Quantum Information Transfer between Topological and Superconducting Qubits

0.21 MB·English

by  Fang-Yu Hong

Quantum Information Transfer between Topological and Superconducting Qubits
Fang-Yu Hong,1, Jing-Li Fu,1 and Zhi-Yan Zhu1
∗
1Department of Physics, Center for Optoelectronics Materials and Devices,
Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China
(Dated: January 22, 2013)
We describe a scheme that enables a strong Jaynes-Cummings coupling between a topological
qubitandasuperconductingfluxqubit. Thecouplingstrengthisdependentonthephasedifference
between two superconductors on a topological insulator and may be expediently controlled by a
phase controller. With this coherent coupling and single-qubit rotations arbitrary unitary opera-
tions on the two-qubit hybrid system of topological and flux qubits can be performed. Numerical
simulationsshowthatquantumstatetransferandentanglementdistributingbetweenthetopological
3 and superconductingflux qubitsmay be performed with high fidelity.
1
0 PACSnumbers: 03.67.Lx,03.65.Vf,74.45.+c,85.25.-j
2 Keywords: topological qubit,superconducting qubit,quantum interface

n
a
J I. INTRODUCTION [22],in generationofentanglementbetweensingle-atoms
atadistance[23]andbetweenaphotonandasolid-state
9
1 The decoherence of quantum states by the environ- spin qubit [24]
ment is the main obstacle in the way towards realizing Thus it is highly desirable to combine the advantages
] quantum computers. To circumvent this difficulty some ofconventionalqubitswiththoseoftopologicalqubitsto
h
p interesting topological quantum computation schemes construct hybrid systems, where the necessary topologi-
- [1, 2] have been suggested, where quantum information callyunprotectedgatescanbeimportedfromtheconven-
nt is stored in nonlocal (topological) degrees of freedom of tionalquantumsystems(CQS)andtopologicalstatescan
a topologically ordered systems. These nonlocal degrees betransferredtoCQSforhighfidelityreadout. Suchhy-
u of freedom are decoupled from local perturbations, en- bridsystemshavebeenconsideredrecentlyfortheanyons
q abling the topological approach to quantum informa- in optical lattices [25, 26] and for the Majorana anyons
[
tion processing to obtain its exceptional fault tolerance coupled to superconducting flux qubits [27–29] or to a

1 and to have a tremendous advantage over conventional semiconductor double-dot qubit [17].
v ones. Thesimplestnon-Abelianexcitationfortopological Here we propose a scheme for quantum information
7 qubits is the zero energy Majorana bound state (MBS) transferbetweenasuperconductingfluxqubit[22,30,31]
3 [3], which is predicted to be exist in the spin lattice sys- and a topological qubit encoded on Majorana fermions
5
tems [1], in the p+ip superconductors [4], in the filling (MFs)atthejunctionsamongthreesuperconductorsme-
4
. fraction ν = 5/2 fractional quantum Hall system [2], in diated by a topological insulator (TI) [6]. The strong
1 the superconductor Sr RuO [5], in the topological in- Jaynes-Cummings(JC)couplingbetweentopologicaland
0 2 4
sulators [6, 7], and in some semiconductors with strong superconducting flux qubits can be obtained on the ba-
3
1 spin-orbit interaction [8–12]. sis of the interaction between two MFs located at the
: However, the local decoupling makes measuring and two ends of a linear superconductor-TI-superconductor
v manipulatingtopologicalstatesdifficultbecausetheycan (STIS) junction, and be coherently controlled by the
i
X only be manipulated by globe braiding operations, i.e., phasedifferencesbetweenthetwosuperconductorsofthe
r byphysicalexchangeoftheassociatedlocalquasiparticle STIS junction. With this strong coupling at hand, arbi-
a non-Abelian excitations [13, 14]. Moreover topologically traryquantuminformationtransferandquantumentan-
protected braiding operations for Ising anyons alone are glementdistributionbetweenthetopologicalandtheflux
not adequate to fulfill universal quantum computation qubits can be accomplished with near unit fidelity.
