Table Of ContentErrors due to Finite Rise/fall Times of Pulses in Superconducting Charge Qubits
Sangchul Oh1,∗
1Department of Physics, Korea Advanced Institute of Science Technology, Daejon 305-701, Korea
(Dated: February 1, 2008)
Westudy numerically thedynamics of two-qubitgates with superconducting charge qubits. The
exactratioofEJ toEL andthecorrespondingoperation timearecalculated inordertoimplement
two-qubitgates. Weinvestigatetheeffectoffiniterise/falltimesofpulsesinrealizationoftwo-qubit
gates. It is found that the error in implementing two-qubit gates grows quadratically in rise/fall
times of pulses.
PACSnumbers: 03.67.Lx,73.23.-b,85.25.Cp
2
0 I. INTRODUCTION qubits. Alsonumericalstudiesoftwo-qubitgatesarepre-
0 sented. InSec.IVweanalyzeerrorsduetofiniterise/fall
2 times of pulses on superconducting charge qubits. Fi-
Building a practical quantum computer with a large
n nally, in Sec. V we summarize the results.
number of qubits has recently attracted much attention
a
since the usefulquantumalgorithms1,2 andthe quantum
J
error-correction codes3,4 were developed. Basic quan-
7
tum logic gates were implemented on trapped ions5,6, II. HAMILTONIAN OF SUPERCONDUCTING
1
QED cavities7, and liquid state NMR8,9,10. Solid-state CHARGE QUBITS
1 devices have advantages of large scale integration, flexi-
v bility in the design, and easy connection to conventional In this paper we consider two qubit systems: ideal
8 electronic devices. Some of solid-state devices proposed qubits and superconducting charge qubits16,18. By com-
6 are as follows: electron spins in quantum dots11, nuclear paring two qubit systems, one can discover differences
0
spins of donor atom in silicon12, and ultrasmall Joseph- and similarities between them, which will be helpful to
1
son junctions13,14,15,16,17,18. improve the design of superconducting charge qubits.
0
2 Two types of superconducting qubits based on ultra- Firstconsideranidealmodelforaquantumcomputer.
0 small Josephson junctions were proposed. One is to use The Hamiltonian of the system of N ideal qubits reads
/
h a number of excess Cooper pairs on a superconduct-
N
p ing Cooper-pair box, called the superconducting charge Hideal(t) = B (t) σ(i)
- qubit14,15,16. And the other is to utilize a single flux q − i ·
nt quantumofasuperconductingloop,calledthesupercon- Xi=1
1
a ducting flux qubit13. Here, we focus on superconducting J (t) σ(i)σ(j)+σ(i)σ(j) , (1)
u charge qubits. In theoretical aspects, the measurement −2 ij x x y y
q of superconducting charge states using a single electron Xi<j (cid:2) (cid:3)
:
v transistoranddecoherenceduetocouplingofqubitswith where σ(i) = (σ(i),σ(i),σ(i)) are Pauli matrices for the
Xi environments were studied17,18. Also the quantum leak- i-th qubit. HerextheyZeemzan coupling terms B (t) and
age of superconducting chargequbits was pointed out19. i
ar In experiments, Nakamura et al.20 demonstrated the co- tbheetwineteenrzqeurobiatncdoufipnliitnegsvaJliuje(ts)incaancboenttruorlnleeddwonaya.nd off
herent oscillations of Cooper pairs on a superconducting
Let us consider a systemof N superconducting charge
Cooper-pairbox. Thiscorrespondstoarotationofasin- qubits 14,16,18. Each qubit consists of a single-Cooper-
gle qubitaboutxaxis. Thusthe realizationoftwo-qubit
pair box with two ultrasmall Josephson junctions of ca-
gatesis animportantstepfor makinga superconducting pacitanceC0 forminga DC-SQUIDringandagate elec-
quantum computer with a medium size. J
trode with capacitance C . The dynamics of a qubit is
g
In this paper, we investigate how the two-qubit gate characterizedby relevant energy scales; superconducting
couldbeimplementedonsuperconductingchargequbits. gap ∆, charging energy E e2/2(C +2C0), Joseph-
Thepulsesequenceandoperationtimes,andthesimula- son coupling energy E0, aCnd≡thermalgfluctuJation k T.
