III. From Majorana to Parafermions - Capri SchoolIII. From Majorana to Parafermions In collabora9on...

DanielLoss

UniversityofBasel

Switzerland

$$:SwissNSF,NanoBasel,QuantumETH/Basel,EU

III. From Majorana to Parafermions

Incollabora9onwithJelenaKlinovaja,Basel

•  Topologicalquantumcompu9ng

•  Majoranafermionsin‘super-semi’nanowiresandchains

•  Majoranaandspinqubitsè2Dsurfacecode

•  Nextgenera9on:Parafermionsindoublenanowires

ècangetnearlyuniversalTQCincludingCNOT

Outline

At T=0: TQC protected against all errors by gap

Motivation: Topological Quantum Computing

Kitaev 2003 braiding of quasiparticles with non-Abelian statistics

Majoranas, parafermions, Fibonacci fermions,…

Motivation: Topological Quantum Computing

Kitaev 2003 braiding of quasiparticles with non-Abelian statistics

Majoranas, parafermions, Fibonacci fermions,…

At T=0: TQC protected against all errors by gap At T>0: errors occur with finite probability è need continuous quantum error correction for TQC

Wootton, Burri, Iblisdir, and Loss, PRX 4, 011051 (2014) Brell, Burton, Dauphinais, Flammia, and Poulin, PRX 4, 031058 (2014) Pedrocchi, Bonesteel, and DiVincenzo, PRL 115, 120402 (2015); PRB 92, 115441 (2015) Hutter, Wootton, and Loss, Phys. Rev. X 5, 041040 (2015) Burton, Brell, and Flammia, arXiv:1506.03815 (2015) Hutter and Wootton, PRA 93, 042327 (2016) Brown, DL, Pachos, Self, and Wootton, Rev. Mod. Phys. 88, 045005 (2016)

decayofgroundstateduetolocalgatefluctua9ons

μloconMajoranachain

FortypicalmetallicgatesoneneedsΔ>20Tforτ>1μs

τ

exponen9alsuppression

butlargeprefactor

Majorana Fermions in Nanowires

Basic ingredients: s-wave superconductivity

& spin texture due to:

•  Spin orbit interaction (SOI) •  Rotating Zeeman field (synthetic SOI, spin helix,…) •  RKKY interaction

‘spin-orbit field’ along z-axis

€

HR = αpxσ z

Rashba(1960)F

Helicalspectrumiscrucial

e.g.[110]zincblendorRashba

‘Helicalspectrum’

✏±(k) =k2

2m� µ⌥

p↵2k2 +�⇤

‘spin-orbit field’ along z-axis

€

HR = αpxσ z

Rashba(1960)F

Helicalspectrumiscrucial

e.g.[110]zincblendorRashba

InAs nanowire λSO ~ 100 nm, Fasth, Fuhrer, Samuelson, Golovach & DL, PRL 98, 266801 (2007)

SOImeasured

inquantumdots

Competing gaps: magnetic field vs. superconductivity

Compe99onbetweengaps;Δz>Δs:topologicalphasewithMajoranaboundstate

Volovik,JETPLef.70,609(1999)

SatoandFujimoto,PRB79,094504(2009)

nanowires:

Lutchynetal.,PRL105,077001(2010)

Oregetal.,PRL105,177002(2010)

Еxperiments: Kouwenhoven group, Science 336, 1003 (2012) Xu group, Nano Letters 12, 6414 (2012) Heiblum group, Nat. Phys. 8, 887 (2012) Rokhinson group, Nat Phys 8, 795 (2012) Marcus group, PRB 2013, Nature (2016), Science (2016) Ando group, PRL 107, 217001 (2011) (topological insulators)

Mourik et al., Science 336, 1003 (2012)


Interpretation: Zero bias peak in dI/dV = Majorana fermion

Similar experiments: Heiblum group, Nat. Phys. 8, 887 (2012) Xu group, Nano Letters 12, 6414 (2012) Rokhinson, Nat Phys 8, 795 (2012) Marcus group: Churchill et al., PRB 87, 241401(R) (2013); Albrecht et al., Nature 531, 206 (2016) Ando group, PRL 2011 (topological insulators)

