Universal Quantum Degeneracy Point for Superconducting Qubits

0.44 MB·English

by  Xiu-Hao Deng

Universal Quantum Degeneracy Point for Superconducting Qubits
X.-H. Deng,1 Y. Hu,1 and L. Tian1,∗
15200 North Lake Road, University of California, Merced, CA 95343, USA
The quantum degeneracy point approach [D. Vion et al., Science 296, 886 (2002)] effectively
protectssuperconductingqubitsfromlow-frequencynoisethatcoupleswiththequbitsastransverse
noise. However, low-frequency noise in superconducting qubits can originate from various mecha-
nismsandcancouplewiththequbitseitherastransverseoraslongitudinalnoise. Here,wepresent
aquantumcircuitcontainingauniversalquantumdegeneracypointthatprotectsanencodedqubit
from arbitrary low-frequency noise. We further show that universal quantum logic gates can be
performed on the encoded qubit with high gate fidelity. The proposed scheme is robust against
small parameter spreads dueto fabrication errors in the superconductingqubits.
1
1 PACSnumbers: 85.25.-j,03.67.Lx,03.67.Pp,03.65.Yz
0
2
n I. INTRODUCTION pearsas a fluctuation in the Josephsonenergy,the qubit
a energy obtains an energy shift proportionalto the noise,
J which generates qubit dephasing.
Decoherence due to the low-frequency noise is com-
5 Inthiswork,weproposeauniversalquantumdegener-
monly considered as the major hurdle for implement-
1 acypoint(UQDP)schemewheretheencodedqubits can
] iinngg qfauublitt-st1o–l3e.raTnthequloawnt-furmequceonmcpyuntionisge,inofstuepnewrciothnd1u/cft-- beprotectedfromgenericlow-frequencynoise. Thephys-
h type of spectrum4,5, is ubiquitous in Josephson junc- ical qubits in this scheme are subject to transverse and
p (or) longitudinal low-frequency noises. We construct en-
tion devices6–9. In the past, extensive efforts have been
- codedqubit in a subspace where the low-frequencynoise
t devoted to study the microscopic origin of the low-
n only generates off-diagonal elements and can be effec-
frequency noise10–16. Most recently, theoretical and ex-
a tivelytreatedastransversenoise. Wewillshowthatuni-
u perimental researches suggested that one source of the
versal quantum logic gates can be implemented in this
q low-frequencynoiseisthespurioustwo-levelsystemfluc-
architecture and are protected from the low-frequency
[ tuators in the substrate or in the oxide layers of the
noise as well. To test the analytical results, we numeri-
Josephson junctions17–21.
1 callysimulate the quantumlogicgatesin the presenceof
v Toprotectthecoherenceofthesuperconductingqubits the low-frequency noise. The gate operations, protected
2 from the low-frequency noise, various approaches have bythe encoding,demonstratehighfidelity inthe simula-
4
been developed during the past few years, including tion. Moreover,wewillshowthattheproposedschemeis
9
2 the dynamic control technique, the quantum degeneracy robustagainsmallfabricationerrorsintheparametersof
. point approach, the calibration of the qubit parameters the Josephson junctions. The paper is organized as fol-
1 by continuous measurement, and the designing of novel lows. In Sec. II, we first present the UQDP scheme and
0 quantumcircuitsandmaterials18,22–34. Amongtheseap- the formationof the encoded qubits. Then, the decoher-
1
1 proaches, the quantum degeneracy point approach27–29 ence of the encoded qubits under generic low-frequency
: has been demonstrated to protect the qubit effectively noise is calculated analytically. In Sec. III, we study the
v fromthe low-frequencynoisethatcoupleswiththe qubit realization of the quantum logic gates on the encoded
i
X through the off-diagonal matrix elements, i.e. the trans- qubits. Numerical simulationof the quantumlogic gates
r versenoise. Thequbitdecoherencetimewasincreasedby is also presented in this section. The discussions on the
a orders of magnitude by operating the qubit at its quan- effects of the parameter spreads, different choices of the
tum degeneracy point, also called the optimal point or coupling between the physical qubits, and comparison
the “sweet spot”, where the first order derivative of the withtheDecoherenceFreeSubspace(DFS)approachare
qubit energy to the noise fluctuation is zero. This ap- presented in Sec. IV together with the conclusions.
proachhasalreadybeenappliedtoboththechargequbit
and the flux qubit35,36.