J B
tion of two-qubit gates are reported. We examine errors Assume thatthe systemis inthe regimeofE ∆and
C
due to finite rise/fall times of pulses in the realizationof k T E in order to suppress quasi-particle t≪unneling
B C
two-qubit gates. orexc≪itation. Alsosupposethatthesystemoperatesun-
Our paper is organized as follows. In Sec. II we intro- der the conditions E E and C V /(2e) 1. Then
J C g g
≪ ∼
duce the Hamiltonians of superconducting charge qubits only two charge states 0 , 1 , no excess Cooper pair
{| i | i}
and of ideal qubits. A numerical method which allows and one excess Cooper pair on the box, play a role and
us to simulate a quantum computer is presented. In representthequbit. Afewwaysofcouplingchargequbits
Sec.IIIweexplicitlyshowthepulsesequencenecessaryto were proposed14,15,16,18. Here we consider the coupling
implement two-qubit gates with superconducting charge betweenchargequbitsviatheLCresonantcircuit,where
2
N charge qubits are connected in parallel to a common Idletime
1
inductorwithinductanceL. TheHamiltonianofthesys-
P t
tem of N superconducting charge qubits is given by rec
( )
e
N 0.5
1
H (t) = E σ(i)+E (Φ )σ(i)
q −2 Ci z Ji Xi x
i=1
X(cid:2) (cid:3)
− EJEiEJjσy(i)σy(j), (2) 00 1 2 3 4
i6=j L Bxt
X ph¯
where E = 4E (C V /e 1) is turn on and off
Ci C gi gi − FIG. 1: Rectangular pulses with finite rise/fall time 2ǫ are
by applying the gate voltage V on the i-th gate elec-
gi plotted as a function of time. Here time is normalized in the
trode. Here E (Φ ) = 2E0cos(πΦ /Φ ) is the effec-
Ji Xi J Xi 0 unit of π¯h/Bx. The role of idle time between two successive
tive Josephson energy of the i-th qubit, and controlled
pulses is to prevent the tails of pulses from overlapping each
by the external flux ΦXi. Φ0 = h¯/2e is a flux quantum. others.
Here is E = Φ20 2CJ0 2 with C−1 =(2C0)−1+C−1.
L π2L Cqb qb J g
The Hamiltonian of Eq. (2) is similar to the ideal
(cid:0) (cid:1) by a kink and anti-kink pair
one of Eq. (1) except one fact. The system of super-
conducting charge qubits has only two kinds of inde- 1 t t t t
a b
pendently controllable parameters, {ECi,EJi}, whereas Prec(t)= 2 tanh ǫ−/2 +tanh ǫ/−2 , (5)
the ideal model has four kinds of controllable parame- (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
ters B ,B ,B ,J . Coupling between two super-
xi yi zi ij where the rise (or fall) time is about 2ǫ. That means
{ }
conductingchargequbitscouldberealizedbyturningon P (t +ǫ) > tanh(2) 0.964 for τ > 2ǫ. The smaller
rec a
both EJi and EJj. This causes somewhat disadvantages ǫ one takes,∼the sharpe≈r rectangular pulse one obtains.
in operating two-qubit gates, which will be discussed in
Fig. 1 illustrates a sequence of two rectangular pulses.
Sec. IV.
