Controllable coherent population transfers in superconducting qubits for quantum computing

0.26 MB·English

by  L. F. Wei

Controllablecoherent population transfers insuperconducting qubits forquantum computing
L.F. Wei,1,2 J.R. Johansson,3 L.X. Cen,4 S. Ashhab,3,5 and Franco Nori1,3,5
1CREST,Japan Science and Technology Agency (JST),Kawaguchi, Saitama 332-0012, Japan
2Laboratory ofQuantum Opt-electronics, Southwest Jiaotong University, Chengdu 610031, China
3FrontierResearchSystem,TheInstituteofPhysicalandChemicalResearch(RIKEN),Wako-shi,Saitama,351-0198,Japan
4Department of Physics, Sichuan University, Chengdu, 610064, China
5Center for Theoretical Physics, Physics Department, CSCS,
The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
(Dated:February3,2008)
We propose an approach to coherently transfer populations between selected quantum states in one- and
two-qubit systems byusing controllable Stark-chirped rapidadiabatic passages (SCRAPs). These evolution-
8 timeinsensitivetransfers,assistedbyeasilyimplementablesingle-qubitphase-shiftoperations, couldserveas
0
elementarylogicgatesforquantumcomputing. Specifically,thisproposalcouldbeconvenientlydemonstrated
0
withexistingJosephsonphasequbits. Ourproposalcanfindanimmediateapplicationinthereadoutofthese
2
qubits. Indeed,thebrokenparitysymmetriesoftheboundstatesintheseartificial“atoms”provideanefficient

n approachtodesigntherequiredadiabaticpulses.
a PACSnumber(s):42.50.Hz,03.67.Lx,85.25.Cp.
J

9
2 Introduction.—Thefieldofquantumcomputingisattract- stronglyrelatedtothetopologicalfeaturesofeitheradiabatic
ingconsiderableexperimentalandtheoreticalattention. Usu- ornon-adiabaticevolutionpaths[5]).

]
h ally, elementarylogic gatesin quantumcomputingnetworks Althoughotheradiabaticpassage(AP)techniques,suchas
p are implemented using precisely designed resonant pulses. stimulatedRamanAPs(STIRAPs)[6],havealreadybeenpro-
- The various fluctuations and operational imperfections that posed to implement quantumgates [7], the presentSCRAP-
t
n exist in practice (e.g., the intensities of the applied pulses based approach possesses certain advantages, such as: (i) it
a and decoherence of the systems), however, limit these de- advantageouslyutilizesdynamicalStarkshiftsinducedbythe
u
q signs. Forexample,theusualπ-pulsedrivingforperforming applied strong pulses (required to enforce adiabatic evolu-
[ a single-qubitNOT gate requiresbotha resonancecondition tions) to produce the required detuning-chirpsof the qubits,
andalsoaprecisevalueofthepulsearea. Also,thedifficulty while in STIRAP these shifts are unwantedand thushave to
1
v of switching on/off interbit couplings [1] strongly limits the beovercomeforperformingrobustresonantdrivings;and(ii)
7 precisedesignoftherequiredpulsesfortwo-qubitgates. it couplesqubitlevelsdirectly via either one-photonor mul-
1 tiphotontransitions,whileintheSTIRAPapproachauxiliary
Here we propose an approach to coherently transfer the
4 levelsarerequired.
4 populationsof qubitstates by using Stark-chirpedrapid adi-
. abatic passages (SCRAPs) [2]. As in the case of geometric The key of SCRAP is how to produce time-dependent
1
phases [3], these population transfers are insensitive to the detunings by chirping the qubit levels. For most natu-
0
8 dynamical evolution times of the qubits, as long as they are ral atomic/molecular systems, where each bound state pos-
0 adiabatic. Thus, here it is not necessary to design before- sessesadefiniteparity,therequireddetuningchirpscouldbe
v: handtheexactdurationsoftheappliedpulsesforthesetrans- achievedbymakinguseoftheStarkeffect(viaeitherreal,but
i fers. This is a convenientfeature that could reduce the sen- relatively-weak,two-photonexcitationsofthequbitlevels[8]
X sitivity of the gate fidelities to certain types of fluctuations. orcertainvirtualexcitationstoauxiliarybosonicmodes[9]).
