Quantum computation in a Ising spin chain taking i

arXiv:0705.3688v1[quant-ph]25May2007QuantumcomputationinaIsingspinchaintakingintoaccountsecondneighborcouplingsG.V.L´opez,T.GorinandL.LaraDepartamentodeF´ısica,UniversidaddeGuadalajara,Blvd.MarcelinoGarc´ıaBarraganyCalzadaOl´ımpica,C.P.44480Guadalajara,Jalisco,M´exicoFebruary1,2008AbstractWeconsidertherealizationofaquantumcomputerinachainofnuclearspinscou-pledbyanIsinginteraction.Quantumalgorithmscanbeperformedwiththehelpofappropriateradio-frequencypulses.Inadditiontothestandardnearest-neighborIsingcoupling,wealsoallowforasecondneighborcoupling.Itisshown,howtoapplythe2πkmethodinthismoregeneralsetting,wheretheadditionalcouplingeventuallyallowstosaveafewpulses.Weillustrateourresultswithtwonumericalsimulations:theShorprimefactorizationofthenumber4andtheteleportationofaqubitalongachainof3qubits.Inbothcases,theoptimalRabifrequency(tosuppressnon-resonanteffects)dependsprimarilyonthestrengthofthesecondneighborinteraction.PACS:03.67.-a,03.67.Lx,03.65.Ta,03.67.Dd,03.67.Hk1IntroductionTheIsing-spinchainhasbeenproposedin(Lloyd1993,Bermanetal.1994,Lloyd1995,Bermanetal.2001)asatheoreticalsystemwhichallowstoimplementaquantumcomputer.Typically,onewouldthinkofachainofspin-1/2nucleonsembeddedintoasolidcrystal,asapossiblephysicalsystem,whosedynamicsmaybewelldescribedbytheIsingHamiltonian.Thissystemmustbesubjectedtoamagneticfield,constantintime,withasufficentlystrongvariationalongthespinchain.AdditionalRF-pulses(radio-frequencypulses)thenallowthecoherentcontrolofthestateofthesystemsuchthataquantumprotocolcanberealized(Bermanetal.2000,Bermanetal.2001,Bermanetal.2002).Thisisaspecialformof1NMRquantumcomputationasdescribedin(Jones2000).UltracoldatomsinopticallatticesmayprovideanalternativephysicalrealizationofaIsing-Hamiltonian(Garc´ıa-RipollandCirac2003).Typically,inthesesystemwefindtwotypesofdipolarinteractions:(i)Intrinsicdipolecouplingsbetweenthenuclearspins.Thosescaleasdistancer−3andcanbecancelledbythe“magicangle”method(Slichter1996).(ii)Mediated(mainlybytheelectrons)dipolecouplingswhichhavenocleardistancedependence.Uptonow,thismodelhasbeendevelopedjusttheoreticallyandhopefullythetechnologicalandexperimentalpartmaystartinanearfuture.However,becausetheHamiltonianofthissystemiswellknown,manytheoreticalstudieshavebeenmade(Bermanetal.1994,Bermanetal.2000,Bermanetal.2001,Bermanetal.2002(a),Bermanetal.2002(b),Bermanetal.2002(c),L´opezetal.2003,Celardoetal.2005)whicharealsoimportantforthegeneralunderstandingofquantumcomputation.Inthismodel,firstneighborIsinginteractionamongthenuclearspinsofparamagneticparticlesofspinonehalfwasconsidered.Thus,inthispaperwewanttoconsideralsosecondneighborIsinginteractionamongthenuclearspins.Thetransversecouplingwillbeneglectedsinceonecouldexpectthattheircouplingconstantstobealeasttwoorthreeordersofmagnitudesmallerthanthelongitudinalcouplingconstant(Ising)forourparticularconfiguration.Witharegisterof4qubits,weperformanumericalsimulationofShor’sfactorizationalgorithm(Shor1994)ofthenumber4andteleportation(Bennettetal.1993)ofanarbitraryqubitinachainofthreequbits,allowingforaninteractionbetweensecondneighborsinthesystem.2ThemodelWeconsideranIsingspinchainwithnearestandnext-nearestneighborinteractionasamodelforaquantumregister.Thespinchainissubjecttoaconstantexternalmagneticfieldinz-direction,aswellastoRF-pulses(withthemagneticfieldvectorinthex-yplane).Thischainisinsideastrongmagneticfieldinthez-directionandmaybesubjecttoRF-pulses(withthemagneticfieldvectorinthex-yplane).TheconstantmagneticfieldB(z),whichmustbeextremelystrong,alsohasafieldgradientinthez-direction,whichallowsindividualaddressabilityofthequbits.DuringanRF-pulse,thewholeexternalfieldmaybewrittenasB=(B0cos(wt+ϕ),−B0sin(wt+ϕ),B(z)),(1)whereB0,wandϕaretheamplitude,theangularfrequencyandthephaseoftheRF-field.Theyareassumedtoremainconstantduringapulse,butaretypicallychosendifferentlyfor2differentpulses.WithoutanyRF-field,theHamiltonianreads:H0=−~nXk=1wkIzk+2Jn−1Xk=1IzkIzk+1+2J′n−2Xk=1IzkIzk+2!,(2)wherewkistheLarmorfrequencyofspink.Wedenotewith|0kithestatewherethenuclearspinkisparalleltothemagneticfieldand|1kiwhereitisanti-parallel.TheRF-fieldinducesthedesiredtransitionsbetweentheZeemanlevelsofthesystems.ThestructureoftheHilbertspaceofthespinchainisparticularlyappropriateforquan-tuminformationstudies,wherethebasisunitofinformationisatwo-levelquantumsystem(“quantumbit”,orqubitforshort).AnysuchstateΨ=C0|0i+C1|1icanberepresentedwithrespecttosomebasisstats|0iand|1ibytwocomplexnumbersC0andC1suchthat|C0|2+|C1|2=1.TheL-tensorialproductofL-basicqubitsformanL-registerofL-qubits.Inthisspace,wedenotetheresultingproductbasisby|αi=|iL−1,...,i0iwithij=0,1forj=0,...,L−1.ApurewavefunctioncanbeexpandedinthisbasisbyΨ=PCα|αi,wherePα|Cα|2=1s.Fornotationalconvenience,werequirethatα=PL−1j=0ij2j.TheHamiltonianH0inEq.(2)isdiagonalinthecomputationalbasis(theproductbasisdefinedabove):H0|αn−1...α1α0i=Eα|αn−1...α2α0iIzk|αki=(−1)αk2|αkiEα=−~2n−1Xk=0(−1)αkwk+Jn−2Xk=0(−1)αk+αk+1+J′n−3Xk=0(−1)αk+αk+2!.(3)Theindexαwithoutsubscriptdenotesthepositiveintegerrepresentedbythestringαn−1...α1α0inthebinarynum