The time-evolutionof N qubits is governedby a time-
dependent Schr¨odinger equation
III. CNOT GATE WITH SUPERCONDUCTING
CHARGE QUBITS
∂
i¯h ψ =H (t) ψ , (3)
q
∂t| i | i
In this section, we explicitly show how the controlled-
where ψ = L a (t) m with L 2N 1 anda not (CNOT) gate with superconducting charge qubits
C. Its| ciomputmat=i0onaml ba|sisiis repre≡sented−by a tenmso∈r could be implemented by applying a sequence of pulses.
P What makes the CNOT gate so important is that the
productofindividualqubits, m = q q q ,
where integer m = 2N−1q +| 2iN−2|q1i+⊗| 2i+⊗·2·0·q⊗|aNnid CNOT gate (or any nontrivial two-qubit gate) and
1 2 ··· N single-qubit gates form a universal set of logic opera-
q 0,1 . Multiplying both sides of Eq. (3) by n, one
i ∈{ } h | tion. Also a general two-qubit controlled-U gate can
gets a set of N coupled first-order ordinary differential
be built up of two CNOT gates and three single-qubit
equations for the function a (t)
n gates. The CNOT gate acting on two qubits i and
j is represented by the unitary matrix in the basis of
L
i
0 0 , 0 1 , 1 0 , 1 1
a˙ = H (t)a , (4) i j i j i j i j
n nm m {| i | i | i | i}
−¯h
m=0
X 1 0 0 0
wwhheicrhe Hallnomw(st)u≡s thon|soHlvqe(t)th|me ia.bWoveedineivteialolpveadluaepprroogbrlaemm UCijNOT =00 10 00 01 , (6)
andto simulate aquantumcomputer. Ourcode is based 0 0 1 0
on the Runge-Kutta method22.
where i and j are indices for a control bit and a target
A quantum logic gate can be realized by controlling
bit, respectively.
thetime-evolutionoperatorwhichactsonselectedqubits
for a fixed period of time. A way to control the time-
evolution operator is to turn on or off each terms in
the Hamiltonians of Eq. (1) or (2). For an example, a A. CNOT gate with ideal qubits
pulse P(t), which is applied as B (t) = B0 P(t) with
αi αi
α = x,y,z, makes it possible to change a term of the First consider the implementation of the CNOT gate
Hamiltonian from zero to finite value B0 for a finite for an ideal system whose Hamiltonian is given by
αi
time τ. In this paper, we consider a rectangular pulse Eq. (1). The primary two-qubit gate acting on the i-
P (t)withwidthτ =t t andaunitheight,modeled th and j-th qubits could be implemented by turning
rec b a
−
3
bility 1 j10i qubits, for an example qubits 1 and 2, can be written as
Proba 0.5 jj32ii Hph = −E2Jσx(1)− E2Jσx(2)−Eintσy(1)σy(2), (9)
0
j i
J12 where E E2/E with an assumption of E =
int ≡ J L J1
E E . For a moment, suppose that switching E ’s
B2 onJ2an≡doffJ couldbe done instantaneously. As willbe dJiis-
cussedlater,a finite time inswitching onE ’sgivesrise
Ji
B to an error in implementing two-qubit gates.
