f oe of te dexes s a eve uber stead of calculatg ter product Te Eq () ca be reduced as: ( ) * So / It = I = I = If s odd or f s odd I / f s eve ad s eve (a) Iterferece operator of Sor s algort Iterferece operator of Sor s algort uses Quatu Fourer Trasforato operator (QFT) [] calculated as: [ QFT ] π J(* ) = e (3) / were: J - agary ut = 0 ad = 0 π π QFT k cos ( * ) s ( * ) = J k/ k + k (5) Etagleet operators of a QAs I geeral etagleet operators are part of QA were te forato about te fucto beg aalyzed s coded as put-output relato Let s dscuss te geeral approac for codg bary fuctos to correspodg etagleet gates Cosder arbtrary bary fucto: f :{ 0} { 0 } suc tat: f( x0 x ) = ( y0 y ) I order to create utary quatu operator wc perfors te sae trasforato frst we trasfer rreversble fucto f to reversble fucto F as followg: + + F :{ 0} { 0 } suc tat: F( x0 x y0 y ) = = ( x x f( x x ) ( y y )) were deotes addto odulo avg reversble fucto F we ca desg a etagleet operator atrx usg te followg rule: B B [ F ] B B F = ff ( ) = 00;; + + B deotes bary codg Actually resulted etagleet operator s a block dagoal atrx of te for: M 0 0 F = (6) 0 M Eac block M = 0 cossts of tesor products of I or of C operators ad ca be obtaed as followg: Iff F( k) = 0 M = (7) k = 0 Cff F( k) = were C stays for NOT operator defed as: 0 C = 0 66 SYSTEMICS CYBERNETICS AND INFORMATICS VOLME - NMBER 3

5 It s clear tat etagleet operator s a sparse atrx sg property of sparse atrx operatos t s possble to accelerate te sulato of te etagleet Exaple 6: Etagleet operator for bary fucto: { } { } f : 0 0 suc tat: f( x ) = 0 x= 0 x 0 Reversble fucto F ts case wll be: 3 3 F :{ 0} { 0} suc tat: ( x y) ( x f( x) y) = = = 0 0 = = = = 0 0 = Te correspodg etagleet block atrx ca be wrtte as: I F = 0 0 C I I Fgure c deostrates te result of te applcato of ts operator Grover s QSA Etagleet operators of Deutsc ad of Deutsc-Jozsa s algorts ave te sae for Exaple 7: Etagleet operator for bary fucto: f :{ 0} { 0} suc tat: f( x ) = 0 00 x= 0 x I I F = 0 0 C I I I C I Etagleet operators of Sor ad of So s algorts ave te sae for 3 RESLTS OF CLASSICAL QA GATE SIMLATION Aalyzg quatu operators preseted te secto we ca do te followg splfcato for creasg perforace of classcal QA sulatos: a) All quatu operators are syetrcal aroud a dagoal atrces; b) State vector s allocated as a sparse atrx; c) Eleets of te quatu operators are ot stored but calculated we ecessary usg Eqs (6) (0) (6) ad (7); d) As terato codto we cosder u of Sao etropy of te quatu state calculated as: + S = p log p (8) = 0 Calculato of te Sao etropy s appled to te quatu state after terferece operato [5] Mu of Sao etropy Eq (8) correspods to te state we tere are few state vectors wt g probablty (states wt u ucertaty) Selecto of approprate terato codto s portat sce QAs are perodcal Fgure 4 sows results of te Sao forato etropy calculato for te Grover s algort wt 5 puts Iterato Fgure 4: Sao etropy aalyss of Grover s QSA dyacs wt fve puts Fgure 4 sows tat for fve puts of Grover s QSA a optal uber of teratos accordg to u of te Sao etropy crtera for successful result s exactly four After tat probablty of correct aswer wll decrease ad algort ay fal to produce correct aswer Note tat teoretcal estato π 5 for 5 puts gves = 444 teratos 4 Sulato results