Towards Efficiently Solving Quantum Traveling Salesman Problem

Transcription

1 Towards Efficiely Solvig Quaum Travelig Salesma Problem Debabraa Goswami, Harish Karick, Praeek Jai, ad Hemaa K. Maji Deparme of Compuer Sciece ad Egieerig Idia Isiue of Techology, Kapur (Daed: November, 004) We prese a framework for efficiely solvig Approximae Travelig Salesma Problem (Approximae TSP) for Quaum Compuig Models. Exisig represeaios of TSP iroduce exra saes which do o correspod o ay permuaio. We prese a efficie ad iuiive ecodig for TSP i quaum compuig paradigm. Usig his represeaio ad assumig a Gaussia disribuio o our-leghs, we give a algorihm o solve Approximae TSP (Euclidea) wihi BQP resource bouds. Geeralizig his sraegy for ay disribuio, we prese a oracle based Quaum Algorihm o solve Approximae TSP. We prese a realizaio of he oracle i he quaum couerpar of PP Lx Iroducio Quadraic speed up achieved by Grover s search algorihm for usrucured daabase is opimal, 3, 4. Direc aemps o solve NP-complee problems are boud o fail uless some srucure is ideified i he search space. Expoeial speedup i may problems is achieved by exploiig some uderlyig propery of he search space, for example, Shor s Facorizaio Algorihm 5, Deusch ad Jozsa s algorihm 6, Childs e. al. 7 radom walk algorihm. Ay search algorihm o solve Approximae Travelig Salesma Problem 8 (Approximae TSP) is boud o fail uless i uses some addiioal iformaio abou he search space. The bes kow classical algorihms for Euclidea Approximae TSP are Chrisofides algorihm 9 ad PTAS 0. Farhi e. al., proposed a quaum opimizaio echique for Saisfiabiliy problem usig adiabaic evoluio, bu his mehod is o able o guaraee bouded error i polyomial ime. Hogg e. al. 3, 4, 5 have proposed a quaum opimizaio echique usig mixig operaors. They have simulaed i for Saisfiabiliy Problem ad Approximae TSP. The mehod performs well o average, bu a bouded error i polyomial ime cao be guaraeed by his algorihm. Hogg e. al. also propose a ecodig ha iroduces saes which do o correspod o ay permuaio. So, efficie represeaio of TSP is aoher issue ha eeds o be addressed. We propose a ew ecodig scheme for permuaios which ca be recursively geeraed. Usig his represeaio scheme ad assumig a Gaussia disribuio o he our leghs we prese a BQP algorihm o solve Approximae TSP (Euclidia). We propose aoher algorihm o solve Approximae TSP usig a oracle which aswers queries abou our-legh disribuio. This algorihm gives correc aswers wih high probabiliy bu is o i BQP. However, his algorihm has he advaage ha i does o assume ay disribuio o he our leghs. Ecodig Scheme for TSP Give a graph o verices, every permuaio of he se {,,,} defies a possible Hamiloia cycle over he graph. So, if we have a eagleme of all possible permuaios, iformaio abou wheher a permuaio defies a Hamiloia cycle over he graph or o ca be associaed wih i. If a

2 permuaio represes a valid Hamiloia cycle he we associae is our legh wih he permuaio. We esablish a bijecio bewee he se of all permuaios of {,,, } ad he se {( a, a,..., a) ai i i i }, which we will call he ecodig se. We give a iducive defiiio of his mappig fucio. For he base case of =, he fucio maps ( a = ) o he permuaio. For >, i maps (a,a,,a ) i he followig maer:. From he elemes ( ) a, a,, a, creae he correspodig permuaio of he se {,,,-}.. Iser bewee he (a -)-h ad a -h elemes of he previous permuaio (for a = we iser i he begiig ad for = we iser i a he ed). A simple O classical algorihm ca be desiged which implemes his mappig. So, here exiss a quaum algorihm o impleme his mappig i polyomial ime (usig sychroizaio lemma 6 ). Noe ha if he i-h elemes differ i wo ecodigs he i is isered a differe posiios ad subseque permuaios which are geeraed are differe. Therefore, his fucio is oe-oe. Moreover, size of he rage ad domai are equal ( =! ). Thus, he fucio is a bijecio ad i suffices o obai a eagleme of all elemes i he ecodig se. Circui o geerae eagleme of all Permuaios ecodigs I order o geerae eagleme of all possible ecodigs, we use ses of regisers R, R o sore iformaio relaed o he eries ( a a ) qubis. i j a,,. The i -h regiser ( R ) has i i + R, represes he j -h qubi of i -h regiser. A ay give ime, exacly oe of he qubis i a regiser is i sae ad res are i sae 0. If R i, j= ad oher qubis i i -h regiser are i 0 sae, he we wrie R = j (his meas ha ai = j ). Our iiial sae is: ( ) Φ = (base codiio of he fucio) Afer ieraios we will have he sae (igorig he ormalizaio cosa): ( ) Φ = a a a = a = = a = a = ( )... a a We apply he gae G + o obai Φ + from Φ. Gae G + ca be described by: Triggered by: R = 0 R, +, = Effecs he chage: R R j j + +, j +, j where R +, j represes compleme of R +, j. R, = 0 implies ha he values of have bee se ad R +, = implies ha values of a + are o ye decided. So, his i a

