Enhanced energy relaxation process of quantum memory coupled with a superconducting qubit

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by  Yuichiro Matsuzaki

Enhancedenergyrelaxationprocessofquantummemorycoupledwithasuperconductingqubit
Yuichiro Matsuzaki1 and Hayato Nakano1
1 NTT Basic Research Laboratories, NTT Corporation, Kanagawa, 243-0198, Japan
Forquantuminformationprocessing,eachphysicalsystemhasadifferentadvantageasregardsimplemen-
tation and so hybrid systems that benefit from the advantage of several systems would provide a promising
approach. Onecommonhybridapproachinvolvescombiningasuperconductingqubitasacontrollablequbit
andanotherquantumsystemwithalongcoherencetimeasamemoryqubit.Theuseofasuperconductingqubit
givesusexcellentcontrollabilityofthequantumstatesandthememoryqubitiscapableofstoringinformation
foralongtime. Ithasbeenbelievedthatselectivecouplingcanberealizedbetweenasuperconductingqubit
andamemoryqubitbytuningtheenergysplittingbetweenthem. However,wehaveshownthatthisdetuning
approachhasafundamentaldrawbackasregardsenergyleakagefromthememoryqubit. Evenifthesuper-
conductingqubitiseffectivelyseparatedbyreasonabledetuning,anon-negligibleincoherentenergyrelaxation
inthememoryqubitoccursviaresidualweakcouplingwhenthesuperconductingqubitisaffectedbysevere
2 dephasing. Thisenergytransportfromthememoryqubittothecontrolqubitcanbeinterpretedastheappear-
1
anceoftheantiquantumZenoeffectinducedbythefluctuationinthesuperconductingqubit. Wealsodiscuss
0
possiblewaystoavoidthisenergyrelaxationprocess,whichisfeasiblewithexistingtechnology.
2

n
a I.INTRODUCTION ble to have a much longer coherence time for a memory su-
J perconductingqubit[14,15]. Bycombiningasuperconduct-

1 ingqubitwithexcellentcontrollabilityandanothersupercon-
A superconducting qubit provides us with excellent con-
1
ductingqubitwithalongcoherencetime,wecanconstructa
trollability of the system for quantum information process-
hybrid system to take advantage of both characteristics. For
] ing. Coherent manipulations of superconducting qubits have
h example, a recent experiment coupling two superconducting
already been demonstrated experimentally, and actually it is
p phase qubits with a resonant cavity [16] showed the possi-
- possibletoperformasinglequbitrotationwithinafewnano
t secondsbyusingaresonantmicrowavefield[1]. Also,ahigh bility of utilizing one of the qubits for control and the other
n
for memory where two phase qubits are entangled through
a fidelity single qubit measurement has already been achieved
u with existing technology [2]. Specifically, a method using a the common quantum bus, namely the resonant cavity [17].
q Another example is a hybrid system consisting of a super-
Josephson Bifurcation Amplifier (JBA) [3, 4] has been used
[ conducting qubit and a microwave cavity. Strong coupling
experimentallytoperformanon-destructivemeasurementon
hasbeenrealizedbetweenthesuperconductingqubitandmi-
2 the superconducting qubit. However, the coherence time of
v crowave cavity [18, 19], which shows a possible application
thesuperconductingqubitisusuallyrelativelyshortwherethe
8 ofahighQcavityasaquantummemoryforstoringtheinfor-
typicaldephasingtimeisoftheorderofamicrosecondatthe
9 mation.