Meanwhile, due to the diverse origins of the low- II. UNIVERSAL QUANTUM DEGENERACY
frequencynoiseinsolid-statesystems,thequbitcancou- POINT (UQDP)
ple with either transverse or longitudinal low-frequency
noise,wherethelongitudinalnoisecoupleswiththequbit The basic idea of the (simple) quantum degeneracy
inthediagonalelementsandshiftsthequbitenergy. The point approach is to use the finite energy separation be-
simplequantumdegeneracypointapproachcanonlypro- tweenthe two eigenstates of a qubit to protect the qubit
tect the qubit from transverse low-frequency noise. For from transverse low-frequency noise. Consider the qubit
longitudinal low-frequency noise, this approach can’t re- coupling with the transverse noise as
duce the decoherence of the qubit. For instance, in the
quantronium qubit27, when the low-frequency noise ap- H =E σ +δV (t)σ (1)
s z z x x
2
wheretheenergyseparationbetweenthequbitstates and gate capacitance C . We then have E = E /2.
g z J
|↑i
and is 2E and δV (t) is the low-frequency noise The qubits are connected by a superconducting quan-
z x
| ↓i
with δV (t) E . The noise couples with the qubit tum interference device (SQUID) with capacitance C
x z m
viath|e σ op|e≪ratorwhichprovidesatransversecoupling and tJosephson energy E 1. It can be derived that
x J2
in the off-diagonal elements. Here, we treat δV (t) as
x
a classical noise for simplicity, but our results can be Emx =Cme2/[(Cm+CJ +Cg)2−Cm2] (5)
applied to quantum noises. The low-frequency nature of
thenoisedeterminesthatitcan’tresonantly(effectively) and Emy = Emz = EJ2/4. Similar couplings can be
−
excite the qubit between its two states due to the large derived for other superconducting qubits such as phase
energy separation between the qubit states. Hence, the qubits and flux qubits.
noise can be treated as static fluctuations. The qubit The eigenstates of the Hamiltonian H0 in Eq. (3) can
energy can be written as be derived as
Hs ≈(Ez +δVx2(t)/2Ez)σz (2) |21i==c−ossiθnθ|↓↓+i+sincoθsθ|↑,↑i,
| i |↓↓i |↑↑i (6)
by second order perturbation approach, i. e. the qubit 3 =( + )/√2,
| i −|↓↑i |↑↓i
Hamiltonian adiabatically follows the time dependence 4 =( + )/√2,
| i |↓↑i |↑↓i
of the noise via the second order term δV2(t)/2E . The
x z
qubit dephasing is determined by this secondorderterm with
and is hence significantly suppressed by a factor of
δVx(t)/2Ez 2. ∼ cosθ =2Ez/ 4Ez2+Em2 (7)
| However,|as we mentioned in Sec. I, in real experi- p
and θ [0,π/2]. The corresponding eigenenergies are
ments,thequbit-noisecouplingcanbemorecomplicated ∈
than that in Eq. 1. In this work, we consider a generic E1 = − 4Ez2+Em2 , E2 = 4Ez2+Em2 , E3 = −Em,
noisemodelwith δVα(t)σα includinganarbitrarycou- and E4 p= Em respectively. pWe can rewrite the Pauli
plingwiththequbPitsinallPaulioperators. Wewillshow operators of the physical qubits in the eigenbasis in the
that an encoded subspace can be constructed in which order from 1 to 4 . For example, in the eigenbasis, we
| i | i
the generic low-frequency noise can be converted to a have
transverse noise for the encoded qubit.
cosθ sinθ 0 0
− −
sinθ cosθ 0 0
σz1 = − 0 0 0 1, (8)
A. The encoded qubit  0 0 1 −0 
 − 
The encoded qubit can be constructed from two iden- and
ticalsuperconductingqubitsconnectedbyacouplingcir-
cuit. The general form of the total Hamiltonian for the 0 0 cosφ sinφ
coupled qubits can be written as σy2 =i c0osφ s0inφ sin0φ −c0osφ . (9)
H0 =Ez(σz1+σz2)+ Emασα1σα2 (3) −−sinφ −cosφ 0 0 
Xα
with the angle φ=θ/2+π/4.
where σαj are the Pauli operators of the j-th qubit and An interesting observation is that the diagonal ele-
α = x,y,z. It can be shown that couplings in this gen- ments of all the Pauli matrices in the subspace of the
eralform can generatethe encoded qubit under the con- states 3 , 4 are zero, i.e.
