Adiabatic Quantum State Generation and Statistical Zero Knowledge Dorit Aharonov Hebrew University Jerusalem Israel doriacs

Adiabatic Quantum State Generation and Statistical Zero Knowledge Dorit Aharonov Hebrew University Jerusalem Israel doriacs - Description

hujiacil Amnon TaShma Tel Aviv University TelAviv Israel amnontauacil ABSTRACT Thedesignofnewquantumalgorithmshasproventobe anextremelydi64259culttask Thispaperconsidersadi64256er entapproachtotheproblembystudyingtheproblemof quantumstategeneration W ID: 25108 Download Pdf

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Adiabatic Quantum State Generation and Statistical Zero Knowledge Dorit Aharonov Hebrew University Jerusalem Israel doriacs

hujiacil Amnon TaShma Tel Aviv University TelAviv Israel amnontauacil ABSTRACT Thedesignofnewquantumalgorithmshasproventobe anextremelydi64259culttask Thispaperconsidersadi64256er entapproachtotheproblembystudyingtheproblemof quantumstategeneration W

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Adiabatic Quantum State Generation and Statistical Zero Knowledge Dorit Aharonov Hebrew University Jerusalem Israel doriacs




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Adiabatic Quantum State Generation and Statistical Zero Knowledge Dorit Aharonov Hebrew University, Jerusalem, Israel doria@cs.huji.ac.il Amnon Ta-Shma Tel Aviv University, Tel-Aviv, Israel amnon@tau.ac.il ABSTRACT Thedesignofnewquantumalgorithmshasproventobe anextremelydifficulttask. Thispaperconsidersadiffer- entapproachtotheproblem,bystudyingtheproblemof ’quantumstategeneration’. WefirstshowthatanyprobleminStatisticalZeroKnowl- edge(includingeg. discretelog,quadraticresiduosityand gapclosestvectorinalattice)canbereducedtoaninstance

ofthequantumstategenerationproblem.Havingshownthe generalityofthestategenerationproblem,wesetthefoun- dationsforanewparadigmforquantumstategeneration. Wedefine’(diabaticState)eneration’((S)),whichis basedonHamiltoniansinsteadofunitarygates. Wede- veloptoolsfor(S)includingaverygeneralmethodforim- plementingHamiltonians(ThesparseHamiltonianlemma), andwaystoguaranteenonnegligiblespectralgaps(The jagged adiabaticpathlemma). Wealsoprovethat(S) isequivalentinpowertostategenerationinthestandard quantummodel.(ftersettingthefoundationsfor(S),we showhowtoapplyourtechniquestogenerateinteresting

superpositionsrelatedto+arkovchains. The(S)approachtoquantumalgorithmsprovidesin- triguinglinksbetweenquantumcomputationandmanydif- ferentareas,theanalysisofspectralgapsandgroundstates ofHamiltoniansinphysics,rapidlymixing+arkovchains, statisticalzeroknowledge,andquantumrandomwalks.We hopethattheselinkswillbringnewinsightsandmethods intoquantumalgorithms. Categories and Subject Descriptors ..2.m[ TheoryofComputation ],(nalysisof(lgorithms and2roblem3omplexity Miscellaneous 5..6.7[ Theoryof Computation ], 3omputationbyabstractdevices com- plexity measures and classes

8.(.wassupportedinpartbythe9nstitutefor:uantum 9nformationthrough;ationalScience.oundation)rant;o. E9(-==8?=78. Permission to ma digital or hard copies of all or part of this ork for personal or classroom use is granted without fee pr vided that copies are not made or distri uted for profit or commercial ad antage and that copies bear this notice and the full citation on the first page. co otherwise, to republish, to post on ser ers or to redistri ute to lists, requires prior specific permission and/or fee. OC’03, June 9–11, 2003, San Di go, California, USA. Co yright 2003 CM

1-58113-674-9/03/0006 ... $5.00. General Terms Theory,(lgorithms Keywords :uantumadiabaticcomputation,quantumsampling,Hamil- tonian,spectralgap,stategeneration,+arkovchains,sta- tisticalzeroknowledge 1. INTRODUCTION :uantumcomputationcarriesthehopeofsolvinginquan- tumpolynomialtimeclassicallyintractabletasks.Themost notablesuccesssofarisShor’squantumalgorithmforfactor- ingintegersandforfindingthediscretelog[A=]. .ollowing Shor’salgorithm,severalotheralgorithmswerediscovered [2B, AA, 67], allheavilyrelyingonthe.ouriertransform. (newblackboxalgorithmwasrecentlydiscoveredwhich

usesadifferentapproachofquantumrandomwalks,but theproblemitsolvesissomewhatcontrived[C].Dnecannot overstatetheimportanceofdevelopingqualitativelydiffer- entquantumalgorithmictechniquesandapproachesforthe developmentofthefieldofquantumcomputation. 9nthis paperweattempttomakeastepinthatdirectionbyap- proachingtheissueofquantumalgorithmsfromadifferent pointofview. 9tisfolkloreknowledgethattheproblemofgraphisomor- phismwhichisconsideredclassicallyhard[72]hasaneffi- cientquantumalgorithmaslongasthesuperpositionofall graphsisomorphictoagivengraph,

canbegeneratedefficientlybyaquantumTuringmachine (forsimplicity,weignorenormalizingconstantsintheabove stateandintherestofthepaper). Toseethisnoticethat fortwoisomorphicgraphs and and are identical,whereasfortwononisomorphicgraphstheyare orthogonal. (simplecircuitcanthendistinguishbetween thesetwocases. Dneistemptedtoassumethat is easytoconstructsincetheequivalentclassicaldistribution, namelytheuniformdistributionoverallgraphsisomorphic toacertaingraph,canbesampledfromefficiently.9ndeed, thestate 〉⊗| canbeefficientlygen-

eratedonaquantumcomputerbythisargument5However, sofarnooneknowshowtogenerate efficiently,because wecannot forget thevalueof 9nthispaperwesystematicallystudytheproblemofquan- tumstategeneration. Wewillmostlybeinterestedina restricted version of state generation, namely generating statescorrespondingtoclassicalprobabilitydistributions, whichwelooselyrefertoas quantum sampling (or Qsam- 20
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pling )fromadistribution. Tobemorespecific, wecon- sider the probability distribution of a circuit, ,whichis thedistributionovertheoutputsoftheclassicalcircuit

whenitsinputsareuniformlydistributed. 8enote def ∈{ . Wedefinetheproblemofcircuit quantumsampling, Definition 1. CircuitQuantumSampling(CQS): Input: ,C )where isadescriptionofaclassicalcircuit from to bits,and= Output: (descriptionofaquantumcircuit ofsize poly )suchthat −| 〉| ConnectiontoStatisticalZeroKnowledge. +ostprob- lemsthatwereconsideredgoodcandidatesforF:2,suchas discretelog(8GD)),quadraticresiduosity,gapversionsof closestandshortestvectorsinalattice,andgraphisomor- phism,belongtothecomplexityclass statistical zero knowl- edge