Majorana zero-mode detected via ‘zero-bias peak’


Interpretation: Zero bias peak in dI/dV = Majorana fermion

Similar experiments: Heiblum group, Nat. Phys. 8, 887 (2012) Xu group, Nano Letters 12, 6414 (2012) Rokhinson, Nat Phys 8, 795 (2012) Marcus group: Churchill et al., PRB 87, 241401(R) (2013); Albrecht et al., Nature 531, 206 (2016) Ando group, PRL 2011 (topological insulators)

Majorana zero-mode detected via ‘zero-bias peak’

CompositeStructureofMajoranaWavefunc9on

exteriorbranches

interiorbranches

interiorbranches:

exteriorbranches:

Exteriorandinteriorbranchesaredecoupled:

Lineariza9on(strongSOIlimit):

KlinovajaandDL,PRB86,085408(2012)

èemergentDiractheory

Defini9onofaMajoranafermion

Fermionbasis:

MFoperator:

then

� = �† �2 = 1

Majorana:‘par9clethatisitsownan9par9cle’

=

0

BB@

" # †

" †

#

1

CCA

Wavefunc9on:

�MF =

0

BB@

f(x)g(x)f

⇤(x)g

⇤(x)

1

CCA

� =

Zdx �MF (x) · = �

†

Defini9onofaMajoranafermion

Fermionbasis:

MFoperator:

then

� = �† �2 = 1

Majorana:‘par9clethatisitsownan9par9cle’

=

0

BB@

" # †

" †

#

1

CCA

Wavefunc9on:

�MF =

0

BB@

f(x)g(x)f

⇤(x)g

⇤(x)

1

CCA

� =

Zdx �MF (x) · = �

†chargeiszero

spiniszero

boundstate

TypicalMajoranawavefunc9on

frominteriorbranch fromexteriorbranch

twolocaliza9onlengths,given

byinnerandoutergap:

twobranchescontributeequally exteriorbranchdominates

Friedel-likeoscilla9ons

KlinovajaandLoss,PRB86,085408(2012)

Rainisetal.,PRB87,024515(2013)

weakSOI

Oscillations of Majorana Splitting

B

Note:amplitudeofoscilla9onincreases!

Marcusgroup:Albrechtetal.,

Nature531,206(2016)

Rainisetal.,PRB87,024515(2013)

weakSOI

Oscillations of Majorana Splitting

B

Note:amplitudeofoscilla9onincreases!

Marcusgroup:Albrechtetal.,

Nature531,206(2016)

?

Amplitudedecreases

Why?

Majorana wire and quantum dot Marcusgroup:Dengetal.,Science354,1557(2016)

Majorana wire and quantum dot Marcusgroup:Dengetal.,Science354,1557(2016)

Twoissues:1)gapalmostclosedwhenZBPappears

2)ZBPmuchhigherconductancethanbulk

SpinandChargeSignaturesofTopologicalPhases

Quasipar9clecharge

Quasipar9clespin

Szumniak,Chevallier,Loss,andKlinovaja,arXiv:1703.00265

Bandstructure:spinpolariza9on

topological phase

trivial phase

Z scΔ < Δ

Z scΔ > Δ

( )X XS k B

Δsc = 0.02, α = 0.3

µ = 0, Eg = Δi

Inversionofbulkspinaroundtopologicalgap

Szumniak,Chevallier,Loss,andKlinovaja,arXiv:1703.00265

Note:SOIisgivenbymaterialproperty

èneedtotunechemicalpoten9alinsidegap

suchthatkF≈2kSO

Arethereself-tuningschemesforMFs?Yes!

•  Normal phase in 1D (RKKY): Braunecker, Simon, and DL, PRL 102, 116403 (2009) •  SC phase in 1D (RKKY): Klinovaja, Stano, Yazdani, and DL, PRL 111, 186805 (2013)

TheSOIinterac9oncanbegauged`away’!

