Ising interaction between capacitively-coupled superconducting flux qubits

0.61 MB·English

by  Takahiko Satoh

Ising interaction between capacitively-coupled superconducting flux qubits
Takahiko Satoh,1,2, Yuichiro Matsuzaki,1 Kosuke Kakuyanagi,1
∗
Koichi Semba,3 Hiroshi Yamaguchi,1 and Shiro Saito1
1NTT Basic Research Laboratories, 3-1, Morinosato Wakamiya Atsugi-city, Kanagawa 243-0198 Japan
2Department of Computer Science, Graduate School of Information Science and Technology,
The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, Japan
3Advanced ICT Research Institute, National Institute of Information and Communications Technology,
4-2-1, Nukuikitamachi, Koganei-city, Tokyo 184-8795 Japan
(Dated: February 2, 2015)
Here, we propose a scheme to generate a controllable Ising interaction between superconducting
5 flux qubits. Existing schemes rely on inducting couplings to realize Ising interactions between flux
1 qubits,andtheinteractionstrengthiscontrolledbyanappliedmagneticfieldOntheotherhand,we
0 havefound a way to generate an interaction between thefluxqubitsvia capacitive couplings. This
2 has an advantage in individual addressability, because we can control the interaction strength by
changingan applied voltage that can beeasily localized. This is acrucial step toward therealizing
n
superconductingflux qubitquantumcomputation.
a
J

0
I. INTRODUCTION while, the bestobservedcoherencetime is anorderof10
3
µs [25, 26]. Furthermore, the tunable coupling schemes

] To realize fault-tolerant quantum computation, it is for two qubit gate operations are proposed and demon-
h crucialtoinvestigateaschemetogenerateaclusterstate strated [27–35]. Quantum non-demolition measurement
p
inascalableway. Theclusterstateisauniversalresource offlux qubit during the coherence time is realizedby us-
-
t for quantum computation, and this state can be used ing Josephson bifurcation amplifier [36–39].
n
a for a fault-tolerant scheme such as a surface code and There are two typical tunable qubit-qubit coupling
u topological code. One can generate a cluster state if we schemes, inductive coupling and capacitive coupling. In
q canturnon/offanIsingtypeinteractionbetweenqubits. fluxqubitsystem,existingschemesrelyoninductivecou-
[ Superconducting circuit is one of the promising sys- plingwiththeexternalmagneticfield. Severalschemesof

1 tems to realize such a cluster-state quantum computa- the tunable qubit-qubit inductive coupling are proposed
v tion. Josephsonjunctions in the superconducting circuit and demonstrated [27–35]. However, it is hard to apply
9 can induce a non-linearity, and so one can construct a magnetic field to a localized region. Due to this prop-
3 two-level system. There are several types of Josephson erty, it is difficult to achieve individual addressability of
7 junction qubit: charge qubit [1], superconducting spin allqubits,becausemagneticfieldmayaffectnotonlythe
7 qubit [2], superconducting flux qubit [3–7], supercon- target qubits but also other qubits as well. Therefore,
0
ducting phase qubit [8–10], superconducting transmon it is important to perform two-qubit gates without af-
.
1 qubit [11, 12], fluxonium qubit [13, 14], and several hy- fecting other qubits by using localized fields for scalable
0 brid systems [15, 16]. quantum computation.
5
Thetransmonqubit[11,12,17],whichisacooper-pair Here, we propose a way to generate and control the
1
box and relatively insensitive to low-frequency charge Ising type interaction between four-junction flux qubits
:
v noise, is considered one of the powerful method of the using capacitive coupling. By using an applied voltage,
Xi qubit implementation by using superconducting circuit. we control the interaction between flux qubits that are
Scheme of the tunable qubit-qubit capacitive coupling is connected by capacitance. Unlike the standard schemes,
r
a proposed and demonstrated [18–20]. The high fidelity our scheme does not require to change the applied mag-
qubit readout using a microwave amplifier is demon- neticfieldonthefluxqubitforthecontroloftheinterac-
strated [21–23]. Furthermore, high fidelity (99.4%) two- tion. This may have advantage to implement two-qubit
qubit gate using five qubits system is achieved. This gates on the target qubits without affecting other qubits
result is the first step toward surface code scheme [24]. because applying local voltages is typically much easier
These results show a good scalability towards the real- than applying local magnetic flux. We take into account
ization of generating a large scale cluster state. ofrealisticnoiseonthistype offluxqubits,andestimate
The flux qubit consist of a superconducting loop con- a qubit-parameter range where one can perform fault-
taining several Josephson junctions. This system has a tolerantquantum computation. Furthermore,we show a
large anharmonicity and can be well approximated to a waytogenerateatwodimensionalclusterstateinascal-
two-levelsystem. Singlequbitgateoperationscanbe re- ableway. Ourresultpavesthewaytoachievethescalable
alizedwith highspeed andreasonablefidelity [7]. Mean- quantumcomputationwith superconductingflux qubits.