1
Thebasictwo-qubitgateisgivenbythetime-evolution
operator under the Hamiltonian of Eq. (9) for a finite
0 1 2 3 4 5 6
time τ
Bx1t
ph¯ Uph(τ)=e−iHphτ/h¯. (10)
FIG.2: Time-dependenceofprobabilities|aij|2’sasafunction Letustransformtheabovetime-evolutionoperatorunder
goaftteheUCn12NorOmTailnizeadntiidmeaelBmxo1dt/eπl.¯hTohnetsheequacetnicoenooffrtehcetaCnNguOlaTr the unitary operator Ry ≡exp −iπ4(σy(1)+σy(2)) as
wpuitlhsesii=sg1i,v2e.ninthebelowpart. HereisBi=(Bxi,Byi,Bzi) Up′h ≡Ry†UphRy =(cid:2)e−iHp′hτ/h¯, (cid:3) (11)
where the Hamiltonian transformed, H′ R†H R
ph ≡ y ph y
on the coupling J (t). It can be written in the basis reads
ij
0 0 , 0 1 1 0 , 1 1 as
{| i ji | i ji| i ji | i ji} H′ = EJσ(1) EJσ(2) E σ(1)σ(2) (12a)
1 0 0 0 ph − 2 z − 2 z − int y y
0 cosγ isinγ 0 a 0 0 1
Uij(γ)= , (7) −
2b 0 isinγ cosγ 0 0 0 1 0
= E . (12b)
0 0 0 1 − int 0 1 0 0
−1 0 0 −a
where γ J t/¯h. Then, the CNOT gate for an ideal
ij
≡ Here a E /E =E /E . Note that the Hamiltonian
qubit system can be realized by combining single-qubit J int L J
≡
gates and the primary two-qubit gate Uij of Eq. (12) does not mix the subspace spanned by 00
2b and 11 with that by 01 and 10 . Then the ti|mei-
UCijNOT = Hieiπ4 e−iπ4σx(j)eiπ4σx(i) beveocloum|tieosin operator of th|e triansfor|meid Hamiltonian Hp′h
×U2ijb(π4)eiπ2σx(i)U2ijb(π4)Hi, (8) U′ =
ph
where Hi is a Hadamard gate acting on the i-th qubit. cosθ+nzsinθ 0 0 inxsinθ
Fig. 2 depicts the time-evolution of qubits under the 0 cosφ isinφ − 0
,(13)
action of CNOT gate for the ideal model. The lower 0 isinφ cosφ 0
part of Fig. (2) illustrates the sequence of rectangular in sinθ 0 0 cosθ n sinθ
pulses which generates the CNOT gate of Eq. (8). Here − x − z
is taken ψ = 3 = 11 , i.e., a 2 = 1, as an initial where the time evolution corresponds to the rotation
in 11
| i | i | i | |
state (denoted by ). After implementing the CNOT about the y-axis by angle φ E τ/¯h in the sub-
gate, one obtains a◦ 2 =1, corresponding to state 10 , space spanned by 01 and 10≡. Alisnot in the subspace
10
| | | i | i | i
labeledby+. InFig.2,ittakesalongtimetoimplement spanned by 00 and 11 , the time evolution gives rise
the single-qubitgate e−iπ4σx(j) because Bαi’s are taken to to the rotati|oniabout|thie axis (nx,0,nz) by angle θ ≡
be always positive or zero. √1+a2E τ/¯h, where
int
a E
L
n = , (14a)
z
B. CNOT gate with superconducting charge qubits ≡ √a2+1 E2 +E2
L J
1 E
n = p J . (14b)
Letus consider the implementationof the CNOT gate x ≡ √a2+1 E2 +E2
L J
with superconducting charge qubits. The coupling be-
tween the i-th and j-th qubits can be switched on by BychoosingtheappropriatevpaluesE ,E ,andτ,one
L J
turning on Josephson couplings, E and E , and by can control the rotation angles φ and θ. Let us consider
Ji Jj
turning off charging energy terms, E = E = 0. specialanglesφ= π(2m 1)withm=1,2, , andθ =
Ci Cj 4 − ···
Then, the interaction Hamiltonian between two charge nπ with n=0, 1, 2, . Note that the case of even n
± ± ···
4
issametothatofoddnuptotheglobalphaseeiπ. These y 1
angles can be obtained by taking the evolution-time τ = bilit j10i
π4(2m−1)Eh¯int, which is givenby the relationof angle φ. oba0.5 j2i
Thenonehasθ = π(2m 1)√1+a2 = π 1+E2/E2 = Pr j3i
4 − 4 L J j i
nπ. Thus we obtain the relation between E and E
L J
p 0
E
J2
2
E 4n
L = 1, (15) Ech2
EJ s(cid:18)2m−1(cid:19) − E
J1
where it could be accomplished by tunning E or E . E
J L ch1
Note that one can not take n = 0. Also the evolution-
time is given by 0 1 2 3 4 5 6 7 8 9
EJt
π¯h ph¯
τ = (4n)2 (2m 1)2. (16)
4E − −
J
p FIG. 3: Time-evolution of probabilities of qubits as a func-
We have the basic two-qubit gate U = R U′ R† tion of time on action of the CNOT gate for superconduct-
ph y ph y
written by ing charge qubits. Time is normalized in the unit of π¯h/EJ.