optimal point and becomes tens of nanoseconds far from the
7
3 optimalpoint[5–7]. However, in this paper, we point out quantitatively that
5. Recently, to overcome the problem of the short coherence such hybrid systems composed of a superconducting qubit
0 time, a hybridapproach hasbeensuggested thatone canuse and a memory qubit could have a potential error caused by
1 anotherphysicalsystemasaquantummemory. Onepromis- unwanted energy leakage from the memory qubit. When we
1
ing system for quantum memory is an atomic ensemble of transfer the quantum information to the memory, we have to
:
v electronic spins such as P-doped Si and nitrogen atoms in tune the energy of the superconducting qubit on resonance
i fullerenecagesC wherethespinensembleiscoupledwith withthememoryqubit,andthenitbecomespossibletoswap
X 60
thesuperconductingqubitthroughamicrowavecavity[8–10]. the information from the controllable qubit to the memory
r
a Magneticcouplingbetweenasuperconductingfluxqubitand qubit. Subsequently,bychangingexternalmagneticfield,we
aspinensemblesuchasnitrogen-vacancycentersindiamond candetunetheenergyofthesuperconductingqubittodecou-
canalsoprovidesuchahybridsystem[11,12].Spinensemble ple from the memory qubit. Importantly, the superconduct-
qubits typically have a long coherence time of, for example, ing qubit is usually affected by severe dephasing [5, 6], and
tensofmilliseconds,whichisamilliontimeslongerthanthat this induces an incoherent energy leakage from the memory
of a superconducting qubit [8–11]. Moreover, for the elec- qubit to the superconducting qubit during information stor-
tronspinsboundtodonorsinsilicon,coherencetimescanbe age. Thisenergyrelaxationcausedbydephasingviolatesthe
as long as several seconds [13]. It is known that a super- energyconservation,andthisphenomenoncanbeunderstood
conductingqubitcouldpotentiallyprovideamemoryqubitif as an occurrence of anti-Zeno effect for quantum transport
thelifetimecouldbeincreased. Thecontrolandthemeasure- [20–23]. With reasonable experimental parameters, we eval-
ment setup used for the superconducting qubit, however, in- uate the actual lifetime of the memory qubit, and this turns
ducesdecoherence,andsothereisatradeoffrelationshipbe- out to be much shorter than the previously expected lifetime
tween efficient control and a long coherence time [14]. This ofthememoryqubits[8–11]. Wewillsuggestpossibleways
means that, by sacrificing controllability, it would be possi- toavoidsuchanenergyrelaxationprocess, whichisfeasible
2
withcurrenttechnology. whichmeansthatwecannotobservequantumZenoeffectfor
Theremainderofthispaperisorganizedasfollows. InSec such an exponential decay system. Interestingly, it is known
II,wereviewtheconceptofquantumZenoandantiZenoef- that unstable systems show quadratic decay initially and ex-
fects. SecIIIpresentsthedetailsofourcalculationstoshow ponentialdecaylater[27]. Thetemporalscaleusedtodenote
how unwanted relaxation occurs in the memory qubit due to thecrossoverfromquadratictoexponentialdecayisknownas
the instability of the detuned but weakly coupled supercon- thejumptime[28]. Therefore, toobservethequantumZeno
ductingqubit. InSecIV,wesuggestsomewaystoavoidsuch effect,itisnecessarytoperformprojectivemeasurementson
relaxationbyusingtheideaofthedecoherencefreesubspace. atimescaleshorterthanthejumptime. Moreover,itwaspre-
SecVconcludesourdiscussion. dicted that, when projective measurements are performed on
atimescalecomparabletothejumptime,thedecayiseffec-
tively accelerated, and this is called the anti quantum Zeno
effect [25, 29]. Recently, it was also predicted that the anti
Residual weak coupling
Dissipation
quantum Zeno effect can be induced when we perform false
Superconducting
Environment Memory qubit measurements[30],namely,thedecaydynamicsoftheunsta-
qubit
blestatecanbeenhancedbyfrequentmeasurementswithan
erroneous apparatus where the energy band of the measure-
ment apparatus is significantly different from the energy of
thesignalemittedfromtheunstablestate.
FIG.1: Schematicoftheenhancedrelaxationofthememoryqubit
viaimperfectdecouplingfromasuperconductingqubit. Weassume
thatthesuperconductingqubitisaffectedbydecoherencewhilethere
III.ENHANCEDENERGYRELAXATIONPROCESS
isnodirectcouplingbetweentheenvironmentandthememoryqubit.