The rest of this paper is organized as follows: In Sec-
tion,wepresentsthedesigndetailsofourfluxqubitand
effectsonafluxqubitfromthechangeoftheparameters.
∗ [email protected] InSectionIII,weproposeourschemeforgeneratingIsing
2
type interaction between capacitively coupled supercon- due to fluxoid quantization around the loop containing
ducting flux qubits. Moreover, we show the relationship phases of Josephson junctions. f denotes the external
betweencouplingstrengthandtwotypesoferrorscaused magnetic flux through the loop of the qubit in units of
by operationaccuracy,the fluctuation of applied voltage themagneticfluxquantumΦ = h. ThetotalJosephson
0 2e
and timing jitter. In Section IV, we present the analysis energy U can be described as follows:
of our scheme for use in multi-qubit system. Addition-
4
ally,wediscusshowtosuppressthenon-nearestneighbor
U = E (1 cosϕ ). (2)
interactions by changing parameters and performing π j(k) − k
pulses. Furthermore,weshowourprocedureforgenerat- kX=1
ing a one and two-dimensionalcluster state using qubits The total electric energy T can be described as follows:
on square lattice in less time.
4 2
1 Φ 1
T = C 0ϕ˙ + C (V V )2 (3)
2 j(k) 2π k 2 g e− i
k=1 (cid:18) (cid:19)
II. VOLTAGE CONTROLLED α-TUNABLE X
FLUX QUBIT where C , V and V denotes the capacitance of the gate
g e i
capacitor, applied external voltage and the electric po-
Letusfirstshowthecircuitofafluxqubitthatwepro- tential of node 1, respectively. Here, node 1 represents
poseinFig.1(a). Here,X-shapedcrossesdenote Joseph- the superconducting island.
Althoughthe systemHamiltonianH hasmanyenergy
levels, the system can be described as a two-level sys-
2525 tem (qubit) due to a strong anharmonicity by choosing
suitable α. We show the α dependence of the energy of
2020 this system Fig. 1(b), where E01 (E12) denotes the en-
ergy splitting between the ground(first excited) and the
Hz)1515 first excited (second excited) state. This clearly shows
G
Energy (1010 E12 tohnalytstyhsetegmrohuansdanstaantehaarnmdonfiircsittyesxocitthedatswtaetceanbycounstirnogl
frequency selectivity.
55 g and e arethegroundandthefirstexcitedstateof
E01 | i | i
the system Hamiltonian H =T +U for f =0.5. In this
00 regime, the ground state and the first excited state of
00..22 00..44 00..66 00..88 11..00
a thissystemcontainsasuperpositionofclockwiseandan-
(a) (b) ticlockwisepersistent currents. Here, L = 1 (g + e )
| i √2 | i | i
corresponds anticlockwise persistent current and R =
FIG.1. (a)Thecircuitofafluxqubitinourdesign. Thisflux 1 (e g ) corresponds clockwise one. | i
qubit has four Josephson junctions (JJ). E and C de- √2 | i−| i
j(n) j(n) While f is around 0.5, due to the anharmonicity, we
notetheJosephsonenergyandcapacitanceofnthJosephson
canconsideronlythegroundstateandfirstexcitedstate
junction JJn. The loop is threaded by an external magnetic
intheHamiltonianH,andsowecansimplify theH into
flux f, and we can control the energy bias of the qubit via
H spanned by g and e as follows:
the magnetic flux. Node 1 represents the superconducting ge
| i | i
island. The electric potential of node 1 is Vi. (b) The α de- 1
pendenceofE01 andE12 whereE01 denotesenergydifference Hge = (∆σZ +εσY) (4)
between the ground state and the first excited state, E de- 2
12
notesenergydifferencebetweenthefirstexcitedstateandthe
where σ = e e g g and σ = ie g +ig e
second excited state. Here, we set Ej(1) =Ej(4) =200 GHz, are PaulZi mat|riiches|,−∆|diehn|otes theYtunn−eli|nighe|nerg|yibhe-|
E =E =40 GHz, and E /E =80(k =1,2,..,4).
j(2) j(3) j(k) c(k)
tween L and R , ε denotes the energy bias between
| i | i
L and R . The energy of the qubit is described as
| i | i
E =√ε2+∆2.
