NMR Quantum Logic Gates for Homonuclear Spin Syste

arXiv:quant-ph/9907003v11Jul1999NMRQuantumLogicGatesforHomonuclearSpinSystems.NoahLindena,b,1,EriksKupˇcec,2,RayFreemand,3aIsaacNewtonInstituteforMathematicalSciences,20ClarksonRoad,Cambridge,CB30EH,UKbDepartmentofAppliedMathematicsandTheoreticalPhysics,SilverStreet,CambridgeCB39EW,UKcVarianLtd,28ManorRoad,Walton-on-Thames,SurreyKT122QF,UKdDepartmentofChemistry,LensfieldRd,CambridgeCB21EW,UK.1email:[e-mail]2email:[e-mail][phone];email:[e-mail]etobeusedaspracticalquantumcomputers,thenumberofcoupledspinswillneedtobesolargethatitisnotfeasibletorelyonpurelyhet-eronuclearspinsystems.Theimplementationofaquantumlogicgateimposescer-tainconstraintsonthemotionofthosespinsnotdirectlyinvolvedinthatgate,theso-called“spectator”spins;theymustbereturnedtotheirinitialstatesattheendofthesequence.Asaresult,ahomonuclearspinsystemwherethereisappreciablecouplingbetweeneverypairofspinswouldseemtorequirearefocusingschemethatdoublesincomplexityanddurationforeveryadditionalspectatorspin.Fortunately,forthemorerealisticpracticalcasewherelong-rangespin-spincouplingscanbene-glected,simplerrefocusingschemescanbedevisedwheretheoveralldurationofthesequenceremainsconstantandthenumberofsoftpulsesincreasesonlylinearlywiththenumberofspectatorspins.Theseideasaretestedexperimentallyonasixqubitsystem:thesixcoupledprotonsofinosine.21IntroductionIfasystemofnuclearspinsistobeofrealpracticaluseasanNMRquantumcomputer[1-13]itshouldconsistoftensofcouplednuclei,otherwisetheaccessiblealgorithmswillaffordnorealadvantageoverthosethatcanbereadilyimplementedonaclassicalelectroniccomputer.Unfortunatelythenumberofsuitable(andnon-radioactive)spin-1/2nucleiisstrictlylimited;theprimecandidatesare1H,13C,15N,19F,29Siand31P,whilechemicalconsiderationsappeartoruleoutthetwonoblegases3Heand129Xe,togetherwithmostoftheheavymetals.Consequently,anNMRquantumcomputerwithaverylargenumberofqubitsisonlylikelytobeattainableifitincludesextensivehomonuclearsystemsofcoupledspins.Thesespinsmustallformpartofthesamemolecule,consequentlychemicalbondingconstraintsfavourthenuclei1H,13C,19F.Thusakeytaskistounderstandhowtoperformalogicoperationonahomonuclearsystem,forexampleanarrayofcoupled13Cspins.Thisisthedifficultpartoftheproblem,whetherornotheteronuclearspinsarealsoinvolved,andisthesubjectofthisLetter.Arecentpaper[12]hasemphasizedthisnew“donothing”aspectofNMRquantumcomputation.ForatotalofNhomonuclearspins,allcoupledtoeachother,weselecteitherone“active”spintoperformarotation,ortwo“active”spinstoevolveundertheirspin-spincouplingoperator,leavingN−1orN−2“spectator”spinstobereturnedtotheirinitialstatesattheendofthesequence.Becauseeachlogicgatehasanappreciableduration,normallymeasuredintensofmilliseconds,the“donothing”featureisnon-trivial,involvingtherefocusingofallchemicalshiftandspin-spincouplinginteractionsofthespectatorspins,andthecouplingsbetweentheactivespin(s)andallthespectators.Afundamentalconstraintisthatnotwocoupledspinsshouldexperiencesimultane-oussoftpulses.Althoughspin-spincouplingcanbeneglectedduringashorthardradiofrequencypulse,thisisnotthecaseforsimultaneouspulsesthatareselectiveinthefrequencydomain.Ifthedurationofthetwosoftpulsesiscomparablewiththereciprocalofthecouplingconstant,undesirableantiphasemagnetizationand3multiple-quantumcoherencesaregenerated[14-17].Thisso-calleddouble-resonancetwo-spineffect“TSETSE”isirreversibleandinterfereswiththeproperoperationofthelogicgate.Thisconstraintdictatestheformoftherefocusingpulsesequence,whichcanbecomehighlycomplexforlargenumbersofspinsiftheyareallcoupledtogetherwithappreciablecouplingconstants[12].Thesoftrefocusingpulseseven-tually“collide”inthetimedomain,settingalowerlimitonthedurationofthelogicgate.Higherappliedmagneticfieldsmitigatetheproblembyincreasingchemicalshiftdifferencesandhencepermittingshorter(lessselective)softpulses.Tworecentpapers[18,19]describepulsesequencesthataremoreefficientinthatthetotalnumberofπpulsesdoesnotincreaseexponentiallywithNforafully-coupledsystemofNspins.InbothreportstheanalogywithHadamardmatricesisstressed,sincetorefocuschemicalshiftorspin-spininteractionsrequiresequalperiodsof“positive”and“negative”evolutionundertheIzor2IzSzoperators,andtheHadamardmatricesprovideanelegantformulationofthisrequirement,suggest-ingthemostefficientrecursiveexpansionprocedure.Unfortunatelytheproposedschemesinvolveseveralpairsofsimultaneousπpulsesandarethereforeunsuitableforhomonuclearsystems.IngeneralthedesignofsuitablehomonuclearpulsesequencescanbebasedontraditionalNMR“refocusing”considerations,ormoreformallyonpatternsderivedfromtheHadamardmatrices.WedrawupasectionofaconventionalHadamardmatrixinwhichtherowsrepresentthedifferentspins(I,S,R,Q,etc.)whilethecolumnsindicatethesenseofnuclearprecession(+or−)inthedifferenttimesegments.Spin-spincouplingbetweenanytworepresentativespinsisrefocusedifthecorrespondingrowsareorthogonal,thecharacteristicpropertyofaHadamardmatrix.Consequently,forafourspinsystem,thepattern:I:++++S:++−−R:+−+−Q:+−−+(1)usingth