Sincewecantunetheenergyofthesuperconductingqubit,itispos-
sibletomaketheenergyofthesuperconductingqubitonresonance Letusstudytheunexpectedrelaxationofthememoryqubit
withtheenergyofthememoryqubittotransfertheinformation. It in a hybrid system shown in Fig. 1 quantitatively. Although
isalsopossibletotunetheenergyofthesuperconductingqubitfar
such relaxation behavior has been studied by [20–23] in an
fromtheresonancewhenkeepingthestoredinformation. However,
antiZenocontextforquantumenergytransport,weintroduce
thistypeofselectivecouplinghasasignificantdrawbackduringthe
asimplersolvablemodelandwederiveananalyticalformof
information storage in the memory qubit, as explained in the main
text. theenergyrelaxationtimeofthememoryqubit. Todescribe
thecouplingbetweenthesuperconductingqubitandthemem-
oryqubit,weusethefollowingHamiltoniancalledtheJaynes-
CummingsmodelortheTavis-Cummingsmodel
ω ω
II.QUANTUMZENOANDANTIZENOEFFECTS H = scσˆ(sc)+ mσˆ(m)+g(σˆ(sc)σˆ(m)+σˆ(sc)σˆ(m)) (1)
2 z 2 z + − − +
Quantum Zeno and anti Zeno effects are fascinating phe- whereω(sc) (ω(m))denotestheenergyofthesuperconducting
nomena predicted by quantum mechanics [24–26]. Let us qubit(memoryqubit)andg denotesthecouplingstrengthof
summarize the quantum Zeno and anti Zeno effects. When the interaction. Note that, although we refer to a supercon-
anunstableexcitedstatedecaystoagroundstate,wecande- ductingqubitasacontrolsystemcoupledwiththememoryin
finethesurvivalprobabilityP (t)asthepopulationremaining this paper, the analysis here can be applied to any system as
e
in the excited state at time t. If this survival probability ex- longastheinteractionwiththememorydeviceisdescribedby
hibits a quadratic behavior in the initial stage of the decay the Jaynes-Cummings model or the Tavis-Cummings model.
suchasP (t)(cid:39)1−Γ2t2forΓt(cid:28)1,itispossibletoconfine Thesemodelsareoffundamentalimportancenotonlyforthe
e
thestatetotheexcitedlevelviafrequentprojectivemeasure- presentsetupbutalsoformanyvariations: couplingbetween
ments. WhenweperformN projectivemeasurementswitha superconductingqubits[31,32]: asuperconductingqubitin-
timeintervalτ = t todeterminewhetherornotthestateis teracting with a microwave cavity [16, 17], a superconduct-
N
stillintheexcitedstate,theprobabilityofprojectingthestate ingresonatorcoupledwithaspinensemble[33–35],orasu-
in the excited level for all N measurements is calculated as perconductingfluxqubitmagneticallycoupledwithnitrogen-
P(t,N) (cid:39) (1−Γ2τ2)N (cid:39) 1−Γ2t2. This success prob- vacancycentersindiamond[11].Sincewecanchangetheen-
N
ability approaches unity as the number of measurements in- ergyofthesuperconductingqubit,itispossibletodetunethe
creases. Sothesystemisfrozenanddecaycanbecompletely energy between qubits when keeping the information stored
suppressed,whichiscalledthequantumZenoeffect. Onthe inthememory. Inthispaper,∆=ω −ω denotesdetuning
sc m
otherhand,ifthesurvivalprobabilityexhibitsanexponential duringsuchastorage. Wechoosetheinitialstateas|0(cid:105) |1(cid:105)
sc m
decaysuchasP (t) = e−Γt, theprobabilityofconfiningthe torepresentthestorageoftheexcitationinthememoryqubit.
e
state to the excited level by performing N measurements is Note that, since the Hamiltonian conserves the total number
calculated as P(t,N) = (e−Γτ)N = e−Γt. Here, the pro- of the excitation, the bases taken into account are |0(cid:105) |1(cid:105)
sc m
jective measurements do not change the success probability, and |1(cid:105) |0(cid:105) , as long as we consider only the dephasing of
sc m
3
thesuperconductingqubitasthedecoherencesource. Inother sionequationsasfollows:
words,thestateofthecoupledsystemisalwaysintheHilbert
1
subspacespannedbythesetwobases. Also,itisworthmen- p = (2g2+∆2p
tioningthat,throughoutthispaper,weassumethatthemem- a,(n+1)τ 4g2+∆2 a,nτ
oryqubitiscoupledonlywiththesuperconductingqubitand + 2g2(p −p )cost(cid:112)4g2+∆2)
a,nτ b,nτ
hasnodirectinteractionwiththeenvironment. Theassump-
1
tionhereisvalidaslongasthelifetimeofthememoryqubit pb,(n+1)τ = 4g2+∆2(2g2+∆2pb,nτ
is much longer than that of a superconducting qubit. This is
(cid:112)
actuallytruefortypicalmemoryqubitsbecausethecoherence − 2g2(pa,nτ −pb,nτ)cost 4g2+∆2).
