Physical implementation of topologically decoherence-protected superconducting qubits

0.12 MB·English

by  Zheng-Yuan Xue

Physicalimplementation oftopologicallydecoherence-protected superconducting qubits
Zheng-Yuan Xue,1 Z. D. Wang,1,∗ and Shi-Liang Zhu2
1Department of Physics and Center of Theoretical and Computational Physics,
The University of Hong Kong, Pokfulam Road, Hong Kong, China
2InstituteforCondensed MatterPhysics, School ofPhysicsandTelecommunication Engineering,
South China Normal University, Guangzhou, China
(Dated:February2,2008)
We propose ascenario tophysically implement a kind of topologically decoherence-protected qubit using
superconductingdevicescoupledtoamicro-wavecavitymodewithunconventionalgeometricoperations. Itis
shownthatthetwoneededinteractionsforselectivedevices,whicharerequiredforimplementingsuchprotected
qubits,aswellassingle-qubitgates,canbeachievedbyusingtheexternalmagneticflux. Theeasycombina-
8 tion of individual addressing withthe many-device setup proposed in the system presents a distinct merit in
0 comparisonwiththeimplementationoftopologicallyprotectedqubitsinatrapped-ionsystem.
0
2
PACSnumbers:03.67.Lx,42.50.Dv,85.25.Cp

n
a
J Physical implementation of quantum computers has at- ricalHamiltonian
1 tracted much attention as they are generally believed to be M M 2 M M 2
1 capable of solving diverse classes of hard problems. Sys-
H =−~χ σx −~χ σy , (1)
temssuitableforhardwareimplementationofquantumcom- x  i,j y i,j!
h] puters should possess certain properties, such as relatively Xi=1 Xj=1 Xj=1 Xi=1
p long coherent time, easy manipulation and good scalability. whichinvolvesonlytwo-bodyinteractionsandcanbeunder-
- Withhighlydevelopedfabricationtechniques,superconduct- stoodasspinsina2-dimensional(2D)lattice. Eachspinwas
nt ingquantuminterferencedevices(SQUIDs)haveshowntheir labeledbyits positionin thelattice with σx,y denotingPauli
i,j
a competence in implementing the qubits for scalable quan- matricesofthespinsituatedattheintersectionoftheithrow
u tum computation [1]. Furthermore, the idea of placing the and jth column. It has a two-fold degenerate ground state
q SQUIDs inside a cavity, i.e., the circuit quantum electrody- whichmaybeconsideredasthetwo statesin a qubit. These
[
namics,hasbeenillustrated[2,3,4,5]tohaveseveralpracti- states are generatedby non-localoperatorsand thus are nat-

4 cal advantagesincludingstrongcouplingstrength, immunity urally protected againstlocal sources of decoherence. How-
v tonoises,andsuppressionofspontaneousemission. ever,comparingwithasize-independentenergygapintheKi-
7
taevHamiltonian,theenergygaphereisslightlydependenton
0 Decoherence and systematic errors always occur in real
thenumberoftheinvolvedspins. Anicefeatureofthemodel
4 quantum systems and therefore stand in the way of physi-
2 calimplementationofquantumcomputers. Generally,larger lies in that it is possible to directly implement the Hamilto-
. nian (1) in a trapped-ion system, as it eliminates a main ex-
0 systemsare moresensitive to decoherence,whichmakesthe
perimentalconstrain-thefour-bodyinteractionsinthe Kitaev
1 scalingofquantumcomputersagreatexperimentalchallenge.
model.Inaddition,theenergygapofHamiltonian(1)remains
7 Therefore,howtosuppresstheinfamousdecoherenceeffects
0 largeas the system size increases, and thusit providesmore
is a maintask forscalable quantumcomputation. A promis-
: efficientprotectionthanthenearestneighbormodel[8]where
v ingstrategyforquantumcomputationina fault-tolerantway
energygapdecreasesrapidly.