ECi/EJ = 2 and EL = √143EJ are taken. The sequence of
1+eiφ 0 0 1 eiφ rectangular pulses is depicted in thebelow.
−
1 0 1+e−iφ 1 e−iφ 0
Uph = 2 0 1 e−iφ 1+−e−iφ 0 , (17)
− [H (t),H (t′)] = 0 for t = t′. This causes an error in
1 eiφ 0 0 1+eiφ ph ph 6 6
− the realization of two-qubit gates with superconducting
where φ = π(2m 1) with m = 1,2, . By combin- charge qubits. If Hamiltonian’s at different times com-
4 − ··· mute,the shapeofthe pulseappliedisofnoimportance.
ing U with single-qubit gates, the controlledphase flip
ph
gate Uij , operatingonthe i-thandj-thqubits, canbe The ideal qubits and single-qubit rotations of supercon-
CPF ducting charge qubits are free from this problem.
realized by
The Magnus expansion23 provides the means of repre-
Uij = e−iφσz(i)eiφσz(j)Uij e−iπσz(i)/2Uij , (18) senting the time-evolution operator of a time-dependent
CPF ph ph Hamiltonian H(t) as
whereφisthevaluegivenabove. Byusingthecontrolled
phaseflipgateUij andtheHadamardgateH onqubit U(t)=Te−h¯i 0τ dτH(τ) =e−h¯iH¯τ , (20a)
CPF j R
j, the CNOT gate can be implemented by
where T denotes the time-ordering operator. Here the
Uij = H Uij H . (19) average Hamiltonian is given by H¯ = H¯(0)+H¯(1)+
CNOT j CPF j where ···
Taking m = 1 and n = 3, one gets EL = 1 τ
√143E . This value satisfies the physical condition H¯(0) = dt H(t ), (20b)
J τ 1 1
EL 10EJ16,18,21. The typical time-scale for operating Z0
itshme∼buachsiclotnwgoe-rqtuhbaintgthaeteoipsegriavteionnbtyimτe=of4πtEhh¯Je√si1n4g3l,e-wqhuibciht H¯(1) = 2−τi¯h τ dt2 t2 dt1[H(t2),H(t1)]. (20c)
Z0 Z0
gate, τ =h¯/E or h¯/E . Fig. 3 shows time-evolution
op J Ci
ofqubitswhen ψ = 11 istakenasaninput(denoted If the rise/fall time is short, then the approximation to
by ). After op|eirnaiting| thie CNOT gate on two qubits, the first term H¯(0) is good. Otherwise the higher order
one◦gets the output ψ = 10 , which is labeled by . terms become significant. If the pulse for the risetime is
| outi | i × modeledbyP(t)=t/2ǫ,thefirsttermH¯(1),whicharises
when the basic two-qubit gate U is implemented, can
ph
IV. ERRORS DUE TO FINITE RISE/FALL be written by
TIMES OF PULSES
E3ǫ
H¯(1) J σ(1)σ(2)+σ(1)σ(2) . (21)
≈−15E ¯h z y y z
Up to now, it was assumed that the effective Joseph- L
(cid:16) (cid:17)
son coupling E (t) could be switched on (or off) in-
Ji The error due to the finite rise/fall times of pulses is
stantaneously. This means a pulse applied is a per-
quantified by the gate fidelity24
fect rectangular one and has no rise/fall times. How-
ever, in reality it takes finite times to turn on (or off) F = ψ U†ρ U ψ = ψ ψǫ 2, (22)
pulses fully. While E (t)’s being switched on (or off), h in| out | ini |h out| outi|
Ji
theHamiltonianofEq.(9)H becomestime-dependent where U is the unitary operator corresponding to the
ph
and the H ’s at different times do not commute, i.e., ideal gate when perfect rectangular pulses are applied.