son junctions (JJ). The first Josephson junctions (JJ1) 01
and the fourth Josephson junctions (JJ4) both have the Inthispaper,unlessindicatedotherwise,wefixparam-
same Josephson energies Ej and capacitances Cj. The eters as α = 0.2 and Ej(1) = 200 GHz and Ej(k)/Ec(k)
second Josephson junctions (JJ2) and the third Joseph- ratio is 80. Here, Ec(k) = e2/2Cj(k) is charge energy
son junction (JJ3) both have the same Josephson ener- of each Josephson junction. In this parameter regime,
gies and capacitances that are α times larger than those E01 is about three times larger than E12 as shown in
of JJ1 and JJ4. Josephson phases ϕ , which is given by Fig. 1(b) so that we could consider this system as an ef-
n
the gauge-invariantphaseofeachJJn,aresubjecttothe fective two-level system. When f is set to be near 0.5,
following equation: the derivative of the qubit energy against the magnetic
flux dE01 takes the minimum value, so that the qubit
| df |
ϕ +ϕ +ϕ +ϕ = 2πf (1) shouldbe welldecoupledfromflux noise,andwe achieve
1 2 3 4
−
3
the maximum coherenttimes. We callthis regime “opti-
Ve=0 V
malpoint”. Onthe other hand, wecancontrolthe value 121212 2525252525 Cg=0.077 fF
∆mct{ohu|fLuarεacrtigbhe,awn|yRliteancirshs}cgttaaaenbmntrgeaatsirshngeeaagna.rndeteHthtitoeechhurevefietaetewtlluiuhdgeneeenwsnoqehifvltuoeifhnbwc.gitntoWtoerhsnsthebeaoeirtdafngesetytphwvhe∆eoeintlH,tedhanteaghSenmereQcgiealpUytpeooIbprfnDsiliaiεi[aess4ntda]εennsiiinndost Qubit energy (GHz)468104681046810 aC=g=0.02.077 fFVVVeee===000 ..m4813V mmVV Qubit energy (GHz)101520101520101520101520101520 aaa=== 000...123
222 55555 a= 0.4
Fig.2. Itisworthmentioningthatwecancontroltheen-
000 00000 a= 0.5
000...444777 000...444888 000...444999 000...555000 000...555111 000...555222 000...555333 00000.....4444477777 00000.....4444488888 00000.....4444499999 00000.....5555500000 00000.....5555511111 00000.....5555522222 00000.....5555533333
External magnetic field External magnetic field
00
11 (a) (b)
D
FIG. 3. (a)The relationship between the external magnetic
flux f and the energy of the qubit E with different voltage
01
55 levels. Here, we set the gate capacitance Cg = 0.16 fF. (b)
The relationship between f and E with different α. Here,
01
we set thegate capacitance Cg=0.16 fF.
z)
H
G
y (00
g e
er
n
E
55 Here,weshowthe circuitforourschemeusing twosu-
−−
a= 0.2 perconductingfluxqubitsinFig.4. Thestructureofeach
Cg=0.077 fF
Ve=0 V
00
11
−−
00..4466 00..4488 00..5500 00..5522 00..5544
External magnetic field
FIG.2. Thetunnelingenergy∆andtheenergybiasεagainst
themagneticfluxf. εdecreasesmonotonicallyasweincrease
f, while ∆ is almost independentof f.
ergy of the qubit by tuning the applied voltage V while
e
operatingattheoptimalpoint. Weshowtherelationship
between ∆ and f with several values of V in Fig. 3.
e
III. ISING TYPE INTERACTION USING
CAPACITIVE COUPLING
FIG. 4. Two flux qubits 1, 2 are coupled via capacitance
Cc(1,2). Each flux qubit is threaded by an external magnetic
A. Generating interaction between two-qubit flux f(l), and we can control the energy bias of the qubit via
system
the magnetic flux. Node 1 and node 2 represent the super-
conductingislands. JJ2andJJ3 ateach qubithavethesame
Inthis section, weshowhowto generateIsingtype in- Josephson energies and capacitances that are α times larger
teraction using charge coupling for superconducting flux thanthoseofallremainingJosephson junctions. Theelectric
qubit. As a novel feature of our scheme, we use only potential of theisland include node1 (2) is Vi(1) (Vi(2)).
externalvoltagesto switchonandoffthe interactionbe-
tweentwofluxqubits. Unlikepreviousschemes,external
magnetic field is not required to control the interaction
qubit is the same as that shown in Fig. 1(a). When we
in our scheme. Since the voltage can be applied locally
compared with the case of applying magnetic field, we apply an external voltage Ve(l) on each qubit, the qubit
may have advantage in this scheme for scalability due to interact with each other across the capacitor C(1,2). We
c
better individual addressability when we try to control describe the details of this circuit in the following sub-
individual qubits. sections.