i is based on the topological idea [6], where gate operations
X In this paper, we propose an intriguingscenario to imple-
depend only on global features of the control process, and
r mentthemodelHamiltonian(1)withsuperconductingdevices
a are therefore largely insensitive to local noises. The Kitaev
(serveasspinsin[6,7])coupledtoamicro-wavecavitymode
model[6]consistsofaclassofstabilizeroperatorsassociated
withunconventionalgeometricoperations.Intrapped-ionsys-
with lattice on the torus, which can be put together to make
tems, the individual addressing is difficult for large arrays
up a Hamiltonian with local interactions. The four-fold de-
(N > 3)sincethedistancebetweentheionsinthe centerof
generate ground state of the Hamiltonian coincides with the
thelineararraygetssmaller[9,10]. Thisconstrainmakesthe
protectedspaceofthestabilizer quantumcode. Sinceallthe
linearconfigurationproposedinRef. [7]difficulttobeexperi-
excitedstates are separatedfromthegroundstates byan en-
mentallyimplemented.Comparingwiththeabovedifficulty,a
ergy gap, the ground states are persistent to local perturba-
distinctfeatureofthepresentimplementationisthatthecom-
tions. However,itisextremelydifficulttodirectlyimplement
binationofindividualaddressingwithamany-devicesetupis
thisnovelideamainlybecausefour-bodyinteractionsareno-
feasible. We will show that, by using the external magnetic
toriouslyhardtogenerateinexperiments.
fluxas theeffectiveswitch tool, the interactionsbetweense-
Veryrecently,Milmanetal.[7]proposedahighlysymmet- lective devicescan be introduced. Furthermore,single-qubit
gates,whicharerequiredintheglobalstateinitialization,can
also be achievedbythe same technology. Duringthe imple-
mentation of the operations, the only parameter needs to be
∗Electronicaddress:[email protected] tunable is the frequency of the external magnetic flux. This
2
ofCooperpairsnontheboxwithitsconjugateϕ= ϕl/4
(a) (b) l
bythe relation[ϕ,n] = i. Attemperaturesmuchlowerthan
P
thechargingenergyandrestrictingthegatechargetotherange
(cid:158) of n¯ ∈ [0,1], only a pair of adjacentchargestates {|0i,|1i}
ontheislandarerelevant. Undertheconditionφ1 = φ2 = 0,
theHamiltonian(2)isthenreducedto
(cid:158)
(cid:158)1 (cid:158)2
ECJJ ECJJ ECJJ J1 J2 J3 J4 ECJJ where E = 2HEs(=1 −−E2cn¯e)σ,zE−2E=ΦσEx, cos(πΦ/φ ) (3=)
n n ce c Φ J 0
E cosφwithφ =hc/ebeingthenormalfluxquantumand
J 0
Cg Cg φ=πΦ/φ0.
When the device is placed in a cavity, the gauge-invariant
Vg Vg pwhhaesreedAifferiesntcheebveeccotomrepsoϕte′mntia=l inϕmthe−sa2φmπ0elgmauAgme o·fdϕlm,.
A maymbe divided into two parts A′ + ARφ, where tmhe
FIG. 1: (Color online) Schematic illustration of the superconduct- m m m
first and second terms arise from the electromagnetic field
ingdeviceastheeffectivespin,thereddashlinedenotetheintegral
of the cavity normal modes and the external magnetic flux,
pathofthemagneticflux. (a)AsingleSQUIDasaneffectivespin.
(b) Device made of two SQUIDs witha common superconducting respectively. For simplicity, we here assume that the cavity
charge box. Thismore flexibledesign willintroduce more control has only a single mode to play a role. Therefore, we have
variablesoftheeffectivespin. 2π A ·dl = 2π Aφ ·dl +2g a+a† ,where
φ0 lm m m φ0 lm m m
2gisthecouplingconstantbetweenthejunctionsandthecav-
R R (cid:0) (cid:1)
ity,a(a†)istheannihilation(creation)operatorforthecavity
is a more efficient way of controlling the system dynamics
mode, with ω being its frequency, the closed path [the red
c
as it is much easier for experimental realization. In achiev- dashlineinFig(1)]integraloftheAφ givesrisetothemag-
ing the implementation of the needed interactions and given
neticfluxΦ. ThentheHamiltonian(3)becomes
the fact that selective couplingcan be effectivelycontrolled,
it is possible to implement topologically protected qubits in H =−E σ −2E cos φ+g a+a† σ . (4)
c ce z J x
this architecture. Moreover, the operations to achieve the
We now show that the requir(cid:2)ed sing(cid:0)le-qubi(cid:1)t(cid:3)gates can be
abovequbitstatesarebasedonunconventionalgeometricop-
achievedby thedc magneticflux. Generallyspeaking, since
erations, which have been illustrated to have a high fidelity
the coupling constant g is very small comparing with φ, we
[11,12,13].
may expand the Hamiltonian (4) up to the first order of g.