,denotedSZK(seesection2forbackground.)Weprove Theorem 1. (ny L SZK canbereducedtoafamilyof instancesof3:S. TheproofreliesonareductionbySahaiandHadhan[7C] fromSZKtoacompleteproblemcalledstatisticaldifference. Theorem6showsthatageneralsolutionforquantumsam- plingwouldimply SZK BQP .Wenotethatthereexists anoracle relativetowhich SZK BQP [6],andso suchaproofmustbenonrelativizing. Theorem6showsthatonecanstartfromaSZKprooffor aproblem,andderiveadescriptionofclassicalcircuitssuch thatefficient:uantumsamplingfromthesecircuitswould thenimplythattheproblemisinF:2.Thederivationofthe

circuitspecificationisingeneralacomplicatedtask,sinceit buildsonthereductionofSahaiandHadhan[7C]fromSZK tothecompleteproblem. Weprovideoneexplicitexamplewhichisofparticular interestasacandidateforF:2,agapversionofclosest vectorinalattice(3H2).Inlikethegeneralconstruction, thisreductionfromSZKto:samplingisfairlystraightfor- ward. WeusetheSZKproofof)oldreichand)oldwasser [26]toderiveanexactspecificationofthecorrespondingcir- cuitswhichoneneedsto:samplefrom,inordertogivea polynomialquantumalgorithmforthisproblem. Explicitspecificationsofthecircuitsto:samplefromcan

bederivedalsofortheproblemsofdiscretelog(8G)and quadraticresiduosity(:J),basedonthetheSZKproofs fortheseproblemsby)oldreichandKushilevitz[2=],and by)oldwasser,+icaliandJackoff[27],respectively.Gikein thecaseof3H2,thederivationsarefairlystraightforward andaresimilarconceptuallytothe3H2case. Wewillnot doitinthispaperduetolackofspace, andsincethese problemsarealreadyknowntobeinF:2. TheAdiabaticStateGenerationParadigm Theprob- lemofwhatstatescanbegeneratedefficientlybyaquantum computeristhusofcriticalimportancetotheunderstand- ingofthecomputationalpowerofquantumcomputers.We

thereforeembarkonthetaskofdesigningtoolsforquantum stategeneration,andstudyingwhichstatescanbegener- atedefficiently. Therecentlysuggestedframeworkofadia- baticquantumcomputation[6C]seemstobetailoredexactly forthispurpose,sinceitisstatedintermsofquantumstate generation5Getusfirstexplainthisframework. Jecallthatthetimeevolutionofaquantumsystem’sstate isdescribedbySchrodinger’sequation, dt (6) where )isanHermitianoperatorcalledthe Hamilto- nian ofthesystem. (diabaticevolutionconcernsthecase inwhich )variesveryslowlyintime5 Thequalitative

statementoftheadiabatictheoremisthatifthequantum systemisinitializedinthegroundstate(theeigenstatewith lowesteigenvalue)of (=), andifthemodificationof intimeisdoneslowlyenough( adiabatically ),thenthefi- nalstatewillbethegroundstateofthefinalHamiltonian ).Jecently,.arhi,)oldstone,)utmannandSipser[6C] suggestedtouseadiabaticevolutionstosolve NP -hardlan- guages. .arhi et. al. ’sideawastofindtheminimumof agivenfunction asfollows, (=)ischosentobesome generic Hamiltonian. )ischosentobethe problem Hamiltonian ,namelyamatrixwhichhasthevaluesof

onitsdiagonalandzeroeverywhereelse. Thesystemis theninitializedinthegroundstateof (=)andevolvesadi- abaticallyontheconvexline )E(6 .Fy theadiabatictheoremiftheevolutionisslowenough,the finalstatewillbethegroundstateof )whichisexactly thesoughtafterminimumof Theefficiencyoftheseadiabaticalgorithmsisdetermined bytwothings. .irst, istakentobe local ,i.e. asumof operators,eachoperatingonaconstantnumberofqubits outofthe qubitsofthesystem. 9twasshownin[6C, 6A]thatadiabaticevolutionsofsuchHamiltonianscanbe simulatedefficientlyonaquantumcircuit,andsodesigning

suchasuccessfulprocesswouldimplyaquantumefficient algorithm fortheproblem. Second, theadiabaticevolu- tionhastobeslowenoughfortheadiabatictheoremto hold. 9tturnsoutthatthisdependsmainlyonthe spec- tral gaps oftheHamiltonians ). 9fthesespectralgaps arenottoosmall,themodificationoftheHamiltonianscan bedone’fairlyfast’,andtheadiabaticalgorithmthenbe- comesefficient. ;otmuchisknownaboutadiabaticcom- putation,sincespectralgapsarehardtoanalyzeingeneral. [6?,66,6B]studynumericallytheperformanceofadiabatic algorithmsonrandominstancesof;2completeproblems.

[6A,78]provedthat)rover’squadraticspeedup[2L]canbe achievedadiabatically,and[6A,6L]alsogivelowerbounds forspecificcasesofadiabaticalgorithms. 9nthispaper,weproposetousethelanguageof(diabatic evolutions,Hamiltonians,groundstatesandspectralgapsas atheoreticalframeworkfor quantum state generation .Dur goalisnottoreplacethequantumcircuitmodel,neitherto improveonit,butrathertodevelopaparadigm,oralan- guage,inwhichquantumstategenerationcanbestudied conveniently.TheadvantageinusingtheHamiltonianlan- guageisthatthetaskofquantumstategenerationbecomes

muchmorenatural,sinceadiabaticevolutioniscastinthe languageofstategeneration. .urthermore,aswewillsee, itseemsthatthislanguagelendsitselfmoreeasilythanthe standardcircuitmodeltodevelopinggeneraltools. 9nordertoprovideaframeworkforthestudyofstate generation using the adiabatic language, we define adia- batic state generation ((S))asgeneralaswecan. Thus, wereplacetherequirementthattheHamiltoniansareona straightline )E(6 ,withHamiltonians onanygeneralpath. Second,wereplacetherequirement 21
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thattheHamiltoniansarelocal,withtherequirementthat theyare simulatable