r Another convenientfeature of our proposalis that the phase Here we show that the breaking of parity symmetries in the
a
factorsrelated to the transfer durations(whichare important bound states in current-biased Josephson junctions (CBJJs)
fortheoperationofquantumgates)needonlybeknownafter providesanadvantage,becausethedesirabledetuningchirps
the populationtransfer is completed, at which time they can canbeproducedbysingle-photonpulses. Thisisbecauseall
be cancelled using easily implementable single-qubit phase- the electric-dipolematrixelementscouldbenonzeroin such
shiftoperations. Therefore,dependingonthe natureof fluc- artificial“atoms”[10]. Asaconsequence,theSCRAP-based
tuations in the system, rapid adiabatic passages (RAPs) of quantum gates proposed here could be convenientlydemon-
populationscouldofferan attractiveapproachto implement- stratedwithdrivenJosephsonphasequbits[11]generatedby
ing high-fidelity single-qubit NOT operations and two-qubit CBJJs. In order to stress the analogy with atomic systems,
SWAPgatesforquantumcomputing.Also,theSCRAP-based we will refer to the energy shifts of the CBJJ energy levels
quantumcomputationproposedhere is insensitive to the ge- generatedbyexternalpulsesasStarkshifts.
ometric propertiesof the adiabatic passage paths. Thus, our Models.— Usually, single-qubitgates are implementedby
approachfor quantumcomputingis distinctly differentfrom usingcoherentRabioscillations. TheHamiltonianof sucha
bothadiabaticquantumcomputation(wherethesystemisal- drivenqubitreadsH (t)=ω σ /2+R(t)σ ,withω being
0 0 z x 0
ways kept in its ground state [4]) and holonomic quantum theeigenfrequencyofthequbitandR(t)thecontrollablecou-
computating (where implementations of quantum gates are plingbetweenthequbitstates;σ andσ arePaulioperators.
z x
2
If thequbitis drivenresonantly,e.g., R(t) = Ω(t)cos(ω t),
0
then the qubit undergoesa rotation R (t) = cos[A(t)/2]−
x
isσinxgslein-q[Aub(tit)/N2O],Tw-giathte,At(hte)p=ulse0taΩre(at′i)sdtr′e.quFiroerdrteoalbizeinpgrea- nergy |1 | +(t) |0 nergy |1 | +(t) |0
R E |1 E |1
cisely designed as A(t) = π, since the population of the | fl(t)
target logic state P(t) = [1 − cosA(t)]/2 is very sensitive |0 |0 | fl(t)
to the pulse area A(t) [in this example, we are assuming an
t t
initially empty target state]. Relaxing such a rigorous con- FIG.1: (Coloronline) SimulatedSCRAPsforinvertingthequbit’s
logic states by certain pulse combinations: (left) a linear detuning
dition, we additionally chirp the qubit’s eigenfrequency ω
by introducing a time-dependent Stark shift ∆(t). There0- pulse∆(t)=vat,combinedwithaconstantRabipulseΩ(t)=Ωa;
and (right) a linear detuning pulse ∆(t) = v t, assisted by a
b
fore,thequbitevolvesunderthetime-dependentHamiltonian Gaussian-shape Rabi pulse Ω(t) = Ω exp(−t2/T2). Here, the
b R
H0′(t)=ω0σz/2+R(t)σx+∆(t)σz/2,whichbecomes solid(black)linesaretheexpectedadiabaticpassagepaths,andthe
dashed(red)linesrepresent theunwanted Landau-Zener tunnelling
1 0 Ω(t) paths.