A SQUID as a superconducting charge qubit that usually
In the rotating frame with respect to H = −E σ −
addressed in the literature is shown in Fig. (1a) [1]. How- dc ce z
2E σ +~ω a†a+ 1 , the strength of the cavity-SQUID
ever, it would be clear thatthis simple configurationcan not Φ x c 2
interactionterminEq. (4)isproportionto1/(ω −ω )with
be used to acheive the coupling described by Eq. (1). To c q
ω =2E (1−2(cid:0)n¯)/~,wh(cid:1)ichisanexperimentallycontrollable
realizethedesiredHamiltonian,wedesignamoreflexiblede- q c
parameterviathegatevoltageV . Inthepresentscheme,we
vice, as shown in Fig. (1b), to serve as the effective spin. g
choose ω = 0, which correspondsto the degeneracypoint.
It consists of two SQUIDs with a common superconducting q
Giventhefactsthatgisrelativelysmallandthefrequencydif-
charge box, which has n excess Cooper-pair charges. Each
ferenceω −ω ischosentosatisfythelarge-detuninglimit,
SQUIDisformedbytwosmallidenticalJosephsonjunctions c q
the cavity mediatedinteraction can thus be safely neglected.
(JJs)withthecapacitanceC andJosephsoncouplingenergy
J
Inotherwords,thedeviceandthecavityevolveindependently
E ,piercedbyanexternalmagneticflux.Acontrolgatevolt-
J
inthiscase. Theexternalfluxismerelyusedtoaddresssep-
ageV isconnectedtothesystemviaagatecapacitorC .The
g g
arately the single-qubit rotations, while the evolution of the
Hamiltonianofthesystemreads[1]
SQUIDisgovernedbyHamiltonian(3).
4 Therequiredinteractionsbetweentheselecteddevicesmay
H =Ec(n−n¯)2−EJ cosϕl, (2) beinducedbyusingtheacmagneticfluxwithdevicesinFig.
l=1 (1b). If N devices are located within a single-mode cavity
X
as shown in Fig. (2a), to a good approximation, the whole
where n is the number operator of (excess) Cooper-pair
systemcanbeconsideredasN two-levelsystemscoupledto
charges on the box, E = 2e2/(C +4C ) is the charging
c g J aquantumharmonicoscillator[3]. TheHamiltonianoftheN
energy,n¯ = C V /2is theinducedchargecontrolledbythe
g g suchdevices,placedinasinglemodecavity,reads[5]
gatevoltageV , andϕl (l = 1,2,3,4)isthegauge-invariant
g
phase differencebetween the two sides of the lth JJ denoted N
as Jl in Fig. (1b). The phase differencesϕl are determined H = Ece(nj −n¯j)2−EJ cosϕlj , (5)
from the flux quantization for three independent loops, that " #
j=1 l
is, ϕ1 −ϕ2 = 2φ1, ϕ2 − ϕ3 = 2φ, and ϕ3 − ϕ4 = 2φ2. X X
Sinceweherefocusonthechargeregime,aconvenientbasis wherewe haveassumedEce,j = Ece andEJl,j = EJ. Here
is formedbythe chargestates, parameterizedbythe number φ1andφ2arethedcmagneticflux,andwehaveneglectedthe
3
Hamiltonianisgivenby[14,15]
H =−~χ J2, (8)
x x x
(cid:120)(cid:120)(cid:120)
whereχ = 4β2/δ. Ifφ = π/2andφ = kπ+π/2,then
1 2 3 8 9 x − +
Eq. (6)reducesto
x
o Hy =2i~β a†eiδt−ae−iδt J , (9)
int y
(a) withtheeffectiveHamilto(cid:0)nianbeinggiven(cid:1)by
1 2 3
H =−~χ J2, (10)
y y y
y
whereχ = χ . ActuallytheconfigurationdescribedinFig.
y x
x (b) 4 5 6 (1a) is exactly equivalent to the device in Fig. (1b) in the
specific case of φ1 = φ2 ≡ 0. So it is now seen that the
z o
required Hamiltonian (1) is unable to be achieved with the
oneSQUID in Fig. (1a) since onlythe interactionH could
7 8 9 x
berealized.
Itisclearthatthespecificchoiceofthedcmagneticfluxcan
leadtothedesignatedinteractiontypeoftheselecteddevice.