and have to be supplemented with topologically unpro-
tected operations [15, 16]. Within a topological system
II. HYBRID SYSTEM
unprotected operations prove to be very challenging be-
causeofsignificantnonuniversaleffects[17]. Ontheother
hand, conventionalquantum informationprocessing sys- The prototypehybrid quantumsystem shownin Fig.1
tems have been advancing steadily, such as the recent ismadeupofasuperconductingfluxqubitandatopolog-
progressesin quantumnetworkusing singleatoms inop- ical qubit encoded on four MFs. The flux qubit consists
ticalcavities[18],inlongcoherencetimesofnuclearspins of a loop of four Josephson junctions (j ) and four
1,2,3,4
in a diamond crystal [19, 20], in high fidelity manipula- superconducting island a,b,c,d, enclosing an externally
tionsontrappedions[21]andonsuperconductingqubits appliedmagneticfluxΦ h. TheMFsaredescribedby
≈ 4e
Majorana fermion operators γ (i = 1,2,3,4), which are
i
∗ Emailaddress:[email protected];Tel:[phone] staeltfi-oHnerremlaittiiaonn, γγi†,=γ γi,=aδnd.fuTlhfiellMfearmjoiroanniacfaenrmticioonmγmuis-
i j ij i
{ }
2
1
a a)
j2 j1 --p2 1 g TuI g 2 --p2 00..68 r22 r11
1 2 0.4
b F d d 0.2
0
g g -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
--p 3 3 TI 4 4 --p tg/p
j j 2 2 1
3 c 4 u 0.75 b) r22
0.5 ir
12
0.25 r
11
FIG. 1. (color online). Schematics for a hybrid system of 0
topological and superconducting flux qubits. A flux qubit is -0.25 ir21
made up of four Josephson junctions (j1,2,3,4) and four su- -0.-50.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
perconductingislands(a,b,c,d)patternedonthesurfaceofa tg/p
topological insulator, enclosing an external fluxΦ h/4e. A
≈
topological qubit consists of two pairs of Majorana fermions FIG. 2. (color online). a) Numerical simulation of the pro-
((γ1,γ2) and (γ3,γ4)). Island d is shared by the topologi- cessofthestatetransfer, 0 i 1 . Thestatetransfer
cal and the flux qubits. Two Majorana fermions (marked |↑ i→− |↓ i
fidelity is F1 = 0.993. b) Numerical simulation of quantum
with circles) at two superconductingtrijunctions arecoupled entanglement generating, 0 ( 0 i 1 )/√2. The
thoughSTISquantumwirewithcouplingstrengthdependent |↑ i → |↑ i− |↓ i
generatedentanglementhasafidelityF2 =0.996. Theparam-
on thephase φd of island d relative toφu =−π. etersusedareg/2π=−2GHz,g′/2π=−1GHz,Tf,1 =900ns,
Tf,2 = 20ns, and ωf/2π = E(φon)/2π = 50 GHz. The
corresponding matrix elements of the density matrix ρ of
localized at trijunction i(i = 1,2,3,4), which comprises
the hybrid system are ρ11 = 1ρ 1 , ρ22 = 0ρ 0 ,
threesuperconductorsdividedbyaTI[6]. ApairofMFs ρ21 = 0ρ 1 , ρ12 = 1ρh↓ 0|.|↓ i h↑ | |↑ i
operatorsγ ,γ connectedbyaSTISwireoflengthLcan h↑ | |↓ i h↓ | |↑ i
i j
formaDiracfermionoperatorf =(γ iγ )/√2,which
ij i j
−
creates a fermion and fi†jfij = nij = 0,1 describes the Expanding the coupling strength E(φd) to first order
occupation of the corresponding state. Combining two in the small parameters θ dE(φ) and ζ dE(φ)
such fermion states gives the two logical states of the gives the Hamiltonian ωf dφ |φ=φc ωf dφ |φ=φc
topological qubit 0 = 0 0 and 1 = 1 1 .