H (t)= (1)
1 2 Ω(t) 2∆(t)
(cid:18) (cid:19)
intheinteractionpicture.Underthecondition aStark-shiftterm∆ (t)σ(2)/2appliedtothesecondqubitand
2 z
evolvethesystemvia(intheinteractionpicture)
1 d∆(t) dΩ(t)
Ω(t) −∆(t) ≪[∆2(t)+Ω2(t)]3/2, (2)
2(cid:12) dt dt (cid:12) −∆2(t) 0 0 0
(cid:12) (cid:12) 1 0 −∆ (t) K(t) 0
thed(cid:12)(cid:12)rivenqubitadiabaticallyevo(cid:12)(cid:12)lvesalongtwopaths—thein- H2(t)= 2 0 K(2t) ∆2(t) 0 . (4)
stantaneouseigenstates|λ (t)i = cos[θ(t)]|0i−sin[θ(t)]|1i
−  0 0 0 ∆2(t)
and |λ (t)i = sin[θ(t)]|0i + cos[θ(t)]|1i, respectively. In  
+  
principle, these adiabatic evolutions could produce arbitrary Obviously, three invariant subspaces; ℜ = {|00i}, ℜ =
0 1
single-qubit gates. For example, a single detuning pulse {|11i}, and ℜ = {|01i,|10i}exist in the above drivendy-
2
∆(t) (withouta Rabipulse)is sufficientto producea phase- namics. This implies that the populationsof states |00i and
shift gate: U (α) = exp(iα|1ih1|), α = − +∞∆(t)dt. |11i are always unchanged, while the evolution within the
z −∞
Furthermore, combining the Rabi and detuning pulses for subspace ℜ2 is determined by the reduced time-dependent
R
rotating the mixing angle θ(t) = arctan[Ω(t)/∆(t)]/2, Hamiltonian(1)withΩ(t) and∆(t)beingreplacedbyK(t)
from θ(−∞) = 0 to θ(+∞) = π/2, another single- and ∆2(t), respectively. Therefore, the APs determined by
qubit gate Ux = exp(iβ+)σ+ − exp(iβ−)σ− (with β± = theHamiltonianH2(t)produceanefficienttwo-qubitSWAP
− +∞µ (t)dt, µ (t) = ∆(t)± ∆2(t)+Ω2(t))can be gate; the populations of |00i and |11i remain unchanged,
−∞ ± ±
adiabaticallyimplementedas: while the populations of state |10i and |01i are exchanged.
R p The passages are just required to be adiabatic and again are
|λ (−∞)i=|0i|λ−−→(t)i|λ (+∞)i=−eiβ−|1i, insensitivetotheexactdetailsoftheappliedpulses.
Ux :(|λ+−(−∞)i=|1i|λ−+→(t)i|λ+−(+∞)i=eiβ+|0i. (3) SCFRigAuPrse. T1hesrheo,wsoslidsc(hbelamcakt)iclindeisaagrreamthsedoefsirtwabolesAimPuplaatthesd,
andthedashed(red)linesaretheunwantedLandau-Zenertun-
Thisisasingle-qubitrotationthatcompletelyinvertsthepop-
nellings[13](whoseprobabilitiesshouldbenegligibleforthe
ulationsofthequbit’slogicstatesandthusisequivalenttothe
presentadiabaticmanipulations).Thesedesignscouldbesim-
single-qubit NOT gate. Note that here the population trans-
ilarlyusedtoadiabaticallyinvertthepopulationsofstates|10i
ferisinsensitivetothepulsedurationandotherdetailsofthe
and|01iforimplementingthetwo-qubitSWAPgate.
pulse shape—there is no need to precisely design these be-
Demonstrations with driven Josephson phase qubits.— In
forehand.Differentdurationsforfinishingthesetransfersonly
principle,theabovegenericproposalcouldbeexperimentally
inducedifferentadditionalphasesβ ,whichcanthenbecan-
± demonstratedwithvariousphysicalsystems[2],e.g.,thegas-
celledbyproperlyapplyingthephaseshiftoperationsU (α).
z phase atoms and molecules, where SCRAPs are experimen-
Similarly, the applied pulses are usually required to be
tally feasible. Here, we propose a convenientdemonstration
exactly designed for implementing two-qubit gates. For
withsolid-stateJosephsonjunctions.