Braunecker,Japaridze,Klinovaja,andDL,PRB82,045127(2010)

Onlyrota9ngfield

andnoSOIanymore!

Gaugetrafo

B≠0

rota9ngB-fieldèeffec9veSOI

Effec9veSOIgeneratedbynanomagnetsèMajoranas

Klinovaja,Stano,andDL,PRL109,236801(2012)

periodofnanomagnets=spin-orbitlength~20nm

richMFphases

Graphene nanoribbon: KlinovajaandDL,PRX3,011008(2013)

MajoranaFermionsinAtomChains

s-wavesuperconductor

Klinovaja,Stano,Yazdani,andDL,PRL111,186805(2013)

BrauneckerandSimon,PRL111,147202(2013)

VazifehandFranz,PRL111,206802(2013)

Nadj-Perge,Drozdov,Bernevig,andYazdani,PRB88,020407(2013)

Pientka,Glazman,andv.Oppen,PRB89,180505(2014)

Nadj-Perge,etal.,Science346,602(2014)

S.Nadj-Perge,etal.,Science346,602(2014)

[Princetongroup]

�F � a

LocalizedmomentsIjembeddedinquasi1Dsuperconductor

MajoranaFermionsinself-tunableRKKYSystems



Interac9onbetweenlocalizedmomentIiatposi9onRiandi9nerantelectronspinσ

β<<EFècanintegrateoutelectronsandobtain...

�F � a

(� ⌘ J)


RKKYinterac9onbetweenmomentsIjgivenbysuscep9bilityof1Dsuperconductor:



i.e.helixatFermimomentum2kF

�F � a



HelixhasperiodofFermiwavelength

èpar9algap~ΔmopensatFermilevelμFwithouttuning!


helicalZeemanfieldseenbyelectrons(backac9on)


Helixopenspar9algapautoma4callyatFermienergyμF:


Wireintopologicalphaseif

Seealso,BrauneckerandSimon,PRL111,147202(2013)

VazifehandFranz,PRL111,206802(2013)





�F � a



Nuclearspinhelix&Majoranasin13CnanotubesChen-HsuanHsu,Stano,Klinovaja,andDL,PRBB92,235435(2015)

an9ferromagne9corderofnuclearspins

addsuperconduc9vityètopologicalregime

with2Majoranasateachendofthenanotube

Aexp~100μeV,T0~100mK,Δs~2K,ξMF~3μm

nuclearspinhelixopensgapsatEF,stronglyrenormalizedbye-einterac9ons:

13C




�F � a



Mono-AtomicFechainsonSuperconduc9ngPb-Surface:

Majoranasignature

mono-atomic zero-energymode

STM

Pawlaketal.,npjQuantumInforma9on(2016)2,16035

•  spin qubits in semiconductors •  topological quantum computing? ‘semi-super devices’

Front-Runners for Quantum Computers

‘small & fast’

‘exotic’ Majorana Para- or Fibonacci fermions?

semiconducting nanostructures è ‘topological spintronics’

Braiding of Majoranas for Topological QC

topologically protected: Hadamard and π/4 gate (Clifford gates) noisy: CNOT (entangling) and π/8 gate (non-Clifford)

Kitaev 2003

Universal QC with Majoranas via noisy gates, Bravyi, PRA 2008; Landau et al., PRL 2016; Vijay & Fu, arXiv:1609.00950; Karzig et al., arXiv:1610.05289 (Station Q)

Are there alternatives (non-braiding) to generate CNOT and π/8 gates ?