{| i | i}
dition: E = 0 and (or) E = 0. In the following,
mx my
6 6
we set E = E = 0 with a finite E for simplicity 3 σ 3 = 4 σ 4 =0 (10)
my mz mx αj αj
h | | i h | | i
of discussion. The low-frequency noise coupled with the
qubits have the general form for α = x,y,z and j = 1,2. The only non-zero matrix
elements in this subspace are the off-diagonal elements
Vn = δVαj(t)σαj (4) h3|σzj|4i and their conjugate elements. We know that
Xαj the qubits couple with the low-frequency noise through
thePaulimatricesasisgiveninEq.(4). Hence,the noise
where δV (t) accounts for the noise coupling with σ onlycoupleswiththissubspacethroughoff-diagonalcou-
αj αj
of the j-th qubit. The total Hamiltonian including the plingelements. Thiscoupling,however,willnotgenerate
system and the noise can be written as H =H +V . effective excitation between the states 3 and 4 due to
en 0 n
| i | i
The coupling Hamiltonian in Eq.(3) can be realizedin the low-frequency nature of the noise and the finite en-
many circuits. For example, consider two charge qubits ergy difference 2E between these two states. We thus
m
biased in their quantum degeneracy points with effec- select 3 , 4 as the subspace for the encoded qubit
{| i | i}
tive Josephson energy E , Josephson capacitance C , and name the parameter point where the Hamiltonian
J J
3
has the form of Eq. (3) as the “universal quantum de-
! #!%&’()
generacy point” (UQDP). Note that because of the gen- !"#!$%&’*)
" $
erality of the form of the Hamiltonian, the UQDP may ! #!%+’&)
" $
! #!%&)
not be just one point, but can be a curve in the param- " $
eter space. At the UQDP, the subspace 3 , 4 cou-
{| i | i}
ples with all the noise δV (t) transversely and suffers
αj
only quadratic dephasing. In addition, due to the same
reasons that protect the encoded qubit from dephasing,
the leakagefrom the encoded subspace to the redundant
space of 1 , 2 due to the perturbation of the noisy is
{| i | i}
alsoprohibitedin the lowestorder. The matrix elements
of σ and σ (hence the noise components δV (t) and
xj yj xj
δV (t) induce only virtual excitations between 3 , 4
yj
{| i | i}
and 1 , 2 . Note we only consider decoherence due to
{| i | i}
the low-frequency noise which is the dominant source of FIG. 1: Dephasing time of the bare qubit (star) and the en-
decoherence for superconducting qubits. High-frequency codedqubitatdifferentratioEm/Ez asislabelledintheplot
noises such as Johnson noise in the electric circuits can (othercurves).
generate transitions between the encoded subspace and
the redundant space.
Tocalculatethedephasingoftheencodedqubit,weuse
the analytical results on dephasing due to Gaussian 1/f
B. Dephasing of the encoded qubit noise29. In the calculation, we assume the noises δVαj(t)
arenotcorrelatedwitheachother37. The parameterswe
choose are E /2π~ = 5GHz, A = 2 10−4E with an
The dephasing of the encoded qubit can be calculated z × z
infrared cutoff for the 1/f noise ω /2π =1Hz. The de-
with perturbation theory. Without loss of generality, we ir
phasing times are plotted in Fig. 1 at different coupling
assume that the noise contains only the x component
ratio E /E . For the bare qubit, the decoherence time
δV (t) and the z component δV (t), both of which are m z
xj zj
decreases rapidly by a few orders of magnitude as the
Gaussian 1/f noises with the spectra
noise distribution changes from transverse noise to lon-
∞ eiωtdt A2cos2η gitudinalnoise. Forthe encodedqubitatEm/Ez =1,in
Sxj(ω)= δVxj(t)δVxj(0) = contrast,thedecoherencetime onlyvariessmoothlyover
Z h i 2π ω
−∞ the whole range of η.