example [12], for a typical two-qubit system described
A CBJJ (see, e.g., [11]) biased by a time-independentdc-
by the XY-type Hamiltonian H12 = i=1,2ωiσz(i)/2 + current I is described by H˜ = p2/2m + U(I ,δ). For-
b 0 b
K(t) σ(i)σ(j)/2, with switchable real interbit- mally,suchaCBJJcouldberegardedasanartificial“atom”,
i6=j=1,2 + − P
couplingcoefficientK(t),theimplementationofatwo-qubit with an effective mass m = C Φ /(2π), moving in a po-
P J 0
SWAPgaterequiresthattheinterbitinteractiontimetshould tential U(I ,δ) = −E (cosδ − I δ/I ). Here, I and
b J b 0 0
be precisely set as tK(t′)dt′ = π (when ω = ω ). This E = Φ I /2π are, respectively,thecriticalcurrentandthe
0 1 2 J 0 0
difficultycouldbeovercomebyintroducingatime-dependent Josephson energy of the junction of capacitance C . Under
J
R
dc-drivingtochirpthelevelsofonequbit.Infact,wecanadd properdc-bias,e.g.,I . I ,theCBJJhasonlyafewbound
b 0
3
Occupation probabilities0.015 PPP021(((ttt))) dsdetwuxuefpserfiiieerncrnagiibemωtlnhe2et1elnSytaaCasnblmRldoyAvaωrlePol1.psb0T)autsohshstua.aimssg,ebptsheleeetmhnaebecnooletnvaesskiipdnargegoreleeptd-ooq.suatFhblieigotuftghrpaeiertrd2efsossshtrahmotoweiuns|lg2dtithhbaieest
−50 0 50
t (ns)
The adiabatic manipulations proposed above could also
FIG.2: (Coloronline) SCRAP-basedpopulation transfersinasin-
be utilized to read out the qubits. In the usual readout ap-
gleJosephsonphasequbit. (Left)manipulatedscheme: CBJJlevels
withdashedchirpedqubitenergysplitting∆(t)iscoupled(solidar- proach [11], the potential barrier is lowered fast to enhance
row) by aRabi pulse Ω(t). Dotted red arrow shows the unwanted the tunnelingand subsequentdetectionofthe logicstate |1i.
leakage transition between the chirping levels |1i and |2i. (Right) Recently[14],aπ-pulseresonantwiththe|1i↔|2itransition
time-evolutions Pj(t) of the occupation probabilities of the lowest wasaddedtothereadoutsequenceforimprovedfidelity;The
threelevels|ji(j =0,1,2)inaCBJJduringthedesignedSCRAPs
tunnellingrateofthestate|2iissignificantlyhigherthanthose
forinvertingthepopulationsofthequbitlogicstates.Thisshowsthat
of the qubit levels, and thus could be easily detected. The
duringthedesirableSCRAPsthequbitleakageisnegligible.
readoutscheme used in [14] canbe improvedfurtherby uti-
lizingtheaboveSCRAPbycombiningtheappliedmicrowave
pulse and the bias-currentramp. The populationof state |1i
states: thelowesttwolevels,|0iand|1i,encodethequbitof
isthentransferredtostate|2iwithveryhighfidelity. Incon-
eigenfrequencyω = (E −E )/~. During the manipula-
10 1 0 trasttotheaboveAPsforquantumlogicoperations,herethe
tionsofthequbit,thethirdboundstate|2iofenergyE might
2 populationtransferforreadoutisnotbidirectional,asthepop-
beinvolved,asthedifferencebetweenE −E andE −E is
2 1 1 0 ulationofthetargetstate|2iisinitiallyempty.Thefidelityof
relativelysmall.Duetothebrokenmirrorsymmetryofthepo-
suchareadoutcouldbeveryhigh,aslongastherelevantAP
tentialU(δ)forδ →−δ,boundstatesofthisartificial“atom”
issufficientlyfastcomparedtothequbitdecoherencetime.