,i.e.,thattheunitarymatrix itH canbeapproximatedbyaquantumcircuittowithinany polynomialaccuracyforanypolynomiallyboundedtime Wethusstillusethestandardmodelofquantumcircuitsin ourparadigmto simulate theadiabaticprocess.Durgoalis thereforetoderivequantumcircuitssolvingthestategener- ationproblem,fromstategenerationalgorithmscastinthe (S)framework. Thefactthatany(S)canbesimulatedefficientlyby aquantumcircuit,followsfromtheadiabatictheoremand fromageneralizationoftechniquesin[6C,6A]. (nalterna- tiveproofwhichdoesnotrelyontheadiabatictheoremis giveninafullerversionofthispaper[2],usingthemuch

simplerZenoeffect[7B]. See[6=]forarelatedapproachof measurementbasedadiabaticcomputation. FoundationsofASG .Thefirstquestionthatoneencoun- tersisnaturally,whatkindofHamiltonianscanbeusedin (S)algorithms.9notherwords,whenisitpossibletosim- ulate,orimplement,aHamiltonianefficiently. Tothisend weprovethe sparse Hamiltonian lemma whichgivesavery generalconditionforaHamiltoniantobesimulatable. ( Hamiltonian on qubitsisrow-sparseifthenumberof non-zeroentriesateachrowispolynomiallybounded. is saidtoberow-computableifthereexistsa(quantumorclas- sical)efficientalgorithmthatgiven

outputsalist( j,H i,j runningoverallnonzeroentries i,j .(sanormforHamil- toniansweusethespectralnorm(seesection7.6.6). Lemma 1. (ThesparseHamiltonianlemma) .9f is arow-sparse,row-computableHamiltonianon qubitsand || || poly ),then issimulatable. Wenotethatthislemmaisusefulalsointwoothercon- texts,first,inthecontextofsimulatingcomplicatedphysi- calsystemsonaquantumcircuit. Second,forefficientim- plementationofcontinuousquantumwalks[62]whichare basedonHamiltonians..orexample,lemma6canbeused tosimplifytheHamiltonianimplementationintherecently

discoveredexponentialquantumspeedupusingquantum walks[C]. Thenextquestionthatoneencountersindesigning(S) algorithmsconcernsboundingthespectralgap. Weneed toolstofindpathsintheHamiltonianspacesuchthatthe spectralgapsareguaranteedtobenonnegligible,i.e.larger than6 /poly ).Durnextlemmaprovidesawaytodothis incertaincases. 8enote )tobethegroundstateof (ifunique.) Lemma 2. (TheJaggedAdiabaticPathlemma) .Get poly =1 beasequenceofsimulatableHamiltonians on qubits, all with polynomially bounded norm, non- negligiblespectralgapsandwithgroundvalues=,suchthat

theinnerproductbetweentheuniquegroundstates and +1 )isnonnegligibleforall . Thenthereisan efficientquantumalgorithmthattakes )towithinar- bitrarilysmalldistancefrom ). Toprovethislemmawedeveloptwosimplebutveryuseful toolsformanipulatingHamiltoniansinthecontextof(S), TheHamiltonian-to-2rojectionGemma,(GemmaA)andthe Two8imensional(diabaticGemma(GemmaL). .inally,weusetheabovedevelopedtoolstoshowthat thequestionofthecomplexityofquantumstategeneration isequivalent(uptopolynomialterms)inthecircuitmodel andinthe(S)model,andsoitissufficienttostudystate generationinthe(S)paradigm. Theorem 2.

canbeefficientlygeneratedinthecircuit modeliffitcanbeefficientlygeneratedby(S). Using ASG for (arkov chain related states 9nthe finalpartofthepaperwedemonstratehowourmethods for (S)work for :sampling from thelimiting distribu- tionsof+arkovchains. (+arkovchainisrapidlymixing ifandonlyifthesecondeigenvaluegap,namelythediffer- encebetweenthelargestandsecondlargesteigenvalueof the+arkovmatrix ,isnonnegligible[A]. Thisclearly bearsresemblancetotheadiabaticconditionofanonnegli- giblespectralgap,andsuggeststolookatHamiltoniansof theform willbeaHamiltonianif is

symmetric5if isnotsymmetricbutisareversible+arkov chain[7A]wecanstilldefinetheHamiltoniancorresponding toit(seesectionA.) ThesparseHamiltonianlemmahas asanimmediatecorollarythatforaspecialtypeof+arkov chains,whichwecall strongly samplable ,thequantumana- logofthe+arkovchaincanbeimplemented, Corroloary 1. 9f isastronglysamplable+arkovchain, then issimulatable. Toapplythe(S)frameworkfor:samplingfromlimiting distributionsof+arkovchains,itisnaturaltoconsider se- quences of+arkovchains,whereeach+arkovchaininthe sequenceisclose(insomewelldefinedsense)tothenext

+arkovchaininthesequence.Suchsequencesappearnat- urallyinthecontextofaverycommonlyusedparadigmin randomizedalgorithms,namelyapproximatecounting[2C]. Weshow, Theorem 3. (Goosely,) Get beanefficientrandomized algorithmtoapproximatelycountasetM,whichusesslowly varying+arkovchainsstartingfromasimple+arkovchain. Thenthereisanefficientquantumalgorithm that:sam- plesfromthefinallimitingdistributionoverM. Westressthatitis;DTthecasethatweareinterestedin aquantumspeedupforsamplingfromvariousdistributions butratherweareinterestedintheefficientgenerationofthe

quantumstatecorrespondingtotheclassicaldistribution. Essentiallyall+arkovchainsthatareusedinapproxi- matecountingthatweareawareofmeetthecriteriaofthe theorem. Thefollowingisapartiallistofstateswecan :samplefromusingTheorem7,wherethecitationsreferto theapproximatealgorithmsthatweuseasthebasisforthe quantumsamplingalgorithm. Iniformsuperpositionover allperfectmatchingsofagivenbipartitegraph[28],allspan- ningtreesofagivengraph[8],alllatticepointscontainedin ahighdimensionalconvexbodysatisfyingtheconditionsof [?],various)ibbsdistributionoverrapidlymixing+arkov

chainsusingthe+etropolisfilter[7A],andlog-concavedistri- butions[?].Wenotethatsomeofthesestates(perhapsall) canbegeneratedusingstandardtechniqueswhichexploit theselfreducibilityoftheproblem(see[2?]).Westresshow- everthatourtechniquesarequalitativelyandsignificantly differentfromprevioustechniquesforgeneratingquantum states,andinparticulardonotrequireselfreducibility.This canbeimportantforextendingthisapproachtootherstates. 22
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Connections to other areas and Open Questions 9n thispaperwesuggesttousethelanguageofHamiltonians andspectralgaps.

Thisdirectionpointsatveryinterest- ingandintriguingconnectionsbetweenquantumcomputa- tionandmanydifferentareas, SZK,adiabaticevolution, rapidlymixing+arkovchainsandtheirspectralgapanalysis [7A],quantumwalks[C],andthestudyofgroundstatesand spectralgapsofHamiltoniansin2hysics,whichisalively areaofresearch(see[A6]andreferencestherein).Hopefully, theseconnectionswillbringtechniquesandinsightsfrom thesefieldstoquantumcomputation.(saninterestingfirst step,itwouldbeinsightfultofindalternativealgorithmsto quadraticresiduosityanddiscretelogby(S)ofthestates

derivedbytheorem6fortheseproblems. Organization of paper Therestofthepaperisdivided tothreesectionswhicharealmostcompletelyindependent. Section2givestheresultsrelatedtoSZK.Section7gives thefoundationsforthe(S)framework, definition,tools, andtheequivalencetostandardstategeneration.SectionA drawstheconnectionto+arkovchainsandshowshowto use(S)to:samplefromapproximatelycountablesets. 2. QSAMPLING AND SZK 2.1 SZK Thecomplexityclass Statistical Zero Knowledge (SZK)is definedusinginteractiveproofssystems.Hereweomitthis definition,sinceitisnotneededforthispaper.9nsteadwe