4
B. Hamiltonian 9.09.29.09.29.09.2 CCgg==00..0057 4f FfF aC=g=0.02.077 fF 0.80.80.8 aC=g=0.02.077 fF Cg=0.085 fF
ergWyeTofn=othwe12ccoi4nrcsuidi2teriCntj(hF(le)ki)ge.le4Φ2ctπa0rsϕi˙cf(kole)lnloe2wrgsy: and potential en- Qubit energy (GHz)8.28.48.68.88.28.48.68.88.28.48.68.8 Cg=0.085 fF Interaction strength (GHz)0.20.40.60.20.40.60.20.40.6 CCgg==00..00574 f FfF
kX=1Xl=1 (cid:18) (cid:19) 8.08.08.0
+1 2 C(l) V(l) V(l) 2 7.87.87.8 000...000 000...111 000...222 000...333 000...444 000...555 0.00.00.0 000...000 000...111 000...222 000...333 000...444 000...555
2 g e − i Voltage (mV) Voltage (mV)
Xl=1 (cid:16) (cid:17) (a) (b)
1 2
+ C(1,2) V(1) V(2) (5)
2 c i − i FIG. 5. The voltage dependence of the qubit energy ∆ and
4 2 (cid:16) (cid:17) the interaction strength g between two qubits of the circuit
U =Xk=1Xl=1Ej((l)k)(cid:16)1−cosϕ(kl)(cid:17) (6) icnouFpilgi.ng4.caHpearcei,tabnoctehCofct=he0.g0a7t7efcFa.pacitance Cgl =0.077 fF,
2
H =T +U = H(l)+H (7)
total A B
Xl=1 C. Effects on interaction from change in electric
1 4 Φ 2 1 2 field
H(l) = C(l) 0ϕ˙(l) + C(l) V(l) V(l)
A 2 j(k) 2π k 2 g e − i
kX=1 (cid:18) (cid:19) (cid:16) (cid:17) To evaluate the performance of our scheme, we focus
4 ontwotypes oferrors. Firstly, we analyzethe dephasing
+ E(l) 1 cosϕ(l) (8)
j(k) − k errors due to the fluctuations of applied voltage. We
Xk=1 (cid:16) (cid:17) define this type of error ǫd and dephasing time T2 as
H = 1C(1,2) V(1) V(2) 2 (9) follows:
B 2 c i − i
t π 1
(cid:16) (cid:17) ǫ = cp,t = ,T = (11)
d T2 cp 4g 2 dE01 δv
whereC(l), f(l), V(l), andV(l) denotesgate capacitance, | dv |
g e i
externalmagnetic flux,appliedexternalvoltage,andthe whereweassumetcp <<T2. Here,tcp denotestheneces-
electric potential of the island including node l for the l sary time to perform a controlled-phase gate with Ising
th qubit respectively. Here, node l represents the super- type interaction, v denotes the external voltage of each
conducting islands. qubit, and δv denotes the fluctuation width of v. It is
worth mentioning that ǫ has a linear relationship with
For an arbitrary f, we can derive the effective four- d
level Hamiltonian Hˆ of the eigenspace spanned by g δv. To make ǫd smaller, We should obtain a parame-
ge l
| i ter set where the absolute value of the gradient of the
and e from H . Here, g and e correspond to
l total l l
the g|roiund state and first ex|citied stat|eiof the lth qubit qubit energy E01 is small and the interaction strength g
without interactions for f(1) = f(2) == 0.5. We expand is large.
H by g and e . The effective Hamiltonian Hˆ Secondly, we investigate the jitter error of a two-qubit
total l l ge
| i | i gateoperation. TheIsingtypeinteractioncanimplement
becomes as follows:
the controlled-phase gate
Hˆge = |v1v2ihv1v2|Hˆtotal|v1v2ihv1v2| U(1,2)(t)=exp i4gt1+σZ(1) 1+σZ(2) , (12)
v1∈(g1,e1X),v2∈(g2,e2) CZ − 2 2 !
2
1
= ∆(l)σ(l)+ε(l)σ(l) +gσ(1)σ(2) (10) where g denotes the interaction strength in Eq. (10),
Xl=1 2(cid:16) Z Y (cid:17) Z Z t = 4πg denotes the time to apply voltages, and UC(1Z,2)
denotes a controlled-phase gate between qubit 1 and 2.
where g denotes the Ising type interaction strength be- By performing the controlled-phase gate on two qubits
tween qubit 1 and 2. We show the change of the qubit which are initialized to ++ state, we can obtain the
12
| i
energyE andtheinteractionstrengthgasafunctionof two-qubit cluster state. But, the applied voltages may
01
appliedvoltagesinFig.5. Largeinteractionstrengthand not create the desired state due to error in the timing
small derivative of qubit energy against voltage can be t = t+δt, where δt is timing jitter. We introduce the
′
achieved by the large coupling capacitance Cc between controlled-phasegateU(1,2)(t)includingthetimingerror
CZ
each qubits. This seems to show that one can suppress to calculate a gate fidelity F = φφ 2 with
CZ ′
errorsbyincreasingCc. Wediscussabouttheerrorsdur- |h | i|
ing controlled-phase gate operation in following section. φ =U(1,2)(t)++ , φ =U(1,2)(t)++ . (13)
| i CZ | i | ′i CZ ′ | i
5
Here,wedefine the timing errorǫ =1 F ,andthe schemeto generatea twodimensionalcluster state using
tim CZ
−
local error ǫ (= ǫ +ǫ ). We show the ǫ against superconducting flux qubits arranged on square lattice.
loc d tim loc
the applied voltage V with the particular values of Cc
e
in Fig. 6. The threshold of local errors for fault-tolerant
A. Generating interaction between multi-qubits
system
0.80.80.80.8 a= 0.2 Here, we discuss the interactions between capacitively
coupled N flux qubits that are arranged in one dimen-
Cg=0.077 fF
sional line as shown in Fig. 7. For simplicity, we assume
d v=0.21 m V homogeneousflux qubits. f(j) denotesthe externalmag-
6666 d t=50 psec
0.0.0.0.