FIG.2: (Color online) (a) Schematicsuperconducting devices (de-
Thecavity-devicecouplinganddecouplingcanbecontrolled
noted as rectangles) array constructed along the cavity direction x
byselectingtheexternalmagneticfluxtobedcorac,notsim-
withN =9.Thechosencoordinateisshownintheleft-lowerpanel.
ply by changing the parameters of the device or the cavity.
Eachdeviceisplacedparalleltothexozplaneandattheantinodes
Sincetheexternalmagneticfluxcanbeeffectivelycontrolled,
ofthesingle-modestanding-wavecavity. Themagneticcomponent
ofthecavitymodeisalongthey direction, whichisperpendicular thecavity-deviceinteractioncanbeimplementintheselected
to the device loop plane, so that it is the only contributed compo- devices. In addition, all the devices are always stay in their
nent. Devicesareplacedattheantinodesofthecavitymode,sothat degeneracypointsinthisimplementation,wheretheypossess
the device-cavity interaction constant (2g) of different devices can longcoherencetimeandminimalchargenoises.Interestingly,
betreatedasthesameconstant. (b)AnN = 9lineararraycorre- theinteractionsinEqs. (8)and(10),whichcorrespondtothe
sponds toa 2D 3×3 array, wherethefilleddotswiththenumber rowandcolumninteractionsinEq. (1),areinsensitivetothe
representsthedevices. Forexample,ifonewantstogeneratethein-
thermal state of the cavity mode. It is notable that the evo-
teractioninrow1,thendevices1,2and3areselectedtoachievethe
lutions governedby the Hamiltonians (8) and (10) may also
J2 coupling viathevirtueexcitationof thecavitymode, whilethe
Jx2 coupling among devices 2, 5 and 8 is implemented in order to be considered as unconventional geometric operations [11],
y which are robust against random operation errors [13], and
achievetheinteractionincolumn2.
thushavebeen essentially used in quantuminformationpro-
cessing[5,10,11,12,15,16].
cavity-mediated interaction parts in these two SQUID loops With the two interactions in Eqs. (8) and (10), and given
consideringthattheycanbemadetobeverysmallcomparing thefactthattheycanbeimplementedonselecteddevices,itis
with the inter-SQUID loop, or we can just put the magnetic straightforwardtoimplementthehighlysymmetricalHamil-
shield to exclude the influence of the cavity mode magnetic tonian (1). The interaction exists only between the devices
fluxinthesetwoloops.Thephaserelationintheinter-SQUID selectedatonetime,thusthespatialpatternofthephysicalar-
loop is now modified as ϕ2 −ϕ3 = 2φ +2g (a+a†) = rayisnotnecessarilyrelatedtothe2Dconfigurationdirectly.
j j j j
2ωt+2g(a+a†). Assumingallthedevicesworkintheirde- Apossibleconfigurationrenderingtheimplementationofthe
generacypoints, underLamb-Dicke(LD) limitand rotating- Hamiltonianeasierinexistingsystemsconsistsofusingalin-
waveapproximationaswellasintheinteractionpicturewith ear array of devices [7]. To implement a topologically pro-
respecttoH =~ω (a†a+1),thecavitymediatedinteraction tectedlogicalqubitconsistsofM×M physicaldevicesin2D
canbedescr0ibedbyctheHam2iltonian lattice, we canuse a 1D lineararraywith N = M2 devices,
wheretheithrowandjthcolumninthe2Dlatticecorrespond
N tothedevices(i−1)×M+1,(i−1)×M+2,...,i×M and
Hint =i~β σj† ηa†eiδt−ξae−iδt +H.c., (6) j,j+M,...,j+(M −1)×M inthe1Darray,respectively.
j=1 Asmall3×32DarrayisillustratedinFig(2b)asaexample.