| it | 12 34i | it | 12 34i
The flux qubit is made up of four Josephson junc- 1 g 1
H =a aω E(φ )σz ′σzσz g(a +a)σz, (3)
tions with Josephson coupling energy EJ,1 =EJ,2 =EJ, † f − 2 c t − 2 f t − 2 † t
E = αE , and E = βE , where 0.5 < α < 1 and
J,3 J J,4 J
β 1. For these parameters and an externally ap- where
≫
plied flux Φ = h/4e, the system has two stable states
ζ dE(φ)(cid:12)
0 and 1 for the flux qubit. Corresponding to these g = (cid:12)
| if | if √2 dφ (cid:12)
twostatesthere arepersistentcirculatingcurrentsofop- (cid:12)φ=φc
posite direction with the corresponding superconducting dE(φ)(cid:12)
phase φ = φ + σzθ + ζa+a† of island d [29], where g′ = θ dφ (cid:12)(cid:12) . (4)
d c f √2 (cid:12)φ=φc
σz = (0 0 1 1) , θ = √4α2 1 is the phase differ-
f | ih |−| ih | f 2αβ− ByrewritingHamiltonian(3)intermsof = 1 (0 +
ence across Josephson junction j , a is the annihilation |↓i √2 | i
otupdeeraotforqufoarntthuemflfluuxcqtuubatitio,nζs=, a(n8dE4EJCth)e14pβh−a12seisφtheofmisalgannid- |w1ai)vteaanpdp|ro↑xiim=a√t1i2o(n|0ain−d|1tih)et ianntderaapctpiloyninpgicthtuerreowtaetionbg--
c
c is fixed relative to the phase φ = π of island u by a tain
u
−
phase controller [29, 32].
1 g
The Hamiltonian for the hybrid system can be writ- HI =−2g(a†σt−+aσt+)− 2′σfz(σt+eiE(φc)t
ten in the form (~ = 1) H = a aω 1E(φ )σz, where
ω =√8E E , σz =(0 0 †1 1f)−,2and tdhetcoupling +σt−e−iE(φc)t), (5)
f J C t | ih |−| ih | t
strength E(φd) has the approximate form [29] where σt+ = |↑ih↓| and σt− = |↓ih↑| are the raising
and lowering operators, respectively, and the resonance
E(φ ) 1.9(Λ 0.5)v /L for Λ 5 (1)
d ≈− φd − F φd ≤− condition ω =E(φ ) has been assumed for simplicity.
f c
and Discussion.—The first term in H (5) describes the
I
JCcoupling betweenthe topologicalandthe flux qubits,
φ
E(φd)≈2∆0sin 2de−Λφd ∼0 for Λφd ≫1, (2) which is just what we want. The last term will cause
the total number of the excitations in the hybrid system
whereΛ ∆0Lsinφd withthe effective Fermivelocity changes and will contaminate the quantum information
φd ≡ vF 2
v and the proximity induced superconducting gap ∆ . transferfidelity, thuswemaycontainits influence bythe
F 0
3
conditions (a) (b)
1 1
√2βα 8EC 1
g/g′ = ( )4 1 (6)
√4α2 1 EJ ≫ 0.95 g'=0
− g'=0
0.95
and E(φ )/g 1. However, because of the factors
c ≫ 0.9
e±iE(φc)t the influence of this non-JC term is very lim-
ited, even for the case g < g′, which is shown in the 0.85 0.9
following numerical simulation.