%) Cg=0.077 fF
(m
ti Cg=0.062 fF
+ed0.40.40.40.4
e
Cg=0.046 fF
2222
0.0.0.0.
FIG. 7. A flux qubit at the site j(1 < j < N) couples with
thenearestneighborqubitsviacapacitanceCc(j,j±1). Forsim-
plicity, we assume homogeneous flux qubits. Each node j
0000
0.0.0.0. represents the superconducting islands. Each qubit has four
0000....0000 0000....2222 0000....4444 0000....6666 0000....8888 Josephson junctions. Two Josephson junctions directly con-
Voltage (mV) nected to the node have the Josephson energies and capaci-
tances that are α times larger than the other two Josephson
junctions.
FIG.6. Thetotallocalerrorǫloc(=ǫd+ǫtim)asafunctionof
voltage with different coupling capacitance Cc. Here, we set
thefluctuationwidthofvoltageδv=0.21 µVandthetiming netic flux through the loop of the j th qubit. When all
jitter δt=50 psec. Dashed line denotes an error of 0.1%. flux f(j) are 0.5, the system Hamiltonian is described as
follows.
quantum computation is known to be around 1%. Also, N N
itthiesnkencoewssnartyhantu,mifbtehreoefrqrourbritastefoirstchloesceotmopthuetatthiorensdhroalds-, Hˆ = l=1 21∆(l)σZ(l)+l,l′=1g(|l−l′|)σZ(l)σZ(l′) (14)
X X
tically increases[40, 41]. Therefore, we set the threshold
to ǫ = 0.1%. As shown in Fig. 5, we can increase the where ∆(l) denotes the energy of the l th qubit, g(l l′)
loc denotes the interaction strength between each pa|i−r o|f
coupling strength g by increasing Cc. Meanwhile, the
qubits at a site (l,l), and l l denotes the site dis-
strong coupling strength causes the large timing error. ′ | − ′|
tance between these qubits (e.g. when qubit l and l are
Therefore, as shown in Fig. 6, the optimal voltage exists ′
nearest neighbor pair, l l =1.).
for each of the Cc which minimizes the total local error. | − ′|
In addition,by increasingCc, the total errortends to be
smaller. This result shows that the large Cc has an ad-
B. Generation of a one dimensional cluster state
vantageforquantumerrorcorrectionagainstlocalerrors.
However, for multi-qubit systems, increasing Cc causes
Non-nearest neighbor interactions cause spatially-
a different problem. Unwanted interaction strength be-
correlated errors that are difficult to correct by quan-
tween non-nearest neighbor qubits increases due to the
tum error correction. In this subsection, we show the
large Cc. For this reason, the Cc should be set to be
way to evaluate this error. We define the ratio between
around 0.075 fF. The detail of this will be discussed in
nearest neighbor interaction g(= g ) and next-nearest
Section IVB. (1)
neighbor interaction g as R = g(2) where all qubits
(2) g(1)
are applied voltage Ve. We s(cid:16)how th(cid:17)at the interaction
IV. MULTI-QUBIT SYSTEM strengthg(l l )decreasesexponentiallyasthesitedis-
′
| − |
tance l l increases, and the Ratio R depends on the
′
| − |
Inthissection,wegeneralizeourschemetomulti-qubit coupling capacitance Cc between each qubit. This is a
system. Firstly, we discuss how to control the capaci- striking feature in our scheme using voltage for the con-
tiveinteractionsbetweensuperconductingfluxqubitsvia trol of the qubit-interaction, because the effect from any
applied voltage. Secondly, we show how to apply our control lines can decrease only polynomially against the
6
site distance if one uses magnetic field for the control. above condition is smaller than the proper range of Cc
We show the interaction strengths of 6 qubits system as discussedinSubsectionIIIC.Therefore,wedonotapply
a function of Cc in Fig. 8. voltageonall qubits but apply voltageon some of them.
We choose pairs of nearest neighbor qubits that we will
apply the voltage, and we set a site distance p between
20 0
1 g( ) 2 the pairs. Then, if R is small enough, ǫ of each qubit
·1 1 0. is the following equation: non
g( )
2
N/2
π 1
plot)1 g(3) 15 ǫ(njo)n = 4R(n−1) ≤ 10000 (16)
z, log -·110 g(4) 0. where p is the site dnXi=stpance between qubits applied by
H g( ) voltage.