willusethecharacterizationofSZKbyacompleteproblem forSZK,calledstatisticaldifference(S8),whichwedescribe shortly.S8wasrecentlyshowntobecompleteforSZKby SahaiandHadhan[7C].ThenicethingaboutS8isthatit doesnotmentioninteractiveproofsinanyexplicitorim- plicitway. .orexcellentsourcesonSZKsee[A7,7C]. 9tis knownthat BPP SZK AM coAM andthatSZK isclosedundercomplement. 9tfollows(see[A7])thatSZK doesnotcontainany;2–completelanguageunlessthepoly- nomialhierarchycollapses. 2.2 Statistical Difference (SD) Weneedsomefactsaboutdistancesbetweendistributions

todefinethecompleteproblemS8..ortwoclassicaldistri- butions definetheir distanceandtheir fi- delity (thismeasureisknownbymanyothernamesaswell), p,q )E Wealsodefinethevariationdistancetobe || || sothatitisavaluebetween=and6. Fact 1. (See[7?])6 p,q ≤|| || p,q WecannowdefinethecompleteproblemforSZK, Definition 2. Statistical,i-erence( SD α, Input ,Twoclassicalcircuits ,C with outputbits. Promise || || or || || Output , WhichofthetwopossibilitiesoccursO ( yes for thefirstcaseand no forthesecond) SahaiandHadhan[7C,A7]showthatforanytwoconstants

β< 6where , SD α, iscompleteforSZK. 2.3 Reduction from SZK to Qsampling. ToproveTheorem6weneedaverysimplebuildingblock whichcanbeprovedbydirectcalculation, Claim 1. Get ,v ,w ),applyaHadamard gateonthefirstqubitandmeasureit. Theprobabilityof answer=is 1+ Real Proof. (ofTheorem6)Weshowthat SD BQP Get ,C beaninputtoan SD problem,andsay ,C arecircuitswith outputs.Getusfirstassumethat wecan:samplefrombothcircuitswith E=error.Wecan thereforegeneratethesuperposition 〉| 〉| ). WethenapplyaHadamardgateonthefirstqubitandmea- sureit.Isingclaim6with ,wehave,

∈{ )E ,D )(2) Wethereforeget=withprobability 1+ ,D .Thus, 9f || || ,thenwemeasure=withprobabil- ity 1+ ,D 1+ −|| || 1+ 876,while, 9f || || ,thenwemeasure=withproba- bility 1+ ,D −|| || 8BL. Jepeatingtheexperiment (log( ))times,wecandecide ontherightanswerwitherrorprobabilitysmallerthan .9f the:samplingcircuithasasmallerror(say < 100 )then thesameargumentholdswithsmallcorrections. 2.4 Reduction of gapCVP to Qsampling (latticeofdimension isrepresentedbyabasis,denoted ,whichisan non-singularmatrixover .Thelat- tice )isthesetofpoints )E Bw ,i.e.,

allintegerlinearcombinationsofthecolumnsof .The distance ,v )betweentwopointsistheEuclideandis- tance . Thedistancebetweenapoint andaset is v, )Emin v,a ).Wealsodenote || || thelengthof thelargestvectoroftheset .Theclosestvectorproblem, 3H2,getsasinputan –dimensionallattice andatarget vector . Theoutputshouldbethepoint ∈L closestto .3H2is;2hard..urthermore,itis;2hardto approximatethedistancetotheclosestvectorinthelattice towithinsmallfactors,anditiseasytoapproximateitto within2 n factor,forevery , =.See[26]foradiscussion. [26]provesthatthefollowingversionof3H2isinSZK. The problem

gapCV, 9nput, (n –dimensionallattice ,avector anddesignateddistance .Weset )E log forsome c, =. 2romise,Either v, )) or v, )) 23
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Dutput,PQesRforfirstcase,P;oRforsecond. The reduction . )ivenaninput( )forthegap3H2 problem,wedescribeaclassicalcircuit ,aninputtothe :samplingproblem. Welet denotethesphereofall pointsin ofdistanceatmost fromtheorigin. The circuit getsasinputarandomstring,andoutputsthe vector ,where isauniformlyrandompointin || ∪{ }|| (Thelatticepointsinsideaverylargesphere), and isauniformlyrandompoint . [26]explain

howtosamplesuchpointswithalmosttherightdistribu- tion,i.e.theygiveadescriptionofanefficientsuch .We remarkthatactually,thepointscannotberandomlychosen fromthereal(continuous)vectorspace, duetoprecision issues,but[26]showsthattakingafineenoughdiscreteap- proximationresultsinanexponentiallysmallerror. Correctness . Weneedtoshowthatefficient:sampling from impliesaF:2algorithmfortheabovegap3H2 problem. 9nfact,wewillgetanJ:2(onesidederror)al- gorithm. Thealgorithmisdefinedasfollows. .irst,define anothercircuit, ,tobelike exceptthattheoutputs areshiftedbythevector

andbecome .9fwecan :samplefrom ,wecanalso:samplefrom byapplying ashiftby attheend.Tosolvethegapproblemthealgo- rithmcreatesthestate 〉| 〉| ]andproceeds asin3laim6. Weshowthatthisindeedgivesthecorrect answerinthetwopossiblecases.9f isfarawayfromthelat- tice ),thenthecalculationat[26]showsthatthestates and havenooverlapandso E=.Dnthe otherhand,suppose isclosetothelattice, v, )) ;oticethatinthiscase,thespheresofradius gd/ 2around alatticepoint ,andaround havealargeoverlap.9n- deed,theargumentof[26]showsthatthevariationdistance betweenthetwodistributions, .Fy fact6wehave ,D .

9terating theabove poly )timeswegetanJ:2algorithm. 3. ADIABATIC STATE GENERATION (ASG) 3.1 Physics Background 3.1.1 Norms Thespectralnormofalineartransformation ,induced bythe norm,is || || Emax =0 T ..or Hermitian || || equalsthelargestabsolutevalueofitseigenvalues..or unitary, || || E6.(lso, || AB ||≤|| |||| || ,and || || max k,& k,& def .Wesayalineartransformation approximatesalineartransformation if || || 3.1.2 Trotter Formula Say witheach Hermitian. Trotter’sfor- mulastatesthatonecanapproximate itH byslowlyin- terleavingexecutionsof itH fordifferent s.Weusea

variantofitwhichcanbeprovedusingstandardtechniques from[7?].8efine, δiH δiH ... δiH (7) Lemma 3. Get beHermitian, =1 .(s- sume || || || || S.Then,forevery t, || itH || )(A) ;oticethatforeveryfixed t,M andS,theerrortermgoes downtozerowith . 9napplications,wewillpick tobe polynomiallysmall,insuchawaythattheaboveerrorterm ispolynomiallysmall. 3.1.3 Time Dependent Schrodinger Equation Theevolutionofthestatefromtime=totime can bedescribedbyintegratingSchrodinger’sequation(6)over time.9f isconstantandindependentoftime,onegets (=) iHT (=) (L) Since isHermitian, iHT