timeofmemoryqubitscanbetensofmilliseconds, whichis
Bysolvingtheseequationswiththeinitialconditionofp =
amilliontimeslongerthanthatofsuperconductingqubits[8– a,0
0andp =1,weobtain
11]. Toobtainananalyticalsolutionofthedynamicsunder b,0
the effect of the dephasing, we adopt a simple model where
1 ∆2
the system is affected by theunitary operation and the deco- p (cid:39) (1−( )n)
herencealternatively,sothatwecanobtainarecursionequa- a,n 2 4g2+∆2
tionasρ = Eˆ(e−iHτρ eiHτ). Here, ρ denotesthe 1 ∆2
(n+1)τ nτ nτ p (cid:39) (1+( )n) (3)
density matrix of the system at time nτ, Eˆ denotes the de- b,n 2 4g2+∆2
phasing process, and τ denotes a period during the unitary
where we use a rotating wave approximation such that
operation. Inthelimitforasmallτ,thissimplificationcanbe (cid:112)
cos 4g2+∆2tshouldvanishduetothehighfrequencyos-
justified by the Trotter expansion [36]. Moreover, the effect
cillation. We define the effective relaxation time induced by
of dephasing can be considered a process of measurements
this anti Zeno effect as the time at which the population of
without postselection, which we refer to as a “non-selective
theexcitationofthememorybecomes pb,0−pb,∞. Sowecan
measurement”. For example, if a pure state α|0(cid:105)+β|1(cid:105) de- 2
calculatethiseffectiverelaxationtimeofthememoryqubitas
coheresduetothedephasing,wefinallyobtainamixedstate
|α|2|0(cid:105)(cid:104)0| + |β|2|1(cid:105)(cid:104)1|, which is the same state as that ob-
αlog2
tainedafterperformingaprojectivemeasurementwithrespect T˜(m) = T(sc)
1 log(1+ 4g2) 2
to σˆz = |0(cid:105)(cid:104)0|−|1(cid:105)(cid:104)1| on the pure state and discarding the ∆2
measurement results [37, 38]. Therefore, our model can be ∆2
(cid:39) αlog2· T(sc). (4)
interpreted as one where the environment “sees” the system 4g2 2
frequentlytodegradethequantumcoherence,whichprovides
us with an intuitive connection between our calculation and Here, we assumed g (cid:28) 1, namely, the coupling is much
∆
the quantum Zeno effect. When the time τ is comparable to smallerthanthedetuning,whichisappropriateforactualex-
thedephasingtimeT(sc)ofthesuperconductingqubitsuchas perimentsinordertodecouplethesystem. Althoughtheen-
2
τ =αT(sc)whereαisafittingparameteroftheorderofunity, ergyrelaxationmightbeconsideredtobeexponentiallysmall
2 foralargedetuning,theeffectiverelaxationtimeofthemem-
theoff-diagonaltermsofthedensitymatrixbecomesmall. So
we consider the superoperator Eˆ to be a non-selective mea- ory is only quadratically dependent on detuning. Moreover,
T˜(m) islimitedbythedephasingtimeofthesuperconducting
surement process for removing out the off-diagonal terms as 1
qubit. Since a detuned superconducting qubit is strongly af-
follows;
fected by the environment [5–7], the dephasing time of the
superconducting qubit becomes as small as tens of nanosec-
onds, which could lead to a severe energy leakage from the
memory qubits. It is also worth mentioning that, even if we
coupleamicrowavecavitywiththememorydeviceinsteadof
Eˆ(ρnτ) (cid:39) (|0(cid:105)sc(cid:104)0|⊗1ˆ1m)ρnτ(|0(cid:105)sc(cid:104)0|⊗1ˆ1m) the superconducting qubit [33–35, 39, 40], any imperfection
+ (|1(cid:105)sc(cid:104)1|⊗1ˆ1m)ρnτ(|1(cid:105)sc(cid:104)1|⊗1ˆ1m) ofthecavityinsuchacoupledsystemwillalsocausesimilar
= Pˆ(sc)ρ Pˆ(sc)+Pˆ(sc)ρ Pˆ(sc) (2) energyleakagefromthememoryqubits.