M 5
ngth (4 R 0.10Ratio HaSminilcteontihaenr,eiatreismdaiffinycupltartaomfietnedrsaonnotphteimiunmterascettioonf
stre-10 parameters that minimize both of local and spatially-
n ·1 correlatederrors. Therefore, we fix the following param-
o
nteracti 0.05 ealattemerdsin:eiαmrru=omrs0.ts2oi,teδbved=iustn0ad.n2ecr1e0µp.V01w, δh%til,=ews5eu0psphpsoreewcs.stiTnhgeoCdthecetearcnmodrirnVee-
I7
-10 dependences of the errors with p = 4,5 in Fig. 9. As
·1
0.00 0.200.200.20 aC=g=0.02.077 fF 0.60.60.6 aC=g=0.02.077 fF
000000......000000000000 000000......C000000o555555upling ca000000p......111111a000000citance (f000000F......)111111555555 000000......222222000000 0.150.150.15 dCdvtc===5000..2p0s17e c7m VfF p=4 %)0.40.50.40.50.40.5 dCdvtc===5000..2 0p17s 7em cVfF pp==45
iFtnhIteGer.cao8cu.tpioTlnihnesgtCrreacntgidotehpRebn(cid:16)edt=weneggec((n21e))o(cid:17)afpwtahhiererioneftqegru(a|blcitt−isoanl′t|)satrdseeintnegot(thless−atlnh′)de. e (%)non0.050.100.050.100.050.10 p=5 e+e (locnon0.10.20.30.10.20.30.10.20.3
0.000.000.00 0.00.00.0
If we apply voltage on all qubits, interaction occurs 000...000000 000...000222 Co000u...p000li444ng cap000a...000c666itance000 (...f000F888) 000...111000 000...000 000...222 Volta000g...444e (mV) 000...666 000...888
between such qubits. The total error ǫ(j) caused by (a) (b)
non
non-nearest neighbor interactions on j th qubit during
controlled-phase operation is calculated as follows: FIG. 9. (a)The Cc dependence of the correlated errors.
Dashed line corresponds to an error of 0.01 %. (b)The V
N/2 N/2π dependenceofthetotalerrors. Dashedlinecorrespondstoan
ǫn(jo)n = g(n)tcpm(n) = 4R(n−1)m(n) (15) error of 0.1 %.
n=2 n=2
X X
where n denotes the site distance betweenthe j th qubit shown in Fig. 9(a), when p = 4, the ǫ exceeds 0.01
non
and the coupled non-nearest neighbor qubits, m(j) de- % around Cc =0.04 fF. We cannot sufficiently suppress
(n)
notes the number of such non-nearest qubits. localerrorsusing coupling capacitancesmaller than 0.07
Suchtheexistenceofthespartially-correlatederrorwill as shown in Fig. 6. Thus, the site distance p should be
increase the threshold for quantum error correction [42]. larger than 5. Meanwhile, when p = 5, the ǫnon exceeds
Largecapacitancetendstodecreaselocalerrorsasshown 0.01 % around Cc=0.09 fF. Then the total error of the
inFig.6,whilelargecapacitanceinducesmorespartially- controlled-phaseoperationcanbe sufficientlysuppressed
correlated errors as shown in Fig. 8. However, when we to be less than 0.1 % using the coupling capacitance Cc
consider the spatially-correlated error, the error thresh- around 0.077 fF as shown in Fig. 9(b). Therefore, it is
old value of the surface code is not well studied. Thus, preferablethatthe sitedistance p=5 be selected. Inor-
we set the upper bound of the spatially-correlated error der to adopt sufficiently large coupling capacitance such
on each qubit ǫnon ≤ 100100 which is an order of mag- that the ǫloc below 0.1 %, we need to choose sufficiently
nitude smaller than the threshold of local error for sur- largepsuchthattheǫnonbelow0.01%. Wediscussabout
face coding scheme. If this condition is satisfied, we as- the way which can further reduce p in the following.
sumethatspatially-correlatederrorisenoughtoperform The p determines the maximum number of controlled-
a fault-tolerant quantum computation. When we apply phase gates that are performed simultaneously on the
voltageonallqubitstoperformcontrolled-phasegatesto same system. For example, we can perform ⌊Np+−12⌋+1
allpairsofnearestneighborqubit, arangeofvaluesthat controlled-phase gates in parallel using N-qubits one di-
thecouplingcapacitanceCccantakewhilesatisfyingthe mensional system. If we can use the smaller p without
7
adding extra errors, we can perform more controlled-
phasegatesinparallel. Sothatwecangenerateacluster
state within a shorter operating time. For this purpose,
we introducethe spinechotechnique where implementa-
tion of a π pulse (single qubit σ rotation) to the target
X
qubit could refocus the dynamics of the spin so that ef-
fects of interactions on the target qubit should be can-
celled out. We apply two π pulses to pairs of qubits to
suppress spatially-correlatederrors. For example, we set
three qubits in a raw and apply voltage V(n) to the n
e
th qubits (n = 1,2,3) as shown in Fig. 10, where V(1)
e
and V(2) are equal, V(3) is an arbitrary voltage,and the FIG. 11. The 3-step procedure for generating a one dimen-
e e
sionalclusterstate. Step1. Weinitialize3n−2thand3n−1
strength of interaction between qubit 1 and 2 is g. We
thqubitsin|+i. Here,n=1,2,··,⌊N+1⌋where⌊x⌋isthein-
3
tegerpartofx. Afterthatweapplyvoltageon3n−2thand
3n−1thqubits. Letthestateevolveforatime tcp,perform
2
π pulses to 6n−2 th and 6n−1 th qubits, and let the state
evolveforatime tcp. Aftertheseoperations,controlled-phase
2
gates havebeen performed between qubit3n−2and 3n−1.