X (cid:0) (cid:1)
Afterscalingtheinteractionsofeachrowandcolumn,there-
wheregEJ/2~=β ≪δ =ωc−ω ≪ω,σ± = 12(σx±iσy), sulting Hamiltonian is the sum of the terms in Eq. (1). The
η = e−iφ− + e−i(2φ+−φ−), ξ = e−iφ− + ei(2φ++φ−), and coupling between rows and columns may be avoided [7] by
φ = 1 φ1±φ2 . If φ = 0 andφ = kπ, thenEq. (6) applyingoperationsthatalternatebetweenrowsandcolumns
± 2 − +
reducesto providedthateachoperationtimeτ satisfiesτχ ≪1.
x;y
(cid:0) (cid:1)
Topologically protected qubits usually consist of many
Hixnt =2i~β a†eiδt−ae−iδt Jx, (7) physicalqubitsandthusseemtoberesource-consumingstrat-
egy. In fact, a small finite array, e.g., M = 3 for Hamilto-
where J = N σ(cid:0)x,y,z. The corre(cid:1)sponding effective nian (1) [7] and M = 5 for nearest neighbormodel[8], are
x,y,z j=1 j
P
4
likelysufficienttoprovidegoodprotectionfromthenoiseby mode (λ = 6 mm), 10 devices may be constructed along
c
suppressingits effectto be manyordersofmagnitudelower, the cavity direction [17]. Typically, a SQUID loop (1 × 1
whichcouldmeetmostpracticaldemands. µm) consists of two small identicalJJs of the size 0.1×0.1
One experimentallyfeasible procedureto generatea topo- µm, and the inter-SQUID loop can be fabricated in the size
logically protected qubit state is as follows. (i) Applying a of 4× 4 µm. Therefore, the distance of the two devices is
large effective field along the x direction (putting n¯ = 1/2 about 103 times of the device size, so that their mutual in-
and φ = 0 or φ = π) when devicesare decoupledfrom the ductionmaybesafelyneglected. Moreover,itisnotablethat
cavity,wegetH =±E σx .Thentheinitialstateislog- other kinds of 4-junction qubits have already fabricated and
1 J i,j
ical|0i = |0i orlogical|1i = |1i dependingonthe usedintheexperiments,e.g.,in[18].Thus,thecurrentdesign
ij ij
P
overallsignintheHamiltonianH . (ii)Adiabaticallyswitch- of qubit impose no extra requirements on the current tech-
1
Q Q
ingoffH andthenswitchingontheHamiltonian(1). After nology of the qubit fabrication process. Finally, we roughly
1
performingthese two steps, the two initial unprotected logi- estimatetheeffectofthenonuniformityindeviceparameters,
calstates|0iand|1ievolveintooneoftheprotectedground E andEl,totheoperationsinourimplementation.Asdif-
ce,j j
statesofthemodelHamiltonian(1). ferentdevicescanbeaddressedandcontrolledindividually,it
We nowbrieflyaddressthe experimentalfeasibilityinim- iseasytotuneeachdeviceinitsdegeneracypointviaitsgate
plementing our scheme. Suppose that the quality factor of voltageseparately,suchthatthenonuniformityofE among
ce
the superconducting cavity is Q = 1 × 106. For the cav- differentdevicesisnotofsignificantimportance.Thenonuni-
ity modewith ω /2π = 50 GHz [17], the cavitydecay time formity of the Josephson couplings with a deviation of ǫE
c j
is τ ≈ 3.2 µs. With a moderate vacuum Rabi frequency willcauseaminorchangeoftheoperationinfidelityintheor-
c
Ω ∼ 10 MHz and the lifetime of the SQUID γ = 2 µs, the derofexp(−∆/ǫχ ),whichisabout1%andthuscanbene-
x
strong-couplinglimitcanbereadilyreached[4]. Meanwhile, glectedforthereasonableexperimentalparametersǫ ∼ 20%
2g = 10−2 [5] to ensure the LD limit. With E = 40 µeV andδ ∼10β,where∆∼χ istheenergygapoftopological
J x
[5],β/2π ≈ 48MHz. Tomaintainthelarge-detuningcondi- phase.
tion, we can chooseδ ∼ 10β, whichin turnsatisfies readily
the requirement of δ ≪ ω ∼ ω . A typical operation time
c
forlarge-detuninginteractionist ≈ 10ns[5],whichensures We thankL.B. ShaoandG.Chenforfruitfuldiscussions.
thatthousandsofoperationsarepossible[4]sincethecoher- This work was supported by the RGC of Hong Kong un-
encetime of theSQUID andcavitymodeare bothinthe or- derGrantNos. HKU7045/05PandHKU7049/07P,the URC
der of µs. When all devices are located at the antinodes of fund of HKU, the NSFC under Grant Nos. 10429401 and
the microwave, the distance of the two neighbor devices is 10674049, NCET and the State Key Program for Basic Re-
a half of the microwave wavelength. For the chosen cavity searchofChina(Nos.2006CB921800and2007CB925204).
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