0 nτ 0 1 nτ 1 This kind of acceleration of the energy relaxation can be
understood in terms of the violation of energy conservation
caused by the dephasing process, which has been discussed
forbiologysystems[20–23]. Also, ifweconsiderthesuper-
conductingqubitasameasurementapparatusforthememory
where Pˆ(sc) (Pˆ(sc)) is the projection operator to a state |0(cid:105) qubit,itwouldbealsopossibletointerpretethisenhancedre-
0 1 sc
(|1(cid:105) ). Under this approximation, the mixed state after per- laxationastheappearanceoftheantiZenoeffectinducedby
sc
forming this superoperator Eˆshould be described as ρ = falsemeasurementsoferroneousapparatus[30]. Thedecayis
nτ
p |10(cid:105) (cid:104)10|+p |01(cid:105) (cid:104)01|wherep andp acceleratedbythedifferencebetweentheenergybandofthe
a,nτ sc,m b,nτ sc,m a,nτ b,nτ
denote the population of each state. So we obtain the recur- measurement apparatus and the energy of the signal emitted
4
from the unstable state [30]. In our case, the detuned super- mation continues to leak during the storage and so the total
conductingqubitwouldbeinterpretedastheerroneousappa- error accumulation will be significant when we try to access
ratustomeasurethesignal,namelytodeterminewhichenergy theinformationinthememoryaftersuchastorage.
etgenstatethememoryqubitisin,sothattheenergytransport We have obtained an analytical formula for the effective
fromthememoryqubitcouldbeacceleratedduetotheimper- relaxation time of the memory qubit with some approxima-
fectionoftheapparatus. Inaddition,asimilarexpressionhas tion. It is possible to obtain a more rigorous result by solv-
alsobeenusedinquantumoptics,andiscalledthescattering ing a Lindblad master equation numerically. So we adopt
rate [41]. For example, when we drive the Rabi oscillations dρ =−i[H,ρ]− 1 [σˆ(sc),[σˆ(sc),ρ]]asthemasterequation
ofanunstabletwo-levelsystemwithadetunedlight,thetotal dt 2T(sc) z z
2
whereρdenotesadensitymatrixforthecomposedsystemof
scattering rate of light from the laser field can be also sup-
asuperconductingqubitandamemoryqubit.Wehaveplotted
pressed only quadratically against the detuning [41]. So it
theeffectiverelaxationtimeofthememoryqubitfromthenu-
wouldbepossibletointerpretetheenhancedrelaxationratein
mericalsolutioninFig. 2. Thenumericalsolutionshowsthe
ourcalculationasthescatteringrateoftheexcitedpopulation,
althoughinourcasethescatteringiscausedbythedephasing quadraticdependencyofT˜1(m) onthedetuning, whichagrees
ofthesuperconductingqubit. withtheanalyticalresultinEq. (4). Byfittingtheanalytical
resultwiththisnumericalresult,weobtainα=0.500andwe
Regardlessoftheinterpretationoftheenhancedrelaxation
also plot the analytical result in Fig. 2. The behavior of our
ofthememoryqubit,ourresultsareofsignificantimportance
fromapracticalpointofview. Sincethethresholdoftheac-
ceptableerrorrateforachievingfaulttolerantquantumcom-
putation is quite small, typically of the order of 1% [42], it
is essential to find ways to store quantum states in reliable
memory devices isolated from the environment. However,
our results show that the standard way to decouple the con-
trolqubit fromthe memoryqubitby detuningmay notsuffi-
cientlysuppressthenoiseinthestoredquantumstates,which
castsadoubtonthefeasibilityofusingmemoryqubitstrate-
giesforscalablequantumcomputation. Therefore,thisresult
motivatesustofindanotherdecouplingmethodtoprotectthe
memoryqubitsfromthenoiseinducedbysuchantiZenore-
laxation,whichwediscusslaterinourpaper.
Itisworthmentioningthatthisincoherentenergyrelaxation FIG.2: Aplotoftherelaxationtimeofthememoryqubitinduced
ismuchmoreseverethanthewellknownerrorscausedbythe bytheantiZenoeffectofasafunctionofthedetuning.Thediscrete
dispersive Hamiltonian. Without decoherence, the Hamilto- plot denotes a numerical result and the continuous line denotes an
nianbetweenthesuperconductingqubitandthememoryqubit analyticalresult.WesetparametersasT(sc) =10nsforthedephas-
2
canberepresentedasadispersiveformH = g2σˆ(sc)σˆ(m) for ingtimeofthesuperconductingqubitandatg/2π = 25MHzfor
∆ z z thecouplingstrengthbetweenthequbits.Theseresultsshowthatthe
alargedetuning[43]. Therefore,ifwetunethesuperconduct-
relaxationtimeisquadraticallydependentontheenergydetuning.