lose their well-defined parities. As a consequence, all the
Similarly, SCRAPs could also be used to implement two-
electric-dipole matrix elements δ = hi|δ|ji,i,j = 0,1,2,
ij
qubit gates in Josephson phase qubits. With no loss of
could be nonzero [10]. This is essentially different from
generality, we consider a superconducting circuit [11] pro-
thesituationsinmostnaturalatoms/molecules,whereallthe
duced by capacitively coupling two identical CBJJs. The
boundstateshavewell-definedparitiesandtheelectric-dipole
SWAP gate is typically performed by requiring that the two
selectionruleforbidstransitionsbetweenstateswiththesame
CBJJs be biased identically (yielding the same level struc-
parity. By making use of this property, Fig. 2 shows how
tures) and the static interbit coupling between them reach
to perform the expectedSCRAP with a single CBJJ by only
the maximal value K . If one waits precisely for an in-
applying an amplitude-controlled dc-pulse I (t) (to slowly 0
dc
teraction time τ = π/2K , then a two-qubit SWAP gate
chirpthequbit’stransitionfrequency)andamicrowavepulse 0
is produced [15]. In order to relax such exact constraints
I (t)= A (t)cos(ω t)(tocouplethequbitstates). Under
ac 01 01
for the coupling procedure, we propose adding a control-
these two pulses, the Hamiltonian of the driven CBJJ reads
H˜1(t) = H˜0−(Φ0/2π)[Idc(t)+Iac(t)]δ. Neglectingleak- lable dc current, Id(2c)(t) = v2t, applied to the second CBJJ.
age, we then get the desirable Hamiltonian (1) with ∆(t) = Thus one can drive the circuit under Hamiltonian H¯12(t) =
∆˜(t) = −(Φ /2π)I (t)(δ − δ ) and Ω(t) = Ω˜(t) = H +(2π/Φ )2p p /C¯ −(Φ /2π)I(2)(t)δ .Here,
0 dc 11 00 k=1,2 0k 0 1 2 m 0 dc 2
−(Φ /2π)A (t)δ . Obviously,foranaturalatom/molecule the last term is the driving of the circuit, and the first term
0 01 01
with δ = 0, the presentscheme forproducinga Stark shift HP =(2π/Φ )2p2/(2C¯ )−E cosδ −(Φ /2π)I δ isthe
ii 0k 0 k J J k 0 b k
cannotbeapplied. Hamiltonian of the kth CBJJ with a renormalized junction-
Specifically, for typical experimental parameters [11] capacitance C¯J = CJ(1 + ζ), with ζ = Cm/(CJ + Cm).
The coupling between these two CBJJs is described by the
(C =4.3pF,I = 13.3µAandI =0.9725I ),ournumer-
icaJlcalculations0showthattheenerbgy-splittings0ofthelowest second term with C¯m−1 = ζ/[CJ(1 + ζ)] being the effec-
tive coupling capacitance. Suppose that the applied driving
three bound states in this CBJJ to be ω = 5.981 GHzand
10
isnottoostrong,suchthatthedynamicsofeachCBJJisstill
ω = 5.637 GHz. The electric-dipole matrix elements be-
21
tween these states are δ00 = 1.406, δ11 = 1.425,δ22 = safely limited within the subspace ∅k = {|0ki,|1ki,|2ki}:
1.450, δ01 = δ10 = 0.053, δ12 = δ21 = 0.077,and δ02 = 2l=0|lkihlk| = 1. The circuit consequently evolves
δ20 = −0.004. If the applied dc-pulse is a linear function Pwithin the total Hilbert space ∅ = ∅1 ⊗ ∅2. Using the
oftime(i.e.,Idc(t) = v1twithv1 constant)andthecoupling interaction picture defined by the unitary operator U0 =
RabiamplitudeΩ(t)=Ω1isfixed,thentheaboveSCRAPre- k=1,2exp(−it 2l=0|lkihlk|),wecaneasilycheckthat,for