(11)
∞ eiωtdt A2sin2η
Szj(ω)= δVzj(t)δVzj(0) = (12) III. PROTECTED QUANTUM LOGIC GATES
Z h i 2π ω
−∞ FOR ENCODED QUBITS
wherethe angleη isaparameterthatdescribesthenoise
powerdistributionandA2isthetotalnoisepower. When In the previous section, we showed that the encoded
qubit is immune to arbitrary low-frequency noise to the
η =0, the low-frequency noise is a transverse noise with
first order of the coupling and the dephasing is domi-
only the x (and y) component; When η = π/2, the low-
natedbyquadratictermsderivedfromperturbationthe-
frequency noise is a longitudinal noise with only the z
ory. As a result, the encoded qubit can be a highly-
component.
coherentquantummemoryforstoringquantuminforma-
WeintroducethePaulioperatorsfortheencodedqubit
tion. In this section, we will further show that universal
as X = 3 4 + 4 3, Y = i3 4 + i4 3, and
| ih | | ih | − | ih | | ih | quantum logic gates on the encoded qubits are also pro-
Z = 3 3 4 4. Whenprojectedtothe encodedsub-
| ih |−| ih | tected from the low-frequency noise with high fidelity.
spaceof 3 , 4 ,theeffectiveHamiltonianofthesystem
{| i | i} The gate operations require manipulations on the physi-
coupling with the low-frequency noise can be written as
calqubits. Wewilltestourtheoreticalresultsforthegate
E ( δV (t))2 ( δV (t))2 operations with a numerical simulation. The numerical
m xj zj
Hen =(cid:18)−Em− P2E2 + P 2E (cid:19)Z results give high gate fidelity for the UQDP.
z m
(13)
by applying secondorder perturbation. The noise enters
thisHamiltonianinquadraticform. Thedephasingofthe A. Single-qubit gates
encodedqubitduetoarbitrarylow-frequencynoisehence
only involves quadratic terms, similar to Eq.(2) for the Quantum logic gates on single encoded qubit can be
simplequantumdegeneracypoint. Eventhelongitudinal performed by manipulating the operators of individual
noiseδV onlycontributestothedephasinginquadratic physical qubits σ or by manipulating the interaction
zj αi
terms. terms between two physical qubits σ σ . In the en-
αi αj
4
coded space, the effective Hamiltonian of the encoded !"./+0 !"./+1
qspuabciet,itsheHo0(ep)er=ato−rEs σmzZ1.caWn bheenexpprroejsescetdedasto this sub- &’()*)+,- EJc &’()*)+,-
% %
P σ P = X (14)
e z1 e − !$#! !"#!
where P is the projection operator onto the subspace
e
3 , 4 . From Eq. (8), it can also be shown that the FIG. 2: (a) Schematic plot of two encoded qubits coupling
{| i | i} via a tunable circuit. The physical qubits are represented
operator σ generates no coupling between the encoded
z1
by the solid circles and are connected by the vertical bars to
subspace and states in 1 , 2 . By pumping the first
physical qubit with a pu{ls|eiH| i}=2λcos(2E t/~)σ for form the encoded qubits. The lower physical qubits in each
X m z1 encodedqubitareconnectedtoachievethecouplingbetween
a duration of θ~/2λ, we can implement the X-rotation
theencodedqubits. (b)Anarrayofencodedqubitsconnected
gate U (θ)=exp(iθX/2).
x with theirnearest-neighbors.
Similarly, the operator σ σ + σ σ can be ex-
y1 y2 z1 z2
pressed as
can also design various geometries to connect arrays of
Pe(σy1σy2+σz1σz2)Pe = 1 Z (15) encodedqubits. For example,in Fig.2b, the lowerphys-
− −
ical qubits of the encoded qubits are connected to their
which generates a rotation in the Z-component of the
nearestneighborsinachain. Inthiscase,thesystemcan
encoded qubit. This operator can be realized in su-
beviewedasaone-dimensionalchainofqubitswithnear-
perconducting circuits. For example, for charge qubits
est neighbor coupling. Universal quantum computation
connected by a coupling SQUID, this operator can be
can be implemented in this configuration.
realized by varying the flux in the SQUID loop. By
Forsuperconductingqubits,theabovecouplingHamil-
applying an ac pumping on the coupling SQUID with
tonian can be constructed using a tunable Josephson
H =2λcos(2E t/~)(σ σ +σ σ ) for a durationof
Z m y1 y2 z1 z2 junction (a SQUID) that connects the physical qubits.