F
According to Eqs.(1, 2) the JC coupling strength g 0.8
can be coherently controlled: g 0 if φ is tuned to φ
satisfying ∆0Lsinφoff 1, and∼g ∆c ζ cosφon if φoff 0.75 0.85
vF 2 ≫ ≈− 0√2 2 c
is5a.dBiaybaatdiicaabllaytiacdajlluystteudrntoonφotnhesactoiusfpyliinnggf∆ovr0FLasdinurφa2otnio≤n 0.7 g'=6g 0.8 g'=6g
−
correspondingtoaπpulseR g(t)dt= π,wecanperform 0.65
−
a unitary transformation
0.75
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
µ 0 +ν 0 µ 0 iν 1 , (7)
|↓ i |↑ i→ |↓ i− |↓ i h /g h /g
1 2
accomplishing a quantum state transfer from the topo-
logicalqubit to the flux qubitby followinga single-qubit
FIG. 3. a) The effect of decoherence sources η1 =1/2Tf,1 on
rotation on the latter, where µ and ν are arbitrary com-
the fidelity of state transfer operation 0 i 1 with
plex numbers satisfying µ2 + ν 2 = 1. If we choose different non-JC couplings g′ (from to|p↑toib→ott−om|↓: gi′/g =
| | | |
R g(t)dt = π/2, we can generate a maximally entan- 0,1,2,3,4,5,6.). OtherparametersareasinFig.2. b)Thesame
−
gled state 0 ( 0 i 1 )/√2. Up to a single- plot for thethe influenceof η1 =1/Tf,2.
|↑ i → |↑ i− |↓ i
qubitrotationa√SWAPgate,thesquaredrootofSWAP
gate, can be obtained by choosing R g(t)dt = 3π/2.
− blad master equation
With √SWAP gates and single-qubit90 rotationabout
◦
zˆdenotedby R (90),we canobtainthe controlled-phase
z ∂ρ 1
(CPt,f) gate ∂t =−i[HI,ρ]+ 2T (2aρa†−a†aρ−ρa†a)
f,1
CPt,f =Rz,t(90)Rz,f(−90)√SWAPRz,t(180)√SWAP(8) + T1 (σfzρσfz −ρ) (9)
f,2
forthehybridsystem. WithCP gatesandsingle-qubit
t,f
where the decoherence from the topological qubit has
rotations an arbitraryunitary transformationon the hy-
been neglected due to this qubit’s great merit of long
brid system is available [33].
coherence time, T and T are the relaxation time
To sufficiently suppress the influence of the non-JC f,1 f,2
and dephasing time of the superconducting flux qubit,
coupling,g/g 1/3isrequired(asweexplainlaterinde-
′
tail),whichma≥ybe fulfilledbychoosingβ 1, α 0.5, respectively [34].
e.g.,wehaveg/g 2forthecasewhereβ =≫15,α→=0.8, To study the quantum information transfer between
′
E /E = 80. Th≈e corresponding flux quantum fluctu- the topological and the superconducting flux qubits un-
J C
ation is ζ = 0.14 and the phase difference of Josephson derrealisticconditionswe numericallysimulatethe mas-
junction j is θ = 0.05, which is within the reach of a ter equation (9). We may set α = 0.8, β = 15,
4
phase controller [32]. Apart from the non-JC coupling, EJ/EC = 80, EJ/2π = 158GHz, ωf/2π = 50GHz,
there areother relevantimperfections forthe hybridsys- Tf,1 = 900ns, and Tf,2 = 20ns for the superconduct-
tem. The tunneling between 0 and 1 with tunnel- ing flux qubit [29, 35]; the parameters for the topo-
| if | if logical qubit may assume to be ∆ /2π = 32.5 GHz,
ing rate r ω exp( pE /E ) decreases the coherence 0
time of th∼e sufperco−nductJingCflux qubit. The coupling vF = 105m/s, L = 5µm [12, 29]. The resonance con-
dition gives E(φ )/2π = ω /2π = 50GHz, resulting in
strength g should be strong enough to repress the un- on f
φ = 1.73 according to (1) with Λ = 7.75. Then
wanted tunneling probability (r/g)2 [29]. Low tempera- on − φon −
equations (4) give the the coupling strength g/2π = 2
tureisrequiredto exponentiallydecreasetheprobability −
GHz and g /2π = 1 GHz. The evolution of the state
oftheoccupationoftheexcitationmodesofthequantum ′ −
wire by the factor exp( −vf ) [29]. transfer
kBTL
Rtf1g(t)dt= π
0 − ψ i 1 (10)
1
|↑ i−−−−−−−−−−→| i≡− |↓ i
III. NUMERICAL SIMULATIONS
and the generating of a maximally entangled state
Considering the decoherence sources, the dynamical
process of the hybrid system is described by the Lind- 0 Rtf2g(t)dt=−π/2 ψ ( 0 i 1 )/√2 (11)
2
|↑ i−−−−−−−−−−−→| i≡ |↑ i− |↓ i
4
are shown in Fig. 2a) and b), respectively, with the be more conveniently accomplished.