ducestothestandardLandau-Zenerproblem[13]. Foratypi- thedynamicsofthepresentcircuit,threeinvariantsubspaces
Q P
caldrivingwithv1 = 0.15nA/nsandA01 = 1.25nA,Fig.2 (relating to the computational basis) exist: (i) ℑ1 = {|00i}
simulatesthetimeevolutionsofthepopulationsinthisthree- correspondingto the sub-HamiltonianH¯1 = E00(t)|00ih00|
levelsystemduringthedesignedSCRAPs. Theunwanted(but with E (t) = −[Φ /(2π)]I(2)(t)δ + (2π/Φ )2p2 /C¯ ,
00 0 dc 00 0 00 m
practicallyunavoidable)near-resonanttransitionbetweenthe pll′ = hlk|pk|lk′iandδll′ = hlk|δk|lk′i;(ii)ℑ2 = {|01i,|10i}
chirping levels |1i and |2i (due to the small difference be- correspondingtothe sub-HamiltonianH¯ (t) takingtheform
2
4
1 ing both conditions simultaneously does not pose any seri-
bilities aosuswdeifhfiacvueltsyhowwitnhatybpoivcea,leexxppeerriimmeennttaalllypaferaamsibelteerAs.PIsncdoeueldd,
a
prob0.5 PP11((tt)) be applied within tens of nanoseconds. This time interval
on P20(t) is significantly longer than the typical period of an experi-
pati 02 mentalRabioscillation,whichusuallydoesnotexceedafew
u
c
c nanoseconds, and could be obviously shorter than the typi-
O
0 −50 0 50 cal decoherencetimes of existing qubits, which might reach
t (ns)
hundredsofnanoseconds,e.g.,fortheJosephsonphasequbits
FIG. 3: (Color online) SCRAPs within the invariant subspace reportedin [11]. Solid-statequbitsofferevidentadvantages
ℑ3 ={|02i,|11i,|20i}forthedynamicsoftwoidenticalthree-level due to their scalability and controllability. Therefore, RAPs
capacitively-coupledCBJJsdrivenbyanamplitude-controllabledc-
in solid-state qubitscould providean attractive approachfor
pulse. (Left)AdiabaticenergiesandthedesirableAPpath(themid-
data storage and quantum information processing. We hope
dlesolid-linewitharrows): A → C1 → C2 → C3 → B. (Right)
that such techniques will be experimentally implemented in
Time evolutions of populations Pα(t),α = 20,11,02, within the
invariantsubspaceℑ3duringthedesignedSCRAPsforinvertingthe thenearfuture.
populationsof|10iand|01i. Itisshownthattheinitialpopulation
of the |11i state (corresponding to the A-regimein the left figure) This work was supported partly by the NSA, LPS, ARO,
isadiabaticallypartlytransferredtothetwostates|20i and|02i in
NSF grant No. EIA-0130383; the National Nature Science
theC1,C2,andC3 regimes,respectively. Notethatthepopulation FoundationofChinagrantsNo.60436010andNo.10604043.
ofthestate|11ivanishesatt = 0andcompletelyreturnsafterthe
passages.
[1] See, e.g., J. Fitzsimons and J. Twamley, Phys. Rev. Lett. 97,
ofEq.(1)withΩ(t) = Ω¯ = 2(2π/Φ )2p2 /C¯ and∆(t) = 090502(2006);L.F.Wei,Yu-xiLiu,andF.Nori,ibid.,246803
0 10 m (2006);X.X.Zhouetal.,ibid.89,197903(2002).
∆¯(t) = [φ /(2π)]I(2)(t)(δ −δ );and(iii)ℑ = {|02i=
0 dc 11 00 3 [2] A. Abragam, The Principles of Nuclear Magnetism (Oxford
|ai,|11i=|bi,|20i=|ci}correspondingto Univ.Press,Oxford,England, 1961); M.M.T.Loy, Phys.Rev.
Lett.32,814(1974);41,473(1978).
E (t) Ω e−itϑ Ω
a ab ac [3] See,e.g.,M.V.Berry,inAsymptoticsBeyondAllOrders,edited
H¯3(t)= Ωbaeitϑ Eb(t) Ωbce−itϑ , byH.SegurandS.Tanveer(Plenum,NewYork,1991).