MajoranaSpinhybrid(`MaSh’)qubit

UniversalQuantumComputa9onwithHybridSpin-MajoranaQubits

oddparity

MFqubit

delocalizedfermion:

Hoffman,Schrade,Klinovaja,andDL,PRB94,045316(2016)

1 0 0 1

|0i |1i

spinqubit

0

1

degeneratestates

‘topologicalprotec9on’

But:parityfluctuatesatMHz-rateè‘Majoranaboxqubit’

Rainis&DL,PRB85,174533(2012);Marcusetal.,arxiv:1612.05748



oddparity

MFqubit

delocalizedfermion:


1 0 0 1

|0i |1i

spinqubit

0

1

degeneratestates

‘topologicalprotec9on’

performperturba9onexpansionintunnelingtbetweendotandwires…


hybridSWAPgate:

ècoherent‘quantumstatetransfer’(fasterthanprojec9vemeasurement)

butgateisnoisyèneedtocontrolitdownto<1%forsurfacecode



UniversalsetofgatesforMajoranaQubitsviaSpinQubit

Hoffmanetal.,PRB94,045316(2016)

MFqubit|0i |1i

spinqubit

Cliffordgates:{H,π/4,CNOT}

braiding(topological)gates

noisygates(need:noise<1%)

qubit-qubitcouplingviafloa9nggates

CNOT

2DSurfacecodebuiltfromtopologicalT-junc9ons

MajoranaSpinhybrid(MaSh)qubit=T-junc9onnanowirewithspin½dot

braiding(H&π/4),

topological

qubit-qubitcouplingviafloa9nggates

andspinexchangeJ(CNOT)

spinrota9on(π/8)

(non-Cliffordgate)

noisy


What is beyond Majorana femions?

`Parafermion’ (= fractional Majorana fermion zero-energy bound state)

Majorana (n=2): can get 2 out of 4 universal quantum gates by braiding Parafermion (n>2): 2 & weak entanglement*. But: CNOT? (no phase gate)

Fibonacci anyon: 4 universal

zero mode

*) D. Clarke, J. Alicea, and K. Shtengel, Nat. Commun. 4, 1348 (2013)

R. Mong, D. Clarke, J. Alicea, N. Lindner, P. Fendley, C. Nayak, Y. Oreg, A. Stern, E. Berg, K. Shtengel, and M. P. A. Fisher, Phys. Rev. X 4, 011036 (2014).

D. Clarke, J. Alicea, and K. Shtengel, Nat. Commun. 4, 1348 (2013)

N. Lindner, E. Berg, G. Refael, and A. Stern, Phys. Rev. X 2, 041002 (2012)

A. Vaezi, PRX 4, 031009 (2014) Y. Oreg, E. Sela, and A. Stern, Phys. Rev. B 89, 115402 (2014).

Parafermions from QHE edge states hybrid between QHE and SC

noB-field!unequalSOI

Double Rashba wire + superconductor

S

KlinovajaandDL,PRL112,246403(2014)&PRB90,045118(2014)


Hofstetter et al., Nat. 461, 960 (2009) Das et al., Nat. Comm. 3, 1165 (2012) Deacon et al., Nat. Comm. 6, 7446 (2015)

S

Double Rashba wire + superconductor KlinovajaandDL,PRL112,246403(2014)&PRB90,045118(2014)

‘Cooperpairsplifer’physics Recher, Sukhorukov & DL, PRB 63, 165314 (2001): quantum dots Recher&DL,PRB65,165327(2002): Luttinger wires

y

x

s-wave

superconductor

d

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

kFd

Eg/γ

•  CrossedAndreevpairing:spin-polarized

wires(quantumHalledgestates)

y

x

s-wave

superconductor

d

•  Direct pairing: decouple one wire

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

kFd

Eg/γ

Crossed Andreev Pairing vs. Direct Pairing

howtodis9nguishbetweendirectand

crossedAndreevcontribu9ons?

C. Reeg, J. Klinovaja and D. Loss, arXiv:1701.07107

Interplay Between Direct and Crossed Andreev

Eg(d) =� sinh(d/⇠s)

cosh(d/⇠s) + | cos kF d|

EDg (d) =

� sinh(2d/⇠s)

cosh(2d/⇠s)� cos 2kF d

ECg (d) =

2� sinh(d/⇠s)

cosh(2d/⇠s)� cos 2kF d| cos kF d|

Eg(d) = EDg (d)� EC

g (d)

Eg

EgD

EgC

0 20 40 60 80 1000.0

0.5

1.0

1.5

2.0

2.5

kFd

Eg/γ

Direct and crossed Andreev pairing interfere destructively

Proximity-induced gap is minimized when pairing is maximized

Comparing three gaps (weak coupling):

C. Reeg, J. Klinovaja and D. Loss, arXiv:1701.07107

Caveat:thestandardmethodofintegra9ng-outSCgiveswrongresultsingeneral!