ing qubit so that it is on resonant with a third party system
such as another qubit for information operations, the super-
conductingqubitinasuperpositionstateinducesphaseerrors
analytical solution matches the numerical solutions, and this
inthedetunedmemoryqubit. Theerrorrateis(cid:15)(cid:39) g2tI where
∆ shows the validity of our approximation. This justifies our
t isthetimerequiredforinformationoperationsonthethird
I interpretationthatthedephasingcorrespondstonon-selective
system. Fortunately,suchinformationoperationscanbeper-
measurementsandtheenergyrelaxationinthememoryqubit
formed in tens of nanoseconds and so this kind of phase er-
iscausedbytheantiZenoeffect.Surprisingly,evenforalarge
ror can be small. Moreover, as long as the superconducting detuningsuchas ∆ (cid:39)50,therelaxationtimeisjustafewmi-
qubit is detuned from any other qubits, the effect of the dis- g
croseconds. Since a typical memory qubit is considered to
persive Hamiltonian on the memory qubit can be negligible
havealonglifetimeof,forexample,tensofmilliseconds[8–
bypolarizingthesuperconductingqubitintothegroundstate.
11], this result shows that the actual relaxation time induced
These resultsseem toshow the suitabilityof this scheme for
by imperfect decoupling is much shorter than previously ex-
the long-term storage of information. However, as we have
pected.
shown above, this naive illustration is no longer valid when
A superconducting qubit can be affected by both dephas-
wetakeintoaccounttheeffectofthedephasingfromtheen-
ing and relaxation. To model both the dephasing and relax-
vironment. In fact, incoherent energy relaxation via the anti
ationonthesuperconductingqubit,weaddarelaxationterm
Zeno effect occurs whenever the information is stored in the
to the Lindblad master equation and we adopt the following
memory qubit. In spite of the fact that the memory qubit is
master equation dρ = −i[H,ρ] − 1 [σˆ(sc),[σˆ(sc),ρ]] −
assumedtoretaintheinformationforalongperiod,theinfor- dt 2T(sc) z z
2
5
1 (σˆ(sc)σˆ(sc)ρ+ρσˆ(sc)σˆ(sc)−σˆ(sc)ρσˆ(sc))whereT(sc)de-
2T(sc) + − + − − + 1
1
notestherelaxationtimeofasuperconductingqubit. Bysolv-
ing this master equation numerically, we are able to plot the
population decay behavior of the memory qubit as shown in
Fig.3.Intheabsenceoftherelaxationinthesuperconducting
FIG. 4: The relaxation time of the memory qubit induced by the
antiZenoeffect. Thehorizontallinedenotesthedephasingtimeof
thesuperconductingqubit.Dotscorrespondtonumericallyobtained
data. Also,continuouslinesaredrawnthroughthepointsasaguide
totheeye.Wesettherelaxationtimeofthesuperconductingqubitat
T(sc) =400nsandthecouplingconstantbetweenqubitsatg/2π=
1
25 MHz. The lowest line is that where the detuning ∆/2π is 600
MHz,andtheotherlinesarewhere∆/2π =800,1000,1200MHz
FIG.3: Thepopulationdecayofthememoryqubitinducedbythe
respectively.
antiZenoeffectasafunctionofatime. Thecouplingconstantbe-
tween qubits is set at 2π ×25 MHz. From the bottom, we plot a
numericalresultwithT(sc) =400nsandT(sc) =10ns,anumerical
1 2
resultwithT(sc) = 10andT(sc) = ∞,andanumericalresultwith
2 1 comesuchenhancedrelaxationproblems. Asanexample,we
T(sc) =∞andT(sc) =400ns.
2 1 discusshowtoavoidthisenhancedrelaxationwhenthemem-
ory qubit consists of an ensemble of microscopic spins with
a long life time. When one uses an ensemble composed of
qubit, the population of the memory qubit decays to half of
N 1 spinsasamemoryqubit,theexcitationofthesupercon-
theinitialpopulationwhilethepopulationdecaystozeroun- 2
ducting qubit is transferred to the ensemble and stored as a
dertheeffectoftherelaxationonthesuperconductingqubit.