Step 2. We initialize 3n th qubits in |+i. After that, simi-
lar to theStep 1, we perform controlled-phase gates between
qubit 3n−1 and 3n. Step 3. We initialize 3n+1 th qubits
in |+i. After that, similar to the Step 1 and 2, we perform
controlled-phase gates between qubit 3n and 3n+1.
Step 2: We apply voltage to (3n 1) th and 3n th
−
qubit for performing controlled-phase gates be-
FIG.10. Whenweperform aπ pulseonqubit1and2att= tween (3n 1) th and 3n th qubit where n =
tcp/2,thenearestneighborinteractionbetweenqubit2and3 1,2, , N+1−.
andthenon-nearestneighborinteractionbetweenqubit1and ·· ⌊ 3 ⌋
3 are cancelled out. In such way, we can perform controlled-
Step 3: We apply voltage to 3n th and (3n+1) th
phase gate without changing thestate of other qubits.
qubit for performing controlled-phase gates be-
tween (3n 1) th and 3n th qubit where n =
set each qubit to be prepared in + state, let the state
| i 1,2, , N+1−.
evolve for a time tcp/2, perform two π pulses to qubit 1 ·· ⌊ 3 ⌋
and2,andletthestateevolveforatimet /2. Thefinal
cp
state become as follows: Ateachstepoftheaboveprocedure,⌊N3−1⌋controlled-
phase gate are performed in parallel. At each step,
1
Uˆ +++ 123 = (+0 12+ 1 12) + 3. (17) we will perform the following procedure to perform the
| i √2 | i |− i ⊗| i
controlled-phasegate. Firstly, prepare the qubit state in
Here, the interactions g(1)σZ(2)σZ(3) and g(2)σZ(1)σZ(3) are |+i. Secondly, let the state evolve for a time t = tc2p ac-
cancelledoutdue totheπ pulsesandweobtainacluster cordingtothe HamiltoniandescribedinEq.14. Thirdly,
state between qubit 1 and 2. perform the π pulses to suppress the non-local interac-
This method canbe appliedwith the case ofarbitrary tion. Finally, let the state evolve for a time t = tcp.
number of qubits. The general rules are follows: let us We show the details of these operations in Fig. 11 and
considera pairofqubits. If weperformπ pulses onboth explain how the non-local interaction is suppressed in
ofqubits,theinteractionbetweenthemisnotaffectedby Fig. 12. When all coupling capacitance are Cc 0.077
≤
these pulses. On the other hand, if we perform π pulse fF, the spatially-correlated error on each qubits become
ononeofthem,theinteractionbetweenthemiscancelled as follows:
out. These properties would be crucial for generating a
N/2
cluster state as we will describe. π π 1
ǫ(j) = R(n 1)m (R4+2R5) .(18)
For generating a large one dimensional cluster state non 4 − (n)≃4 ≤10000
using N qubits of the circuit in Fig. 7, we show the pro- nX=5
cedure as follows:
The k th qubit is affected by mainly three non-local
Step 1: We apply voltage to (3n 2) th and (3n 1) interactions as shown in Fig. 12. The strength of the
th qubit for performing contr−olled-phase gates−be- largest interaction is gR4, and the strength of the other
tween (3n 2) th and (3n 1) th qubit where twointeractionsaregR5. The remainingnon-localinter-
n=1,2, ,−N+1 . − actions are negligibly small.
·· ⌊ 3 ⌋
8
between each pair of qubits at site (l,m) and (l ,m),
′ ′
and l l + m m denotes the site distance between
′ ′
| − | | − |
these qubits.
FIG. 12. The influence of non-local interactions. During the
controlled-phase gate, each target qubit are affected by non-
localinteractions. Weshowthestrengthofmainlythreenon-
local interactions with k th qubit. Theseinteractions arenot
cancelled out by π pulses.
C. Generation of a two dimensional cluster state
Next,weshowhowtogenerateatwodimensionalclus-
terstate usingN2 flux qubits arrangedonN N square
lattice. We show a part of the circuit in Fig×. 13. f(j,k)
FIG.14. Schematicofourprocedureforgeneratingatwodi-
mensionalclusterstatebygraphstaterepresentation. Circles
correspond to qubits, dashed lines correspond to electrically
connection via a capacitance, solid-lines correspond to en-
tanglement between qubits, and numbers show the order in
whichcontrolled-phasegatesareperformedbyourprocedure.
Whitecircles denote separable qubit,and gray circles denote
qubitsconstituent of cluster state(s).