isunitary.Theresultofthein- tegrationisunitaryalsofortimedependentHamiltonians, whichgivesthefamiliar unitaryevolutionfromquantum circuits. ThegroundstateofaHamiltonian istheeigen- statewiththesmallesteigenvalue,andifitisuniquewe denoteitby ).ThespectralgapofaHamiltonian is thedifferencebetweenthesmallestandsecondtosmallest eigenvalues,andwedenoteitbyT( ). 3.1.4 The adiabatic Theorem 9nthestudyof adiabatic evolution oneisinterestedinthe longtimebehavior(atlargetimes )ofaquantumsys- teminitializedinthegroundstateof attime=whenthe Hamiltonianofthesystem, )changesveryslowlyintime, namely

adiabatically .Tostatetheadiabatictheorem[B,7=, 7L],itisconvenientandtraditionaltoworkwithare-scaled time where isthetotaltime. TheSchrodinger’s equationrestatedintermsofthere-scaledtime thenreads ds (?) where dt ds canbereferredtoasthe delay schedule ,or the total time Theorem .. (Theadiabatictheorem.adaptedfrom [31. 13]) .Get )beafunctionfrom[= 6]tothevector spaceofHamiltonianson qubits.(ssume )iscontin- uous,hasauniquegroundstate,forevery [= 6],andis differentiableinallbutpossiblyfinitelymanypoints. Get , =andassumethatthefollowingadiabaticcondition holdsforallpoints (=

6)inwhichthederivativeisde- fined, T ds (T( )) (B) Then,aquantumsystemthatisinitializedattime=withthe groundstate (=)),andevolvesaccordingtoEquation(?) with ),endsupatre-scaledtime6atastate (6) that iswithin distancefrom (6))forsomeconstant c, =. WewillrefertoEquation(B)asthe adiabatic condition Theproofoftheadiabatictheoremisratherinvolved. .or intuition,considerSchrodinger’sequationforeigenstatesof 59ftheeigenvalueis ,theeigenstateevolvesbyamulti- plicativefactor iλt ,whichrotatesintimefasterthelarger theabsolutevalueoftheeigenvalue is,andsotheground- staterotatestheleast.

Thefastrotationsareessentially responsibletothecancellationsofthecontributionsofthe vectorswiththehighereigenvalues,duetointerference. 3.2 Adiabatic Quantum State Generation 9nthissectionwedefineourparadigmforquantum state generation ,generalizingtheideasofadiabaticquantum com- putation (andtheadiabatictheorem).Wedefine, 24
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Definition 3. (SimulatableHamiltonians). Wesaya Hamiltonian on qubitsis simulatable ifforevery t, andeveryaccuracy= <α< 6theunitarytransformation )E iHt canbeapproximatedtowithin accuracyby aquantumcircuitofsize poly n,t, / ). 9f

issimulatable,thenbydefinitionsois cH forany poly ). 9tthereforefollowsbyTrotter’sformula (Gemma7)thatanyconvexcombinationoftwosimulat- able,boundednormHamiltoniansissimulatable. (lso,9f issimulatableand isaunitarymatrixthatcanbeef- ficientlyappliedbyaquantumcircuit,then UHU isalso simulatable,because itUHU Ue itH .Theinterested readerisreferredto[7?,C]foramorecompletesetofrules forsimulatingHamiltonians.Wenowdescribeanadiabatic path,whichisanallowablepathintheHamiltonianspace, Definition .. (Adiabaticpath). (function from [= 6]tothevectorspaceofHamiltonianson qubits,is an

adiabatic path if )isalwayscontinuous,differentiable exceptforfinitelymanypoints,forevery sH )hasaunique groundstate,andforevery sH )issimulatablegiven Definition 5. (AdiabaticQuantumStateGeneration [ASG]). (nadiabatic:uantumState)enerator( ,T, hasforevery ∈{ anadiabaticpath ),suchthat forthegiven T, theadiabaticconditionissatisfiedforall [= 6]whereitisdefined.Wealsorequirethatthegen- eratorisexplicit,i.e.,thatthereexistsapolynomialtime quantummachinethat Dninput ∈{ outputs (=)),theground- stateof (=),and, Dninput ∈{ [= 6]and δ, =outputs a

circuit )approximating iδH Wethensaythegeneratoradiabaticallygenerates (6)). Remark1 Wenotethatinpreviouspapersonadiabaticcom- putation,eg.in[6A],adelayschedule )whichisafunc- tionof wasused.Wechosetoworkwithonesingledelay parameter, ,instead,whichmightseemrestrictive5How- ever,workingwithasingleparameterdoesnotrestrictthe modelsincemorecomplicateddelayschedulescanbeen- codedintothedependenceon 3.3 Circuit simulation of ASG (n(S)canbesimulatedefficientlybyaquantumcircuit, Claim 2. (CircuitsimulationofASG) .Get( ,T, bean(S).(ssume poly ).Then,thereexistsaquan- tumcircuitthatoninput

generatesthestate (6))to within accuracy,withsize poly T, /,n ). Proof. (5ased on Adiabatic Theorem) Thisproof isverysimilartotheproofgivenin[6A]forthefactthat adiabaticevolutioncanbesimulatedbyquantumcircuits efficiently. Thecircuitisbuiltbydiscretizingtimetosuffi- cientlysmallintervalsoflength ,andthenapplyingtheuni- tarymatrices iH δj . 9ntuitivelythisshouldwork,since theadiabatictheoremtellsusthataphysicalsystemevolv- ingfortime accordingtoSchrodinger’sequationwiththe givenadiabaticpathwillendupinastatecloseto (6)), andthediscretizationintroducesonlyasmallerrorif is

smallenough. Theformalerroranalysiscanbedoneby exactlythesametechniquesthatwereusedin[6A]. Wesketchanotherproofherewhichdoesnotrelyonthe adiabatictheoremandcanbederivedfromfirstprinciples, Proof. 5asedontheZenoe-ect) .orthefullproof see[2]. (sbefore,webeginatthestate (=)),andthe circuitisbuiltbydiscretizingtimetosufficientlysmallinter- valsoflength .(teachtimestep E6 ,...,R ,insteadof simulatingtheHamiltonianwemeasurethestateinabasis whichincludesthegroundstate )).Thiscanbedone usingGemmaAbelow.9f issufficientlylarge,theadi- abaticconditionensuresthatsubsequentHamiltoniansare

verycloseinthespectralnorm..urthermore,because is polynomiallybounded,thespectralgapsarenonnegligible. 9tcanbeshownthatthesetwofactsimplythatsubsequent groundstatesareveryclose. )iventhatattimestep the stateisthegroundstate )),thenextmeasurementre- sultswithveryhighprobabilityinaprojectiononthenew groundstate +1 )). TheZenoeffectguaranteesthat theerrorprobabilitybehaveslike6 /R ,i.e.quadraticallyin (andnotlinearly),andsotheaccumulatederrorafter stepsisstillsmall,whichimpliesthattheprobabilitythat thefinalstateisthegroundstateof (6)isveryhigh,if istakentobelargeenough. 3.4 The