 
Ω Ω eitϑ E (t) [4] E. Farhi et al., Science 292, 472 (2001); arXiv:
ca cb c
  quant-ph/0001106 (2000); See also, S. Ashhab, J. R. Jo-
withE (t)=−[Φ /(2π)]I(2)(t)δ +(2π/Φ )2p p /C¯ , hansson,andF.Nori,Phys.Rev.A74,052330(2006).
a 0 dc 22 0 00 22 m
E (t)=−[Φ /(2π)]I(2)(t)δ +(2π/Φ )2p2 /C¯ ,E (t)= [5] L.X.Cenetal.,Phys.Rev.Lett.90,147902(2003);P.Zanardi
b 0 dc 11 0 11 m c andM.Rasetti,Phys.Lett.A264,94(1999).
−[Φ0/(2π)]Id(2c)(t)δ00+(2π/Φ0)2p22p00/C¯m;Ωab =Ωba = [6] See,e.g.,K.Bergmann,H.TheuerandB.W.Shore,Rev.Mod.
Ω = Ω = (2π/Φ )2p p /C¯ , Ω = Ω = Phys. 70, 1003 (1998); N.V. Vitanovet al., Annu. Rev. Phys.
bc cb 0 01 12 m ac ca
(2π/Φ )2p2 /C¯ , and ϑ = ω − ω . Under the APs Chem.52,763(2001); Z.KisandE.Paspalakis,Phys.Rev.B
0 02 m 10 21
for exchanging the populations of the states |10i and |01i, 69,024510(2004).
[7] C. Menzel-Jones and M. Shapiro, Phys. Rev. A 75, 052308
we can easily see that the population of |00i remains un-
(2007);X.Lacouretal.,Opt.Comm.264,362(2006).
changed. Also, after the desired APs, the population of the
[8] T.Rickesetal.,J.Chem.Phys.113,534(2000);L.P.Yatsenko
state |11i should also be unchanged. Indeed, this is verified
et al., Opt. Commun. 204, 413 (2002). A.A. Rangelov et al.,
numericallyinFig.3forthetypicalparametersζ =0.05and Phys.Rev.A72,053403(2005).
v = 3.0 nA/ns. Therefore, the desirable two-qubit SWAP [9] M.Amniat-Talab,R.Khoda-BakhshandS.Guerin,Phys.Lett.
2
gate could also be effectively produced by utilizing the pro- A359,366(2006).
posedSCRAPs. [10] J.Clarke,Science239,992(1988);Yu-xiLiuetal.,Phys.Rev.
Lett.95,087001(2005);J.Q.YouandF.Nori,Phys.Today58,
Discussions and Conclusions.— By using SCRAPs, we
No.11,42(2005).
haveshownthatpopulationscouldbecontrollablytransferred
[11] A.J. Berkleyet al., Science300, 1548 (2003); R.McDermott
between selected quantum states, insensitive to the details
etal.,Science307,1299(2005); J.Claudonetal.,Phys.Rev.
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populationtransferscouldbeusedtogenerateuniversallogic [12] N.SchuchandJ.Siewert,Phys.Rev.A67,032301(2003); J.
gates for quantum computing. Experimentally existing su- KempeandK.B.Whaley,ibid.A65,052330(2002);F.Meier,
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perconducting circuits were treated as a specific example to
[13] H.Nakamura,NonadiabaticTransitions(WorldScientific,Sin-
demonstratetheproposedapproach.
gapore,2002);M.Wubsetal.,NewJ.Phys.7,218(2005).
Like other RAPs, the adiabatic nature of the present
[14] E.Luceroetal.,unpublished.
SCRAPsrequiresthatthepassagesshouldbesufficientlyslow [15] See,e.g.,D.P.DiVincenzo,Phys.Rev.A51,1015(1995);J.H.
(comparedtotheusualRabioscillations)andsufficientlyfast Plantenbergetal.,Nature447,836(2007);F.W.Strauchetal.,
(compared to the decoherence times of the qubits). Satisfy- Phys.Rev.Lett.91,167005(2003).