θ~/2λ, we have a Z-rotation gate U (θ) = exp(iθZ/2).
z For a SQUID with Josephsonenergy E and capacitive
Combining the operations in Eq. (14) and Eq. (15), we Jc
energy E , we have
achieveacompleteSU(2)generatorsetthatgeneratesar- cc
bitrary single-qubit quantum logic gates on the encoded E
Jc
H = (σ τ +σ τ )+E σ τ (20)
qubit. cgate − 4 z2 z2 y2 y2 cc x2 x2
where both the Josephson energy and the capacitive en-
B. Controlled quantum logic gates ergy can be tunable38–41. The coupling includes the
extra terms σ τ and σ τ , which only contain off-
y2 y2 x2 x2
resonant leakage terms such as 1 3 3 1. Un-
Two-qubitgatesontheencodedqubitscanbeachieved | iσh | ⊗ | iτh |
derthetime-dependentmodulationcos(ω t)bychoosing
by connecting the circuits of the encoded qubits as is d
properpumpingfrequency,thesetermscanbeneglected.
shown in Fig. 2a. We consider two encoded qubits and
useσ asthePaulioperatorsforthefirstencodedqubit
αj
andτ as Paulioperatorsfor the secondencoded qubit.
αj C. Numerical simulation
Assume that the lower physical qubits in each encoded
qubit are connected with the coupling Hamiltonian
To test the analytical results in the previous subsec-
H = 2λ cos(2(E E )t/~)σ τ . (16) tions, we numerically simulate the quantum logic gates.
cgate c m1 m2 z2 z2
− −
In the simulation, we made the following assumptions
The operators can be projected as on the 1/f noise: 1. all the physical qubits couple
withthe environmentalnoiseinthe formofδV (t)σ +
xj xj
Peσz2Pe =X1 (17) δV (t)σ ; 2. the noise power A2 and the noise distri-
zj zj
bution angle η are the same for all physical qubits. The
simulation of the 1/f noise can be implemented as
P τ P =X (18)
e z2 e 2
ωuv
ontotheencodedsubspaces. Hence,intherotatingframe
V (t)= a (ω)cos(ωt+φ)∆ω (21)
αj αj
the above coupling Hamiltonian can be written as
ωX=ωir
λ
Hc(groatt)e =− 2c(X1X2+Y1Y2), (19) awnhderseataisαfijesis a gaussian distribution with zero average
which provides a swap-like interaction between the en-
a (ω )a (ω ) =A2cos2(η)δ(ω +ω )/ω (22)
codedqubits. Combiningthisinteractionwiththesingle- h αx 1 αx 2 i 1 2 1
qubit gates U (θ) and U (θ), we obtain a universal set
x z
of quantum logic operations for the encoded qubits. We a (ω )a (ω ) =A2sin2(η)δ(ω +ω )/ω (23)
αz 1 αz 2 1 2 1
h i
5
FIG. 4: The gate fidelity FC of two-qubit gate UC versus η
at various Ecc/2π~ as is labelled in the plot.
For the two-qubit operation U = exp[ iπ(X X +
C 1 2
−
Y Y )/4], the gate fidelity can be defined as42
1 2
FIG. 3: Gate fidelity Fx of single-qubit gate Ux(π) versus η 1 1
and Em. Here, EJ = 2Ez. Bottom left, Fx versus Em at FC = 5 + 80 Tr(UC(Σi⊗Ωj)UC†ε(Σi⊗Ωj)) (25)
various η as is labelled in the plot. Bottom right, Fx versus Xi,j
η at various Em/Ez as is labelled in theplot.
where Ω = X, Y, Z for i = 1,2,3 are the Pauli opera-
i
torsforthesecondencodedqubitandthesuper-operator
inconnectionwiththedefinitionsinEq.(11)andEq.(12). gives ε(Σ Ω )=P (Σ Ω )P†. We choose the cou-
Here, discrete noise components are used to replace pling of tih⊗e firjst encoedLediq⊗ubitjtoebe E /2π~=5GHz
m1
the continuous integral on the spectral density with and the coupling of the second encoded qubit to be
∆ω/2π = 10−4MHz. The phase φ is a random number E /2π~ = 2GHz. The operating Hamiltonian is given
m2
withtheuniformdistributionbetween0and2π. Thepa- by Eq. (20) andis appliedfor a durationof π~/4λ with
c
rameters we choose for the physicalqubits and the noise λ /2π~ = 300MHz. The gate fidelity F for U versus
c c C
areEz/2π~=5GHzforthe physicalqubits, theinfrared η is plotted in Fig. 4 at various capacitive coupling Ecc.