Here,weshowthe12-stepprocedureasfollowsforgen-
erating a two dimensional cluster state.
Step 1-3: Weperform(N 1) N controlled-phasegate
− ⌊4⌋
to generate N one dimensional cluster states us-
⌊4⌋
FIG. 13. Physical circuit for generating a two dimensional ingqubitslocatedinthe4m 3(m=1,2, , N+3 )
− ·· ⌊ 4 ⌋
cluster state. These four qubits correspond to the qubits rowinthesamewayasshowninFig.11. Thenthe
surrounded by dot line in Fig. 14. Two Josephson junc- spatially-correlatederrorofeachqubitinthe4m 3
tions directly connected to a node (the superconducting is- row is smaller than 1 . We show the outline−of
10000
lands)havetheJosephson energiesandcapacitancesthatare these steps in Fig. 14(a).
αtimeslargerthantheothertwoJosephsonjunctions. Every
flquubxitqsuvbiiatactapsaitceit(ajn,kc)ecCouc(p(lje,ks)(wj±it1h,kt±h1e))f.ournearest neighbor Stepg4a-t6e: tWoegpenerefroartme (NN −2 1)o⌊nNe4−d2i⌋mceonnstioronlaleldc-lpuhsateser
⌊ 4− ⌋
states using qubits located in the 4p 1(p =
denotes the external magnetic flux through the loop of 1,2, , N+1 ) row in the same way as ab−ove. We
·· ⌊ 4 ⌋
the qubit at site (j,k). Here, (j,k) corresponds to the show the outline of these steps in Fig. 14(b).
lattice point. When all flux f(j,k) are 0.5, the system
Step 7-9: Weperform(N 1) N controlled-phasegate
Hamiltonian is described as follows: − ⌊4⌋
togenerateatwodimensionalgraphstateasshown
Hˆ = ∆(2l,m)σZ(l,m) icnoluFmign. 1a4cr(co)ssusNingoqnuebditismleoncsaitoendalinclutshteer4smta−tes3.
(Xl,m) We show the ou⌊t2li⌋ne of these steps in Fig. 14(c).
+((l,mX),(l′,m′))g(|l−l′|+|m−m′|)σZ(l,m)σZ(l′,m′) (19) Stepg1a0t-e12to: Wgeenepreartfeoramt(wNo−d1im)⌊eNn4−si2o⌋ncaolnctlruosltleerd-spthaateses
where ∆ denotes the energy of the qubit at site usingqubitslocatedinthe4p 1column. Weshow
(l,m)
−
(l,m), g(l l′+m m′) denotes the interaction strength the outline of these steps in Fig. 14(d).
|− | | − |
9
During each step, a part of the non-local interactions
are not cancelled out by π pulses. When all coupling
capacitance are Cc 0.077 fF, the spatially-correlated
≤
error on each qubits become as follows:
N/2
π π 1
ǫ(j,k)= R(n 1)m (R4+4R5) .(20)
non 4 − (n)≃4 ≤10000
n=5
X
The qubit at site (j,k) is affected by mainly five non-
local interactions as shown in Fig. 15. The strength of
the largest interaction is g R4, and the strength of the
(1)
other four interactions are g R5. The remaining non-
(1)
local interactions are negligibly small.
FIG. 15. Operations and the influence of non-local interac-
tions in generating a two dimensional cluster state. In this
step, we apply voltage to qubit at site (3n−2,4m−3) and V. CONCLUSION
(3n−1,4m−3). Letthestateevolveforatime tcp,perform
π pulsestoqubitatsite(6n′−5,8m′−7),(6n′−24,8m′−7), In conclusion, we suggest a new way to generate Ising
(6n′−2,8m′−3),and(6n′−1,8m′−3),andletthestateevolve interaction between capacitively-coupled superconduct-
for a time tcp. So that controlled-phase gates can be imple- ing flux qubits by using an applied voltage, and we also
2
mentedbetweenthepairofqubitsatsite(3n−2,4m−3)and show architecture about how to make a two-dimensional
(3n−1,4m−3). Here,m=1,2,··,⌊N+3⌋,m′=1,2,··,⌊N+3⌋,
4 8 cluster state in this coupling scheme. Unlike the stan-
n=1,2,··,⌊N+1⌋,and n′ =1,2,··,⌊N+1⌋. Each target qubit
3 6 dardschemes,ourschemedoesnotrequiretochangethe
is affected by non-local interactions from qubits on the same
applied magnetic field on the flux qubit for the control
row and other rows. We show mainly five non-local interac-
of the interaction. Since applying local voltages is typi-
tions with the qubit at site (j,k). These interactions are not
cally much easier than applying local magnetic flux, the
cancelled out by π pulse.
scheme described in this paper may have advantage to
perform two-qubit gates on target qubits without affect-
ing any other qubits. Our result paves the way for scal-
We show the details of each step of above procedure able quantum computation with superconducting flux
for generatinga two dimensionalcluster state in Fig.15. qubits.
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