corresponding fidelity F = ψ ρ(t )ψ = 0.993 and
1 1 f1 1
h | | i
F = ψ ρ(t )ψ = 0.996. Fig.3 shows the influence
2 2 f2 2
of thehdec|oheren|ceisources η = 1/2T , η = 1/T , IV. CONCLUSIONS
1 f,1 2 f,2
and g on the state transfer fidelity F . From Fig.3 we
′ 1
see that the influence of g′ on the state transfer is small: In summary, we have presented a scheme for quantum
F1 = 0.982 for the case where g′ = 3g = 6(2π)GHz, information transfer between topological and supercon-
−
E(φon), ωf, Tf,1, Tf,2, vF, L, and Λφon remain the same ductingfluxqubits. AstrongJaynes-Cummingscoupling
as in fig.2, while other parameters are α=0.97, β =10, between topological and flux qubits is achieved. With
EJ/EC = 30000 [32, 36], θ = 0.086, ζ = 0.04, EJ = 3.1 this scheme, quantum state transfer, quantum entangle-
THz, φon = 0.646,and ∆0 =78GHz. mentgenerating,andarbitraryunitarytransformationin
−
the topological-flux hybrid system may be accomplished
Apart from the aforesaid decoherence sources, there
with near unit fidelity. This quantum interface enable
exist processes which may influence the interaction be-
us to store quantum information on topological qubits
tweenthetwoMajoranafermions,suchasdynamicmod-
for long-time storage, to efficiently read out of topolog-
ulations of the superconducting gap and variationof the
ical qubit states, to implement partially protected uni-
electromagnetic environment owing to charge fluctua-
versal topological quantum computation, where single-
tions. We estimate their influence on the operation fi-
qubitstateofflux qubitcanbe preparedwithhighaccu-
delity by assuming unknown errors in E(φ ), g , and g,
on ′ racy and is transferred to topological qubit to compen-
andfindthatthecorrespondingfidelityF decreasesfrom
1 sate topological qubit’s incapability of generating some
0.993 to 0.968 for even 10% unknown errors in E(φ ),
on single-qubit states.
g , and g.
′ This workwassupportedbythe NationalNaturalSci-
The recent proposal [29] applies in the parameter ence Foundation of China ( 11072218and 11272287),by
regime g g, in contrast our scheme works well in the ZhejiangProvincialNaturalScienceFoundationofChina
′
≫
parameter regime g 3g. With Jaynes-Cummings cou- (Grant No. Y6110314),and by Scientific Research Fund
′
≤
plingquantumstatetransferandquantumentanglement ofZhejiangProvincialEducationDepartment(GrantNo.
distribution between the topological and flux qubits can Y200909693).
[1] A.Y.Kitaev, Ann.Phys.(N.Y.) 303, 2 (2003). G. Rempe,Nature (London) 484, 195 (2012).
[2] C. Nayak, S.H. Simon, A. Stern, M. Freedman, and S. [19] M.V.GurudevDutt,L.Childress, L.Jiang,E.Togan,J.
Das Sarma, Rev.Mod. Phys.80, 1083 (2008). Maze, F. Jelezko, A.S. Zibrov, P.R. Hemmer, and M.D.
[3] F. Wilczek, NaturePhys 5, 614 (2009). Lukin, Science316, 1312 (2007).
[4] N.Read and D.Green, Phys.Rev.B 61, 10267 (2000). [20] P.C. Maurer, G. Kucsko, C. Latta, L. Jiang, N.Y. Yao,
[5] S.DasSarma,C.Nayak,andS.Tewari,Phys.Rev.B73, S.D.Bennett,F.Pastawski,D.Hunger,N.Chisholm,M.
220502R (2006). Markham, D.J. Twitchen, J.I. Cirac, and M.D. Lukin,
[6] L. Fu and C.L. Kane, Phys. Rev. Lett. 100, 096407 Science 336, 1283 (2012).