limit of the noise frequency ωir/2π = 1Hz, the upper ForzeroEcc,thegateUC canbeaccomplishedwithhigh
bound of the noise frequency ωuv/2π = 0.1MHz, and fidelity and shows only weak dependence on the distri-
the noise power A/E =2 10−4s/rad. bution angle η, which proves the “universality” of our
z
×
For the single-qubit operation Ux(π) = exp(iπX/2), scheme. However, for non-zero Ecc, the fidelity drops
which generates a bit-flip gate for the encoded qubit, we quickly, and hence the implementation of the two-qubit
use the following definition for the gate fidelity42 gate requires the capacitive coupling to be small.
1 1
F = + Tr(U (π)Σ U†(π)ε(Σ )) (24)
x 2 12i=X1,2,3 x i x i D. State preparation and detection
where Σ = X, Y, Z for i = 1,2,3 are the Pauli opera-
i To implement the abovequantum logicgates,we need
torsforthe encodedqubitandε(Σ )=P (Σ )P† is the
i eL i e topreparethe initialstateofsysteminthe encodedsub-
projection of the final state onto the encoded subspace
space. Thiscanbeachievedbylettingthecoupledqubits
after applying the quantum process on the initial den-
relaxtotheirgroundstates 1 viathermalization. Then,
sity matrix Σ . The coupling constant is chosen to be | i
i an ac driving
λ/2π~ = 300MHz. The gate fidelity F of U (π) versus
x x
η and Em is plotted in Fig. 3. For finite coupling Em, Hprep =2λpcos[(E1 Em)t/~]σx1 (26)
wheretheencodedsubspaceisprotectedbythecoupling, −
the fidelity only varies smoothly as the noise varies from canbeappliedonthefirstphysicalqubit,whichgenerates
transversenoisetolongitudinalnoise. WhileforE =0, aRabioscillationbetweenthestates 1 and 3 giventhe
m
| i | i
i.e. the uncoupled (bare) qubits, the fidelity decreases non-zero matrix element 3σ 1 = sin(θ/2 +π/4).
x1
h | | i −
sharply as η increases to π/2. At fixed noise distribu- Afteraperiodofπ/2λ sin(θ/2+π/4),thestatebecomes
p
tion η, the fidelity first rises rapidly when the coupling 3 which is now in the encoded subspace.
| i
E increasesfromzero,thenbecomessaturatedandeven State detection of the encoded qubits can be imple-
m
showsoscillatorybehaviorasE furtherincreases. With mented with the assistance of single-qubit rotations as
m
E 0.4E , F exceeds 0.997 for η [0,π/2]. In this well. To measure the probability of an encoded qubit in
m z x
∼ ∈
regime,the optimal E for a particularη usually canbe the state 3 or 4 , apply single-qubit gate exp[ iπX/4]
m
| i | i −
found within the range of (0.4E , E ). as discussed above which converts the state 3 to the
z z
| i
6
state and converts the state 4 to the state . We would like to mention that the proposed scheme
| ↑↓i | i | ↓↑i
Then, measuring the physical qubits can give us infor- is different from the Decoherence Free Subspace (DFS)
mation of the encoded states. approachthathasbeenwidelystudiedinquantuminfor-
mation processing43. The DFS approachprotects qubits
from spatially correlated noises by choosing a subspace
IV. DISCUSSIONS AND CONCLUSIONS that is immune to such noises, i.e. the dephasing is sup-
pressed by the noise correlation. While in our scheme,
Theencodedqubitproposedaboveismadeoftwoiden- we explore the energy separation of the encoded sub-
ticalphysicalqubitswithqubitenergyE . Forsupercon- space and the low-frequency nature of the noise (which
z
ducting qubits, this parameter can be the Josephson en- can’t excite transitions between states with large energy
ergy E of a chargequbit in the degeneracypoint or the separationin the first order)to protect the quantum co-
J
energygapinthe fluxqubit. Thisparameterdepends on herence.