3 sep chages R + from 0 o he sae + a + i he ( )-h a + = + ieraio. A he ed of ieraios, we obai he sae represeig he eagleme of all possible permuaio ecodigs: Φ = a a a = a = =... a a a = a = We modify he previous se of gaes G o obai gaes which resul i a fial wave where resul i a fial wave where he probabiliy of observig a paricular -our permuaio is proporioal o legh, >. The G gaes rasformed a permuaio of pois io a equal superposiio of all possible permuaios resulig from iserig he (+)-h poi. Give a permuaio of pois, we kow he exac icreme i each our due o he iserio of he (+)-h poi a each of + possible posiios (i all cases he legh icreases because he space is Euclidea). So, we ca redisribue he probabiliies i he resulig + saes such icreme ha hey are proporioal o. These gaes, afer ieraios, resul i a superposiio of he ecodig se, such ha he probabiliy of observig a paricular -our legh legh ecodig is proporioal o. So, we obai he wave: a τ τ. τ is a permuaio Algorihm for solvig Approximae TSP assumig a disribuio Give a graph, wih a Euclidea orm, we scale i o fi wihi he smalles axis parallel square. Sice here is o smaller square which ca coai he graph compleely, here mus be a leas wo verices o opposie sides of he square. We ow scale he graph such ha his square is a ui square. Observe he followig wo properies for such graphs i ui squares:. Tour leghs are O : Maximum edge legh i a ui square is, so he ( ) maximum our-legh ha is possible i a ormalized graph is. This boud is igh ad ohig beer ca be obaied (cosider = k pois i a ui square wih all verices lyig o oe verex of he square ad all eve verices lyig o he diagoally opposie verex of he square. The ay our which aleraely chooses a odd umbered ad eve umbered verex is of legh ).. Tour leghs are Ω() : Wihou loss of geeraliy, assume ha he wo verices which lie o opposie sides of he square are o edges which are parallel o he X-axis. Cosider ay our, ad projec is edges o he Y-axis. Edge leghs are greaer ha heir projecios, so legh of ay our is greaer ha is projecio o he Y-axis. The projeced our has legh, s all ours have our legh. Oe ca also verify ha ha his boud is igh (cosider pois lyig o a sraigh lie parallel o he Y-axis ad he X- coordiaes of he verices of a our form a bioic sequece, he i has legh ). Cosider a fully coeced Euclidea TSP isace. We assume ha he our-

4 legh disribuio is Gaussia wih he hump of he disribuio lyig bewee he miimum ad maximum our-leghs. For his TSP isace we prepare he wave τ legh Ψ = τ τ is a permuaio Give a close o 0, we would like o choose such ha he probabiliy of observig a our wih our-legh (by readig Ψ ) wihi ( + ) of opimal ourlegh is greaer ha a o-zero cosa. Assume g( x) is a Gaussia ceered a some poi i he rage xmi, xmax. x mi is miimum our-legh ad x max is maximum our legh. is a suiable parameer o be decided i he prepared wave. + imes he opimal The probabiliy of observig a our wih our-legh a mos ( ) our legh is a fucio of ad could be expressed as: σ = xmi ( + ) xmi xmax xmi gxdx ( ) gxdx ( ) We will make wo assumpios regardig he aure of σ (he variace of he Gaussia). We will assume ha σ grows slower ha a polyomial i ad / σ grows slower ha a polyomial i. Now, if we subsiue a ew variable for x/ σ, he we ge he followig properies i he ew coordiae sysem:. σ p'( ) eveually: So x mi xmi p'( ) p( ) i ew coordiae sysem.. q'( ) eveually: So xmax q'( ) xmax q( ) i ew coordiae σ sysem. Heceforh, we will assume ha σ = ad prove our resuls wih he above cosrais. x (l a) x Now, observe ha e e kx = =, where k = la> 0. The fucio kx x k ( x k) ha we ied o iegrae becomes e e = e e + x. Muliplyig by gives aoher Gaussia wih ceer shifed o he lef by k > 0. As varies, k varies wih i ad he resulig se of fucios is he se of all Gaussias shifed lef from he origial oe. Now, we reduce he problem o aalyzig he followig fucio: x+ xmi e dx x hx ( ) = x+ ( xmax mi ) e dx x kx We are jusified i doig so, because e serves o shif he ceer of g( x) o he lef by k. This ew ceer is ake as 0 ad x mi goes o xmi k i he ew co-ordiae frame, which is ake as he variable x o accou for variable k i he origial problem. Now, we se x = 0, i.e. we choose k such ha he ceer of he shifed Gaussia is a O q( ). Therefore, ( ( )) x of he origial problem. So, he shif is ( ) mi k Oq = e = e. We recall he wo iequaliies xmi p ( ) p ( ) ad q ( ) / ad xmax q ( ) (where are fixed polyomials). Now, we see:

5 0+ xmi p ( ) e dx e dx 0 0 h(0) > xmax q( ) e dx e dx 0 0 p( ) e > q ( ) e p ( ) x whe 0 x < x e x 4 Algorihm : BQP algorihm for Approximae TSP (Euclidia) assumig Gaussia disribuio o he our-leghs. Ipu a graph over verices ad ( ( )). Se = e Oq, ad prepare Ψ 3. Read he wave ad fid ou our-legh of he observed our 4. Repea Sep ad p ( ) 3 ( ) O imes 5. Reur he our wih miimum our-legh Prepare he wave Ψ wih O( q( ) ) = (as described i las secio) ad p ( ) read i o fid a our. If we repea his procedure ( ) r ( ) O polyomial i ) wih se as our i O( q( )) e eighborhood of he opimal our-legh i γ p( ) r( ) imes ( r ( ) is a, he he probabiliy ha we do o ge ay p ( ) r ( ) γ repeiios of he γ r( ) experime is: e <. p ( ) Therefore, we ge a our i a eighborhood of he opimal our wih a probabiliy greaer ha r( ) e. Hece, i ime which is polyomial i ad γ we ge a our which wihi ( + ) imes he opimal our wih very high probabiliy. Oracle Algorihm ad implemeaio I he previous secio, he assumpio ha he ours of a radomly geeraed Euclidea TSP isace i a ui square have a Gaussia disribuio helped us o solve he problem efficiely. Now, we cosider he quesio of wheher we ca solve TSP efficiely if we have a oracle ha ca give us iformaio abou he possible disribuio of our leghs i a paricular TSP isace. Assumig a oracle exiss ha ca ell wheher here are ay ours wih leghs i he rage [a,b) we give a efficie algorihm o solve Approximae TSP. Assume ha we have a equal superposiio of all possible permuaio of he se {{,,, }. We kow ha give a permuaio i is possible for a classical Turig Machie o verify i polyomial ime wheher ha permuaio is valid ad has a our legh i he rage [ ab), i.e., he our legh is acually i he rage [ ab, + δ ). So, here exiss a quaum Turig Machie which rus i polyomial ime ad solves his problem. We doveail his machie ad he circui ha produces a equal

6 superposiio of all permuaios. This composie machie M pus a validiy qubi i sae if i is a valid our wih desired our legh, oherwise i sae 0. We creae aoher machie M which pus is qubi i he sae 0 + if i is a valid our wih desired our legh else pus i i he sae 0. We ow cosider he machie M as described i he followig algorihm: Algorihm : PP oracle realizaio. Creae wo idepede equal superposiios of all possible permuaios.. Apply M o he firs wave ad apply M o he secod wave. 3. Read he wo validiy bis of hese machies usig he measureme operaors{ 0 0, }. Allow oher qubis o decohere. 4. If boh are 0 reur false else r ue. Assume ha here are m ha boh qubis o readig give 0 is ( ) m valid ours ou of a oal of N =! ours. Probabiliy N m m N N. If here are o valid ours, he M reurs false wih probabiliy else reurs rue wih probabiliy >. Here wih high probabiliy he aswer give by M is correc, bu his probabiliy ca o be direcly amplified as he gap bewee he probabiliies (whe m = 0 ad m = ) decreases faser ha iverse of ay polyomial i. So, his oracle is i he quaum couerpar of classical PP. We divide he rage [, ] io size rages ad umber hem serially from o. We search liearly wih δ = o hese rages ad ry o fid ou he firs rage ha has a valid our i i. Le i 0 be he firs rage which has a valid our such ha is our-legh is i is rage. We sar wih a equal superposiio of all permuaios. We doveail a machie ha fids he rage o which a our legh belogs. This resuls i a equal superposiio of all permuaios ad is correspodig rage umber i a separae regiser. The as a fial sep, we projec he -h rage ad he decohered bis of he wave give us a permuaio. i 0 Sice o bi wih idex less ha i 0 has a our, he opimal our also lies i he i0 -h bi. The our observed i he previous algorihm has legh a mos + δ = greaer ha he opimal our. We kow ha every our has our legh greaer ha. + imes he So, we ge he resul ha he obaied our s legh is less ha ( ) opimal our-legh. So, he obaied our is a ( + ) approximaio of he opimal our. Each ieraio uses a call o he oracle ad has O ( ) ime requireme ad X here are O ( ) ieraios. So, we ge a algorihm i he class P, where X is he class i which oracle lies. Our algorihm gives correc aswers wih high probabiliy,