(2008). [21] R. Blatt and D. Wineland, Nature (London) 453, 1008
[7] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, (2008).
3045(2010). [22] J.ClarkeandF.K.Wilhelm,Nature(London)453,1031
[8] J.D.Sau, R.M. Lutchyn,S.Tewari, andS.Das Sarma, (2008).
Phys.Rev.Lett. 104, 040502(2010). [23] D.L. Moehring, P. Maunz, S. Olmschenk, K.C. Younge,
[9] J. Alicea, Phys.Rev. B 81, 125318 (2010). D.N.Matsukevich,L.-M.Duan,andC.Monroe,Nature
[10] Y.Oreg, G. Refael, and F. von Oppen,Phys.Rev.Lett. (London) 449, 68 (2007).
105, 177002 (2010). [24] E. Togan, Y. Chu, A.S. Trifonov, L. Jiang, J. Maze, L.
[11] R.M.Lutchyn,J.D.Sau,andS.DasSarma,Phys.Rev. Childress, M.V.G. Dutt, A.S. Søensen, P.R. Hemmer,
Lett.105, 077001 (2010). A.S.Zibrov,andM.D.Lukin,Nature(London)466,730
[12] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. (2010).
A. M. Bakkers, L. P. Kouwenhoven, Science 336, 1003 [25] L. Jiang, G.K. Brennen, A.V. Gorshkov, K. Hammerer,
(2012). M.Hafezi,E.Demler,M.D.Lukin,andP.Zoller,Nature
[13] D.A. Ivanov,Phys.Rev.Lett. 86, 268 (2001). Phys. 4, 482 (2008).
[14] A.Stern, Nature( London) 464, 187 (2010). [26] M.Aguado,G.K.Brennen,F.Verstraete,andJ.I.Cirac,
[15] P.Bonderson, Phys. Rev.Lett. 103, 110403 (2009). Phys. Rev.Lett. 101, 260501 (2008).
[16] P. Bonderson, D.J. Clarke, C. Nayak, and K. Shtengel, [27] F. Hassler, A.R. Akhmerov, C.-Y. Hou, and C.W. J.
Phys.Rev.Lett. 104, 180505(2010). Beenakker, New J. Phys. 12, 125002 (2010).
[17] P. Bonderson and R.M. Lutchyn, Phys. Rev. Lett. 106, [28] J.D.Sau,S.Tewari,andS.DasSarma,Phys.Rev.A82,
130505 (2011). 052322 (2010).
[18] S. Ritter, C. N¨olleke, C. Hahn,A. Reiserer, A. Neuzner, [29] L. Jiang, C.L. Kane, and J. Preskill, Phys. Rev. Lett.
M. Uphoff, M. Mu¨cke, E. Figueroa, J. Bochmann, and 106, 130504 (2011).
5
[30] J.E. Mooij, T.P. Orlando, L. Levitov, L. Tian, C.H. van andQuantumInformation,(CambridgeUniversityPress,
derWal, and S.Lloyd, Science 285, 1036 (1999). Cambridge, England, 2010).
[31] X.Zhu,S.Saito,A.Kemp,K.Kakuyanagi,S.Karimoto, [34] Y. Makhlin, G. Sch¨n, and A. Shnirman, Rev. Mod.
H. Nakano, W.J. Munro, Y. Tokura, M.S. Everitt, K. Phys. 73, 357 (2001).
Nemoto, M. Kasu, N.Mizuochi, and K. Semba, Nature( [35] I.Chiorescu,Y.Nakamura,C.J.P.M. Harmans,andJ.E.
London) 478, 221 (2011). Mooij, Science 299, 1869 (2003).
[32] L.Jiang,C.L.Kane,andJ.Preskill,arXiv: 1010.5862v2. [36] J. M. Martinis, S. Nam, J. Aumentado, and C. Urbina,
[33] M.A. Nielsen and I.L. Chuang, Quantum Computation Phys. Rev.Lett. 89, 117901(2002).