Sparse Hamiltonian Lemma ThemainideaoftheproofofGemma6istoexplicitly write as a sum of polynomially many bounded norm Hamiltonians whichareallblockdiagonal(inacom- binatorialsense)andsuchthatthesizeoftheblocksineach matrixisatmost2 2.WethenshowthateachHamiltonian issimulatableanduseTrotter’sformulatosimulate 3.4.1 The reduction to block matrices. Definition 2. (3ombinatorialblock.) Get beamatrix withrows )andcolumns ).Wesay( R,C )isacombinatorialblockif ,forevery c,r )E=,andforevery c,r )E=. isblockdiagonalinthecombinatorialsenseiffthereis

somerenamingofthenodesunderwhichitbecomesblock diagonalintheusualsense.Equivalently, isblockdiagonal inthecombinatorialsenseiffthereisadecompositionof itsrowsinto )E =1 ,andofitscolumns )E =1 suchthatforevery ,( ,C )isacombinatorial block.Wesay is2 2combinatoriallyblockdiagonal,if eachcombinatorialblock( ,C )isatmost2 2,i.e.,for every either E6or E2. Claim 3. (,ecomposition lemma) .Get bearow- sparse,row-computableHamiltonianover qubits,withat most non-zeroelementsineachrow.Thenthereisaway todecompose into +1) =1 where, Each isarow-sparse, row-computableHamilto- nianover qubits,and, Each

is2 2combinatoriallyblockdiagonal. Proof. (Df3laim7)Wecoloralltheentriesof with K6) colors..or( i,j ]and i (i.e.,( i,j isanupper-diagonalentry)wedefinethecoloring col i,j tobethetuple( k, i mod k, j mod k, rind i,j ,cind i,j )) where 25
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9f weset E6,otherwisewelet bethefirst integerintherange[2 ..n ]suchthat (mod ), andweknowtheremustbesucha 9f i,j E=weset rind i,j )E=,otherwisewelet rind i,j )betheindexof i,j inthelistofallnon- zeroelementsinthe ’throwof cind i,j )issim- ilar,butwithregardtothecolumnsof .orlower-diagonal entries, i,j , wedefine col i,j )E col

j,i ).(ltogether,weuse( K6) colors. .oracolor ,wedefine i,j ]E i,j col i,j ,m ,i.e., is ontheentriescoloredby andzeroeverywhere else. 3learly, andeach isHermitian. (lsoas isrow-sparseandrow-computable,thereisasim- ple poly )-timeclassicalalgorithmcomputingthecoloring col i,j ),andsoeach isalsorow-computable.9tisleft toshowthatitis2 2combinatoriallyblock-diagonal. 9ndeed,fixacolor .Get NZ bethesetofallupper- triangular,non-zeroelementsof .Saytheelementsof NZ are ,j ,..., ,j ..oreveryelement( ,j NZ weintroduceablock.9f thenweset whileif thenweset ,j Say (the

caseissimilarandsimpler).(sthe color containstherow-indexandcolumn-indexof( ,j ), itmustbethecasethat( ,j )istheonlyelementin NZ fromthatrow(orcolumn). .urthermore,as mod mod ,andboth k, i mod and mod areincludedin thecolor ,itmustbethecasethattherearenoelements in NZ thatbelongtothe rowor column(see.igure 6).9tfollowsthat( ,C )isablock.Wethereforeseethat is2 2combinatoriallyblock-diagonalasdesired. Figure 1: 3n the upper diagonal side of the matrix 1 the rowand column of ,j are empty because the color con- tains the row-index and column index of i,j ,andthe 4th rowand 4th column are empty

because contains k, i mod k, j mod and mod mod .The lowerdiagonalsideof is5ustare6ec- tion of the upper diagonal side. 3tfollows that ,j isa combinatorialblock. Claim .. .orevery || ||≤|| || Proof. .ixan iscombinatoriallyblockdiagonal andthereforeitsnorm || || isachievedasthenormofone ofitsblocks.;ow, blocksareeither 6,andthentheblockis( i,i )forsome ,andit hasnorm i,i ,or, 2,andthentheblockis k,& k,& forsome k,) ,andhasnorm k,& 9tfollowsthatmax || || max k,& k,& . Dntheother hand || || max k,& k,& .Theprooffollows. 3.4.2 block matrices are simulatable. Claim 5. Every2

2combinatoriallyblockdiagonal,row- computableHamiltonian issimulatabletowithinarbi- trarypolynomialapproximation. Proof. Get t, =and α, =anaccuracyparameter. Thesimulatingquantumcircuit is2 2combinatorially blockdiagonal. Get beabasisstate,andlet belong tothe2 2block k,) in .Wenotethat leavesthe subspacespannedby invariant. Set E2(fora 2block)and Emin( k,) ), Emax( k,) ).Wethen set tobethepartof relevanttothissubspace ,m ,M ,m ,M and itA .9f belongs toa6 6blockwesimilarlydefine E6, E( k,k )and E( itA ). Durapproximation circuitsimulatestheapplicationof on .Weneedtwo

transformations.Wedefine k, ,m ,M ,k where isourapproximationtotheentriesof and isourapproximationto it ,andwherebothmatrices areexpressedbytheirfour(orone)entries. Weuse (1) accuracy. Having ,m ,M ,k writtendown,wecansimulatethe actionof . Wecanthereforehaveanefficientunitary transformation ,m ,M ,m ,M for Span ,M .Duralgorithmisapplying fol- lowedby andthen forcleanup. Correctness .Getusdenote8iffE itA .Dur goalistoshowthat || 8i || . Wenoticethat8iff is also2 2blockdiagonal, andthereforeitsnormcanbe achievedbyavector belongingtooneofitsdimension

oneortwosubspaces,sayto Span ,M .Get and .Weseethat ismappedto ,m ,M , ,m ,M ,m ,M ,m ,m ,M ,M wherethefirstequationholdssinceitholdsfor andbylinearityitholdsforthewholesubspacespannedby them.Weconcludethat 8i 〉| 〉| andso || 8i || max || || . However,byourconstruction, || || (1) andso || || asdesired. Weprovedtheclaimformatriceswith2 2combinato- rialblocks. Weremarkthatthesameapproachworksfor matriceswith combinatorialblocks,for poly ). 3.4.3 Proving the Sparse Hamiltonian Lemma Proof. (DfGemma6.) Get berow-sparsewith poly )non-zeroelementsineachrow, and || || ES poly ).Get

t, =.Durgoalistoefficientlysimulate itH towithin accuracy. Weexpress =1 asin3laim7, 6) poly ).Wechoose suchthat t .;otethat poly t,n )forsomelargeenoughpolyno- mial. Wethencompute towithin accuracy, us- ingasinGemma7,ourapproximationsto iδH (where 26
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each iδH iscomputedtowithin Mt/ )accuracy.)Fy Gemma7ourcomputationis closeto itH ,asdesired (usingthefactthatforevery || ||≤|| || ES poly )by3laimA).Thesizeofthecomputationis poly δ,M,n, )E poly n,t, )asrequired. 3.5 The Jagged Adiabatic Path Lemma Proof. (ofGemma2)WeconsiderthesequenceofHamil- tonians

whereU isaprojectiononthespaceor- thogonal to the groundstate of , and we connect two neighboringprojectionsbyaline.WeproveinGemmaA,us- ingKitaev’sphaseestimationalgorithm,thatthefactthat issimulatableimpliesthatsoisU . (lso,asprojec- tions, U haveboundednorms, || || 6. 9tfollows then,bytheresultsmentionedinSection7.6,thatallthe Hamiltoniansonthepathconnectingtheseprojectionsare simulatable,asconvexcombinationsofsimulatableHamil- tonians. WenowhavetoshowtheHamiltoniansonthepathhave nonnegligiblespectralgap. FydefinitionU hasaspec- tralgapequalto6. 9tremainstoshow,however,thatthe