7 however, a bouded error i probabiliy cao be guaraeed i polyomial ime. Whe ierpreed i classical complexiy heory, he resul is i lies of Toda s heorem, bu a ieresig fac is ha we ca ry o use more ha differe bases (here we used { 0, } ad, bases ses i he oracle). This may o be possible i he sadard Quaum Turig Machie Model, bu here may be a bouded error algorihm i polyomial ime for oher quaum compuaio models. Algorihm 3: Geeral Algorihm o solve TSP. Divide he rage [, ] io rages of size ad umber hem from o.. Se δ = 3. Sequeially for each rage from o query he oracle wih is search rage. 4. If i is he firs idex for which oracle reurs rue he se 0 = i. 5. Creae he wave τ is a permuaio τ # τ, #τ is he rage umber i which τ s our-legh lies. 6. Projec he qubis sorig rage iformaio o i 0 i 0 ad le he qubis sorig he our-legh iformaio decohere. 7. Reur he τ obaied i he qubis sorig he permuaio iformaio. Coclusio To he bes of our kowledge, here is o quaum or classical algorihm which guaraees bouded error performace i polyomial ime for ay geeric class of Travelig Salesma Problem. The Leer shows ha if we assume a Gaussia disribuio o he our-leghs of all possible Hamiloia cycles, he we ca solve Approximae TSP i BQP resource bouds. Exac disribuio of our leghs may o be kow. Oracle algorihm preses a mehod where we use a oracle o aswer simple queries abou Hamiloia cycles properies. We prese a PP algorihm o realize a oracle which provides sufficie iformaio o help solve Approximae TSP. Alhough his meas ha he algorihm does o guaraee bouded error i polyomial ime, bu i gives correc aswer wih high probabiliy. The resuls preseed here ca be cosidered amogs he few opimisic resuls o TSP. The mehodology preseed here provides a geeral framework wihi which oe ca use beer oracles o obai performace ehaceme. There are a couple of evide exesios of he work preseed i his Leer. We ca aalyze he effec of usig muliple bases isead of wo bases used i he oracle circui preseed i his Leer. Oherwise oe ca sudy oracle realizaios i oher models of quaum compuaios. If here are oher models of quaum compuaio i which we ca efficiely solve Approximae TSP wih bouded error, he NP problems could be efficiely solved i hose models of quaum compuaio. L.K. Grover, Phys. Rev. Le. 79, 35 (997). R. Beals, H. Buhrma, R. Cleve, M. Mosca, ad R. de Wolf, Proc. FOCS p35 (998).

8 3 C.H. Bee, E. Bersei, G. Brassard, ad U.V. Vazirai, SIAM J. Compu (997). 4 H. Buhrma ad R. dewolf, qua-ph/98046 (998). 5 P.W. Shor, Exra Volume ICM: Proc. Ier. Cogress of Mahemaicias, I, 467 (998). 6 D. Deusch ad R. Jozsa, Proc. of Royal Sociey of Lodo, A 439, 553 (99). 7 A.M. Childs, R. Cleve, E. Deoo, E. Farhi, S. Guma, ad D.A. Spielma, Proceedigs of he 35 h ACM symposium o Theory of compuig, (003). 8 E.L. Lawler, J.K. Lesra, A.H.G. Riooy-Ka, ad D.B. Shmoys, The Travellig Salesma Problem (Joh Wiley & Sos, 985). 9 N. Chrisofides, Algorihms ad Complexiy: New Direcios ad Rece Resuls, page 44, Academic Press, S. Arora, Joural of he ACM, 45, 753 (998). E. Farhi, J. Goldsoe, S. Guma, ad M. Sipser, qua-ph/ E. Farhi, J. Goldsoe, S. Guma, J. Lapa, A. Ludgre, ad D. Preda, quaph/ T. Hogg, Phys. Rev. A, 6, 053 (000). 4 T. Hogg, Phys. Rev. A, 67, 034 (003). 5 T. Hogg ad D. Porov, Quaum opimizaio, Iformaio Scieces, 8, 8 (000). 6 Eha Bersei ad Umesh Vazirai, SIAM J. Compu. 6, 4 (997). 7 D. Goswami, Phys. Rev. Le. 88, 7790 (00).