ph
5
On the other hand, the unitary operator generated by Fig.4showsthenumericalstudyoftheerrorduetofi-
pulses with finite rise/fall times transforms the input niterise/falltimes inimplementing theCNOTgatewith
state ψ intothedensityoperatoroftheimperfectout- superconducting charge qubits. For ideal qubits, if fi-
in
| i
put state ρ . In our case the gate fidelity is nothing nite idle times betweensuccessivepulses areapplied, the
out
but the square of the overlap between the perfect out- correct CNOT gate is realized. However, for supercon-
putstate ψ =U ψ andtheimperfectoutputstate ductingchargequbits,althoughafiniteidletimereduces
out in
| i | i
ρ = ψǫ ψǫ . Assuming that Eq. (21) is small, it the error, there still exits the error due to finite rise/fall
out | outih out|
is straightforwardto calculate the gate fidelity times. As shown in Fig. 4 (b), the probability of getting
thecorrectstatereducesquadraticallyinrise/falltime2ǫ
F ψ e−iH¯(1)τ/h¯ ψ 2 1 ǫ2 ∆η2 (23) in agreement with the theoretical prediction of Eq. (23).
in in
≈|h | | i| ≈ − h i
where ∆η2 ψ η2 ψ ( ψ η ψ )2 is the dis- In Nakamura et al.’s experiment20, the rise/fall times
in in in in
persionhof ηi ≡H¯h(1)τ|/¯hǫ|. Fio−r smhall|ris|e/faill time, error of the pulse was about 30 40 ps at the top of the cryo-
−
≡ stat. If the effective Josephsoncoupling is E 50 µeV,
grows quadratically in rise/fall time 2ǫ. J
≈
then the timescale of single-qubit x rotation is about
(a)Idealqubits ¯h/EJ 1 ps. This means that the rise/fall times of
≈
the pulsesshouldbe lessthan1psinorderfortwo-qubit
gates to be implemented correctly. It should be noted
bility 1.0 tbheatwtetehnisttwypoeqoufbeitrsr.orTsihsucsatuhsiesdpbryobthleemcsoucapnlinbgescsohlevmede
a
b by improving the design of devices, i.e., by introducing
Pro0.98 new ways of two-qubit couplings.
noidling
0.5e idling
0.96
e idling
0.94
0 0.2 0.4 0.6 0.8 1.0
V. SUMMARY
e t
op
=
(b)Superconductingqubits We have studied the dynamics of qubits on the ac-
tionofCNOTgatesforidealandsuperconductingcharge
y qubits. We have explicitly shown how the CNOT gate
bilit 1.0 couldbeimplementedforsuperconductingchargequbits.
a
b Itisfoundthattheerrorinimplementingtwo-qubitgates
o
r with superconducting charge qubits grows quadratically
P
0.9 noidling in finite rise/fall times. Thus it is necessary to keep the
0.5e idling rise/fall times small or to find new ways of coupling two
e idling qubits other than the coupling scheme via the common
3e idling inductor.
0.8
0 0.2 0.4 0.6 0.8 1.0
e t
op
=
FIG. 4: Probability of success on the action of the CNOT
Acknowledgments
gate as a function of the ratio of the rise/fall time to the
operation time, ǫ/τop: (a) ideal qubits (b) superconducting
chargequbits. Herethetimescaleofoperationisτop =¯h/Bz
TheauthorwouldliketothanksY.Makhlin,A.Shnir-
for the ideal case and τop = ¯h/ECi for the superconducting
case. Idle times prevent the tails of pulses from overlapping. man, and G. Scho¨n for helpful discussions. This work
The solid line is the fitting function 1 0.027(ǫ/τop)2 which was partially supported by Korea Science and Engineer-
−
shows thequadraticgrowth of error in rise/fall times. ing Foundation.
∗ Electronic address: [email protected] on the Foundation of Computer Science, ed. by S. Gold-
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