of fast Grover QSA are suarzed Table Nubers of teratos for fast algort were estated accordg to terato codto as u of Sao etropy of quatu state vector Te followg approaces were used sulato: Approac : Quatu operators are appled as atrces eleets of quatu operator atrces are calculated dyacally accordg to Eqs (6) (0) ad (7) Classcal ardware lt of ts approac s aroud 0 qubts caused by expoetal teporal coplexty Approac : Quatu operators are replaced wt classcal gates Product operatos are reoved fro sulato accordg to [4] State vector of probablty apltudes s stored copressed for (oly dfferet probablty apltudes are allocated eory) Wt secod approac t s possble to perfor classcal effcet sulato of Grover s QSA wt arbtrary large uber of puts (50 qubts ad ore) Wt allocato of te state vector coputer eory ts approac perts to sulato 6 qubts o PC wt GB of RAM Fgure 5 sows eory requred for Grover algort sulato we wole state vector s allocated eory Addg oe qubt requre double of te coputer eory eeded for sulato of Grover's QSA case we state vector s allocated copletely eory SYSTEMICS CYBERNETICS AND INFORMATICS VOLME - NMBER 3 67

6 Table : Teporal coplexty of Grover s QSA sulato o Gz coputer wt two CPs Nuber of teratos Approac (oe terato) Teporal coplexty secods Approac ( teratos) ~ ~ ~ ~ ~ ~ approaces [] gve us ew effectve possblty for sulato of quatu cotrol algorts usg classcal coputers Fgure 6: Teporal coplexty of Grover's QSA 5 REFERENCES Fgure 5: Spatal coplexty of Grover QA sulato Teporal coplexty of Grover's QSA s preseted Fgure 6 I ts case state vector s allocated eory ad quatu operators are replaced wt classcal gates accordg to [3 4] Fastest case s we we copress state vector ad replace quatu operator atrces wt correspodg classcal gates accordg wt [34] I ts case we obta speedup accordg to Approac 4 CONCLSIONS Effcet sulato of QAs o classcal coputer wt large uber of puts s dffcult proble For exaple to operate oly wt 50 qubts state vector drectly t s ecessary to ave at least 8TB of eory (for te oet largest supercoputer as oly 0TB [6]) I preset report for cocrete portat exaple as Grover s QSA [] t s deostrated te possblty to overrde spato-teporal coplexty ad to perfor effcet sulatos of QA o classcal coputers Coparso wt sparse atrx based [] P Sor Wy ave't ore quatu algorts bee foud? Joural of te ACM (JACM) Vol 50 Issue pp ; L G Valat Quatu crcuts tat ca be sulated classcally polyoal te SIAM J of Coputg Vol 3 No 4 pp ; L Grover Quatu ecacs elps searcg of te eedle a aystack Pys Rev Lett Vol 79 No pp ; M Nelse ad I Cuag Quatu Coputato ad Quatu Iforato Cabrdge v Press 00 [] J Nwa K Matsuoto ad Ia Geeral-purpose parallel sulator for quatu coputg Pys Rev A Vol [3] SV lyaov F Gs S Paflov I Kurawak ad LV Ltvtseva Sulato of quatu algorts o classcal coputers verstà degl Stud d Mlao Polo Ddattco e d Rcerca d Crea Note del Polo Vol 3 Crea 999 [4] P Aato S lyaov D Porto SA Paflov ad G Rzzotto ardware arctecture syste desg of quatu algort gates for effcet sulato o classcal coputers Proc SCI 003 Vol 3 pp Orlado 003 [5] SV lyaov SA Paflov I Kurawak AV Yaze Iforato aalyss of quatu gates for sulato of quatu algorts o classcal coputers Proc QCM&C000 Kluwer Acadec/Pleu Publ pp [6] ttp://wwwesastecgop 68 SYSTEMICS CYBERNETICS AND INFORMATICS VOLME - NMBER 3