collectivemode. Astatewithasinglecollectivemodeinthe
From the numerical solution, we plot T˜(m) as a function of
T(sc) in Fig. 4. Even for a long depha1sing time and huge ensembleisrepresentedas|W(cid:105) = √1N (cid:80)Ll=1σˆ+(l)| ↓↓ ··· ↓(cid:105)
de2tuning such as T(sc) = 35 ns and ∆ = 44, the effective where σˆ+(l) denotes the raising operator of a spin and | ↓(cid:105)
2 g denotes the ground state of a single spin. This strategy of
relaxation time T˜(m) is around 14 µs, which is much shorter
1 utilizing the spin ensemble directly coupled with the super-
thanthetypicallifetimeofthememoryqubit[8–11]. There-
conducting qubit as a memory is suggested theoretically in
fore,ourresultsshowthatthestandardstrategytodetunethe
[11]. However, if we adopt their strategy straightforwardly,
superconducting qubit with the memory qubit actually fails
thememoryensemblewillsufferfromtherelaxationinduced
duetotheenergyleakage fromthememoryqubitduringthe
by the anti quantum Zeno effect as we have discussed. So
storageoftheinformation. Itshouldbenotedthat, sinceour
our purpose here is to decouple this excitation in the ensem-
model is quite general, the result here is significant for ev-
ble from the superconducting qubit. To achieve this, we can
ery hybrid system where a device having a short coherence
apply a spatial magnetic field gradient dB (T/m) with some
time couples with a memory device, as long as the coupling dx
timedurationtothestate|W(cid:105)oftheensemblesothatweob-
is represented by the Jaynes-Cummings model or the Tavis-
Cummingsmodel. tainthestate|Wθ(cid:105) = √1N (cid:80)Nl=1eiθlσˆ+(l)| ↓↓ ··· ↓(cid:105). Here,we
have θ = τµdB∆x where µ denotes the magnetic moment
dx
of the spin, τ denotes application time of such a field gradi-
IV.OVERCOMINGENERGYRELAXATIONPROCESSBY ent, and ∆x denotes the distance between the spins. Since
USINGADECOHERENCEFREESUBSPACE we have (cid:104)W|W (cid:105) = 1 (cid:80)N eiθl, the state |W (cid:105) becomes
θ N l=1 θ
orthogonal with the state |W(cid:105) for θN = 2π, and there-
Finally,wediscusspossiblesolutionstotheproblemofthe fore we can decouple the ensemble from the superconduct-
energyrelaxationcausedbytheantiZenoeffect.Asdiscussed ingqubit. Forreasonableparameterssuchas2π×28GHz/T
in the previous section, significant dephasing of a supercon- for the Zeeman splitting and N∆x = 20 µm for the ensem-
ductingqubitcannullifytheadvantageofthelonglifetimeof ble length, we need a field gradient 10 T/m to achieve the
thememoryqubitduetotheindirectrelaxation,andtherefore orthogonal state in hundreds of nanoseconds. This idea of
ifwearetorealizeahybridsystemitiscrucialthatweover- applying a field gradient was developed in the field of holo-
6
graphicquantumcomputationfordifferentpurposes[39,40]. superconductingqubitwouldnotbepromisingunlesswecan
Inholographicquantumcomputation,weutilizeahybridsys- succeed in making the coherence time of the superconduct-
tem composed of a superconducting qubit, a spin ensemble, ingqubitlongerthanthepresentvalue. However,wecanfind
and a microwave cavity. A field gradient will be applied to a possible solution to this problem, for example, via decou-
transfer the collective excitation of the ensemble to another plingfromthesuperconductingqubitusingadecoherencefree
spinwavemodesothatwecanstoremanyqubitsinoneen- subspaceforthememoryqubits. Ourmodelisquitegeneral,
semble. However, the effect of relaxation enhanced by the andthereforetheresultsreportedherecanbeappliedtomany
anti Zeno effect, in other words, indirect relaxation caused systems. We thank K Semba, B. Munro, X. Zhu, K. Fujii,
by the imperfect decoupling, has not been discussed in pre- S.Saito,H.Tanji,andK.Kakuyanagiforusefuldiscussions.
vious research. We use the technique to apply a field gradi- ThisworkispartiallysupportedbyKAKENHI(Grant-in-Aid
ent for the efficient decoupling of the superconducting qubit forScientificResearchA22241025).
fromthespinensemblesothatalongtimestorageofquantum
states should be possible in the memory. Although we have
mentionedasolutionforthespinensemblememoryhere,the
basic idea could be applied to many other systems. We dis-
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