the parametersof the Josephsonjunctions in the circuit. InSec. IIandIII,westudiedtheencodedsubspaceand
Inrealdevices,thejunctionparametersusuallyhavefab- the gate operationsusing the coupling Hamiltonianwith
rication errors with an error spread on the order of 5%. Emx = Em, Emy = Emz = 0 for the simplicity of our
Belowwediscusstheeffectoftheparametererroronthe discussions. It can be shown that the UQDP approach
proposed scheme. We introduce a factor a as the ratio can be applied to the general form of coupling given in
0
between the energy of the second qubit and the energy Eq. (3) as far as either Emx = 0 or Emy = 0 can be
6 6
of the first qubit with a 1 1. The Hamiltonian in satisfied. As an example, consider the situation
0
| − |≪
Eq. (3) becomes
E =E , E =E =b E (32)
mx m my mz 0 m
H =E (σ +a σ )+E σ σ . (27)
0n z z1 0 z2 m z1 z2
withacoefficientb . Inrealsystem,thiscouplingcanbe
0
The eigenbasis of this Hamiltonian still includes the en- obtainedfromaJosephsonjunctionwithenergyE that
J2
coded subspace 3 , 4 , but with the new eigenstates connects the two physical qubits and b = EJ2. It can
and eigenenergie{s| i | i} 0 −4Em
befoundthatthestates 3,4 definedinEq. (6)stillform
| i
thesubspacefortheencodedqubit. Theenergiesofthese
3 =cosϕ3 +sinϕ4 , E = E ;
||4iinn =−sin|ϕi|3i+cos|ϕi|4i, E34nn =−Em,mn;,n (28) astastheisftbdecuoemteoǫt3h=e fi−nEitme−b02.b0IEtmcaanndalǫs4o=beEmshoinwcnludthinagt
theprojectionstotheencodedsubspaceare P σ P =
where E = (1 a )2E2+E2 and − e z1 e
m,n − 0 J m Peσz2Pe =X and PeσxiPe =PeσyiPe =0. The encoded
p
qubit is protected against any first order dephasing by
ϕ=1/2sin−1[(a 1)E /E ] (29)
0− z m,n thelow-frequencynoise. Thisobservationshowsthatthe
UQDPschemecanbeappliedtovarioussuperconducting
respectively. The operators σ , σ , σ , and σ
x1 x2 y1 y2 qubits such as flux qubits and phase qubits as far as we
only contain non-vanishing matrix elements connecting
can construct the coupling in Eq.(3).
1 , 2 and 3 , 4 . However, σ and σ now in-
z1 z2
{| i | i} {| i | i} In conclusion, we have proposed a scheme of a univer-
clude non-trivial diagonal terms
sal quantum degeneracy point (UQDP) that can protect
3σ 3 = 4σ 4 = −(1−a0)Ez, the superconducting qubits from generic low-frequency
nh | z1| in −nh | z1| in Em,n (30) noise. Using coupled qubits to form the encoded qubits,
3σ 3 = 4σ 4 = (1−a0)Ez,
nh | z2| in −nh | z2| in Em,n we find a subspace where the low-frequency noise only
affects the qubit dephasing to quadratic order. We have
whichinduceresiduallongitudinalnoisesonthe encoded shown that universal quantum logic gates can also be
qubit. The residual coupling can be derived as performed on the encoded qubits with high fidelity. The
schemeisrobustagainparameterspreadsduetofabrica-
(a 1)E
V 0− z(δV δV )Z (31) tion errors. The scheme can be applied to systems with
res z1 z2
≈ E −
m,n very general form of couplings and provides a promis-
ing approach to protect superconducting qubits against
in terms of the effective Pauli operator Z, which gener-
low-frequency noise.
ates dephasingin firstorderterms ofδV . However,the
zj
residual coupling contains the ratio a 1 of the error
0
| − |
spreadwhichislessthan5%orevensmallerwiththead-
vance ofcurrenttechnology. The dephasing by this term V. ACKNOWLEDGMENTS
ishencereducedbyafactorof a 12 <2 10−3. Given
0
| − | ×
the recent experimental measurement28 of δV /E < This work is supported by the National Science Foun-
zj z
10−2, the longitudinaldephasing due to the|errorspr|ead dation under Grant No. NSF-CCF-0916303 and NSF-
canbecomparablewithorevenlowerthanthequadratic DMR-0956064. XHD is partially supported by Scholar-
dephasing due to the transverse noise in Eq. (13). ship from China.
7
∗
Electronic address: [email protected] B 79, 094520 (2009).
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