HamiltoniansonthelineconnectingU andU +1 have largespectralgaps,whichweproveinthesimpleGemmaL. WecannowapplytheadiabatictheoremandgetGemma 2. 9ndeed,alineartimeparameterizationsufficestoshow thatthisalgorithmsatisfiestheadiabaticcondition. WenotethatGemmaAiscrucialfortheaboveproof,since ifweusetheHamiltoniansdirectlyandnottheirprojections, thepathconnectingthemmighthave=spectralgap. We nowturntotheproofsofGemmasAandL. Lemma .. (Hamiltonian-to-projectionlemma) .Get beaHamiltonianon qubitssuchthat iH canbeap- proximatedtowithinarbitrarypolynomialaccuracybya

polynomialquantumcircuit,andlet poly ). GetT( )benonnegligible,andlargerthan6 /n ,andfur- therassumethatthegroundvalueof is=.Thenthepro- jectionU ,issimulatable. Proof. .irstapplyKitaev’sphaseestimationalgorithm [76,7?]. (sthespectralgapisnon-negligiblewecande- cidewithexponentiallygoodconfidencewhetheraneigen- statehasthelowesteigenvalueoralargereigenvalue. We canthereforewritedownonebitofinformationonanextra qubit,whetheraninputeigenstateof isthegroundstate ororthogonaltoit. Second,applyaphaseshiftofvalue it tothisextra qubit,conditionedthatitisinthestate (ifitis we

donothing).Thisconditionalphaseshiftcorrespondstoap- plyingfortime aHamiltonianwithtwoeigenspaces,the groundstateandthesubspaceorthogonaltoit,withre- spectiveeigenvalues=and6,whichisexactlythedesired projection. .inally, toerase theextraqubitwrittendown, were- versethefirststepanduncalculatetheinformationwrit- tenonthatqubitusingKitaev’sphaseestimationalgorithm again. .oravector ,theHamiltonian −| isthe projectionontothesubspaceorthogonalto .Weprove, Lemma 5. TheTwo,imensionalAdiabaticLemma Get betwovectorsinsomesubspace, −| and −| . .oranyconvexcombination (6 )(

−| )K −| , [= 6],ofthetwo Hamiltonians ,H ,T( ≥| 〉| Proof. Dbservethattheproblem istwodimensional, write ,writethematrix inanorthonor- malbasiswhichcontains and ,anddiagonalizeto findtheeigenvalues. 3.6 Equivalence of Standard and Adiabatic State Generation Theproofoftheorem2consistsoftwodirections. We alreadysawonedirectioninclaim2,andnowwegivethe otherdirection. Claim 2. let bethefinalstateofaquantumcircuit with gates,thenthereisan(S)whichoutputsthis state,ofcomplexity poly n,M ). Proof. W.l.o.g. thecircuitstartsinthestate .We

firstmodifythecircuitsothatthestatedoesnotchange toomuchbetweensubsequenttimesteps. Thereasonwe needthiswillbecomeapparentshortly.Tomakethismod- ification,letusassumeforconcretenessthatthequantum circuit usesonlyHadamardgates,Toffoligatesand;ot gates. ThissetofgateswasshowntobeuniversalbyShi [A2,7]. (Durproofworkswithanyuniversalsetwithob- viousmodifications.) Wereplaceeachgate inthecircuit bytwo gates. .or wecanchooseanyofthepossi- blesquarerootsarbitrarily,butforconcretenesswenotice thatHadamard,;otandToffoligateshave 6eigenvalues, andwechoose 6E6and 6E

.Wecallthemodified circuit .Dbviously and computethesamefunction. The path .Welet bethenumberofgatesin ..or integer= ,weset )E −| where isthestateofthesystemafterapplyingthe first gatesof ontheinput ..or [= 6), define )E(6 )K ηH K6). The spectral gaps are large . 3learlyalltheHamiltonians )forinteger= ,havenon-negligiblespectral gaps,sincetheyareprojections.Weclaimthatforanystate andanygate | 〉| . 9ndeed,represent as where belongstothe6-eigenspaceof and belongstothe -eigenspaceof .Weseethat | 〉| || .(s E6,alittle algebrashowsthatthisquantityisatleast

.9nparticular, setting )weseethat | K6) 〉| .9t thereforefollowsbyGemmaLthatalltheHamiltonianson thelinebetween )and K6)havespectralgaps largerthan The Hamiltonians are simulatable . )ivenastate we canfirstapplytheinverseofthefirst gatesof ,thenif weareinstate x, applyaphaseshift i ,finallyapply thefirst gatesof .Thisimplements iδH The 7diabatic Condition is Satisfied .Wehave dH ds )E lim . Weignorethefinitelymanypoints 27
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where is an integer in [= ,M ]. .or all other points ,when goesto=both )and )be- long to the same interval.

Say they belong to the ’th interval, ,= <η< 6. Then, )E(6 )K ηH K6)and )E )E (6 )K( K6). 9tfollowsthat )E ζH K6) ζH )and dH ds )E K6) )].Weconcludethat || dH ds || and tosatisfyEquation(B)wejustneedtopick ). 4. QSAMPLING AND MARKOV CHAINS 4.1 Markov chain Background Weconsider+arkovchainswithstatesindexedby bit strings.9f isanergodic(i.e.connected,aperiodic)+arkov chain,characterizedwiththematrix operatingonproba- bilitydistributionsoverthestatespaceM,and isaninitial probabilitydistribution,thenlim pM . Thelim- itingdistribution isindependentof (+arkovchain

haseigenvaluesbetween 6and6. correspondstoeigenvalue6. 9tisconvenienttoassume thatalleigenvaluesarenonnegative(byaddingselfloops whichslowthechainbyafactorof2.) (+arkovchainis rapidly mixing ifstartingfromanyinitialdistribution,the distributionafter poly )timestepsiswithin variation distancefrom . [L]showsthata+arkovchainisrapidly mixingiff6 /poly ),where isthesecondlargest eigenvalue.6 iscalledthesecondeigenvaluegap. (+arkovchainis reversible if i,j j,i (symmetric+arkovchain isreversible. (lso,foran ergodic,reversible+arkovchain M> =forall

9napproximatecountingalgorithmsoneisinterestedin sequencesofrapidlymixing+arkovchains,wheresubse- quent+arkovchainshavecloselimitingdistributions. .or morebackgroundregarding+arkovchains, see[7A]. .or morebackgroundregardingapproximatecountingsee[2C]. 4.2 Markov chains and Hamiltonians .orareversible wedefine Diag Diag ). (directcalculationshowsthat issym- metrici isreversible. Wecall the Hamiltonian corresponding to .Thepropertiesof and arevery muchrelated,bythefollowingclaim(Theproofisstraight forwardandomittedhere), Claim 8. 9f isareversible+arkovchain,wehave, isaHamiltonianwith || ||

6. Thespectralgapof equalsthesecondeigenvalue gapof 9f isthelimitingdistributionof ,thentheground stateof is )E def Thisclaim givesadirectconnection betweenHamilto- nians, spectral gapsandgroundstates onone hand, and rapidlymixingreversible+arkovchainsandlimitingdis- tributionontheotherhand. 4.3 Simulating ;oteveryHamiltoniancorrespondingtoareversible+arkov chaincanbeeasilysimulated.However,wewillshortlysee thattheHamiltoniancorrespondingtoasymmetric+arkov chainissimulatable. .orgeneralreversible+arkovchains weneedsomemorerestrictions.Wedefine, Definition 8.

(reversible+arkovchainis strongly sam- plable ifitisrowcomputable,and,)iven i,j M,thereis anefficientwaytoapproximate Jowcomputabilityholdsinmostinterestingcasesbutthe secondrequirementisquiterestrictive. Wenotethatifwe couldrelaxit,thetechniquesinthissectioncouldhavebeen usedtogiveaquantumalgorithmforgraphisomorphism. Still,wenotethatthesecondrequirementholdsinmany interestingcasessuchasall+etropolisalgorithms(see[2A]). 9talsotriviallyholdsforsymmetric ,wherethelimiting distributionisuniform.Wecannowprovecorollary6, Proof. (ofcorollary6)Since i,j ]E i,j ]we seethatif

isstronglysamplablethen isrow-computable. hasboundednormandsothesparseHamiltonianlemma applies. 4.4 From Markov chains to QSampling Weareinterestedinstronglysamplablerapidlymixing +arkovchains, sothattheHamiltoniansaresimulatable andhavenonnegligiblespectralgapsbyclaimB.Toadapt thissettingtoadiabaticalgorithms,andtothesettingofthe jaggedadiabaticpathlemmainparticular,wenowconsider sequencesof+arkovchains,anddefine, Definition 8. (Slowly9arying(arkovChains) .Get =1 beasequenceof+arkovchainsonM, .Get bethelimitingdistributionof .Wesaythe sequenceis slowlyvarying ifforall c,

=,foralllargeenough ,forall6 +1 /n Weprovethatwecanuseasequenceofslowlyvarying rapidlymixing+arkovchainsto:samplefromthelimiting distributionofthefinal+arkovchain. Thisistheorem7, whichwecannowstateprecisely. Theorem3: Let =1 be a slowly varying sequence of strongly samplableMarkov chainswhichare allrapidlymix- ing, and let be their corresponding limiting distributions. Then ifthere isan e:cient Qsamplerfor , then there is an e:cient Qsampler for Proof. WealreadysawtheHamiltonians aresim- ulatable and have boundednorm. (lso, as the +arkov chainsinthesequencearerapidlymixing,theyhavelarge

spectralgaps,andthereforesodotheHamiltonians Tocompletetheproofweshowthattheinnerproductbe- tweenthegroundstatesofsubsequentHamiltoniansisnon negligible, andthenthetheoremfollows fromthejagged pathlemma. 9ndeed, +1 +1 +1 +1 andthereforeisnon- negligible. 4.5 Qsampling from Perfect Matchings Weillustrateourtechniquewiththeexampleofhowto :sample from all perfect matchings in a given bipartite graph .9nthissubsectionweheavilyrelyontheworkof Sinclair,VerrumandHigoda[28]whorecentlyshowedhow 28
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toefficientlyapproximateapermanentofanymatrixwith

nonnegativeentries,usingasequenceof+arkovchainson thesetof+atchingsofabipartitegraph. Thisworkisfar tooinvolvedtoexplainherefully,andwereferthereader to[28]formoredetails. 9nanutshell,theideain[28]isto applya+etropolisrandomwalkonthesetofperfectand nearperfectmatchings(i.e. perfectmatchingsminusone edge)ofthecompletebipartitegraph.Weightsareassigned totheedgessuchthatedgesthatdonotparticipateinthe inputgraph areslowlydecreasinguntiltheprobability theyappearinthefinaldistributionpracticallyvanishes. Theweightsoftheedgesareupdatedusingdatathatiscol-

lectedfromrunningthe+arkovchainwiththepreviousset ofweights,inanadaptiveway.Thefinal+arkovchainwith thefinalparametersconvergestoaprobabilitydistribution whichisessentiallyconcentratedontheperfectandnear perfectmatchingsoftheinputgraph,wheretheprobability oftheperfectmatchingsis6 /n timesthatofthenearperfect matching.Hence,ifwecan:samplefromthefinallimiting distribution,wecanprojectontheperfectmatchingswith polynomialsuccessprobability 9tremainstocheckthatwecanapplytheorem7. 9tis easytocheckthatthe+arkovchainsbeingusedin[28]

areallstronglysamplable,sincetheyare+etropolischains. +oreover, the sequence of +arkov chains is slowly vary- ing. 9tremainstoseethatwecanquantumsamplefrom thelimitingdistributionoftheinitialchainthatisusedin [28].Thislimitingdistributionisadistributionoverallper- fectandnearperfectmatchingsinthecompletebipartite graph,wheretheweightofeachnearperfectmatchingis timesbiggerthanthatofaperfectmatching,where isthe numberofnodesof . 9tisasimpleexerciseinquantum computationto:samplefromthisdistributionefficiently. 5. ACKNOWLEDGMENTS WewishtothankImeshHazirani,Wimvan8am,Zeph Gandau, Dded Jegev,

8ave Facon, +anny Knill, Eddie .arhi,(shwin;ayakandVohnWatrousformanyinspir- ingdiscussions. 9nparticularwethank8aveFaconforan illuminatingdiscussionwhichledtotheproofofclaim?. 6. REFERENCES [1] S. Aaronson, Quantum lower bound for the collision problem. STOC 2002, pp. 635-642 [2] D. Aharonov and A. Ta-Shma. quant-ph(02100)). A longer version of this paper. [3] D. Aharonov, A simple proof that Toffoli and ,adamard are quantum universal, quant-ph 0301040 [4] -. Alon, .igenvalues and .xpanders. Combinatorica 6011263, pp. 23-16. [5] -. Alon and 4. Spencer, The 5robabilistic 6ethod. 1111.

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