Consider inter- and intra-wire pairing

‘crossedAndreev’termΔc


usualΔτ

è Two competing gap mechanisms

èKramerspairsofMajoranasincrossedAndreevdominatedregime

S

duetointerac9onsRecher&DL,

PRB65,165327(2002)

Double Rashba wire + superconductor KlinovajaandDL,PRL112,246403(2014)&PRB90,045118(2014)



Parafermions:needinterac9ons!

Consider inter- and intra-wire pairing plus e-e interactions (gB)

‘crossedAndreev’termΔc usualΔτ

4 Fermi points:



Parafermions:needinterac9ons!

Consider inter- and intra-wire pairing plus e-e interactions (gB)

‘crossedAndreev’termΔc usualΔτ

4 Fermi points:

Note:gapopensonlyduetointerac9ons

èfrac9onalizedexcita9ons

èparafermionsatboundaries

•  Parafermionsindoublewireemergeonlyduetostronginterac9ons;

withoutinterac9onsthesystemisgapless.

•  topologicalstabilityofgroundstate…?

Klinovaja,Mandal,Simon,DL

•  Fundamentallydifferentcase:startfromsystemwithpairinggapΔ

andthenaddweakinterac9onsgsuchthatgapdoesnotclose(g<<Δ)

èonlytopologicalphaseswithMajoranafermionsareallowed

Fidowsky&Kitaev2010/11

Turner,Berg&Pollmann2011

TopologicalStabilityofParafermionPhase?

Two Z3-Parafermions 3-fold degeneracy (fractional ‘charges’): 0, 2/3, 4/3 (mod 3)

N+2

2

4/3

2/3

Note:assumethatonlythespin-upKramerspartnercanbeoccupied

(andthespin-downKramerspartnerisalwaysempty)

0

2/3

4/3


N+2

2

4/3

2/3

Note:assumethatonlythespin-upKramerspartnercanbeoccupied

(andthespin-downKramerspartnerisalwaysempty)

0

2/3

4/3

N+2

2

4/3

2/3

usethe3degeneratestatesasaqu-trit(3-levelsystem)


0

2/3

4/3

|0 >L= |0; 0 >, |1 >L= |2/3; 4/3 >, |2 >L= |4/3; 2/3 >4PFs=1qutrit:

KramersPairsofMajoranaFermionsandParafermions

inFrac4onalTopologicalInsulators

Boundstatesatinterfacesbetweenregionsof

dominantcrossedAndreevandusualsuperconduc9ngpairingterms

topological

insulator

superconductor

Klinovaja,Yacoby,andDL,PRB90,155447(2014)

Braiding of Zd Parafermions for TQC

CNOTgateCX:

Hutter and DL, PRB 93, 125105 (2016)

Ford=3weneedS2,i.e.2x12=24braidingsforoneCNOTgate

4PFs=1qudit

[SeealsoproofbyPFdiagramma9cs,Jaffe,Liu,andWozniakowski,arXiv:1602.02671]

(missingnon-CliffordT-gatefordoddwell-studied)

|0 >L= |0; 0 >, |1 >L= |2/3; 4/3 >, |2 >L= |4/3; 2/3 >

e.g.d=3inparity-0sector,thelogicalqutritbecomes:

dodd

•  Topologicalquantumcompu9ng

•  Majoranafermionsin‘super-semi’nanowiresandchains

•  Majoranaandspinqubitsè2Dsurfacecode

•  Nextgenera9on:Parafermionsindoublenanowires

ècangetnearlyuniversalTQCincludingCNOT

Summary

Hutter and DL, PRB 93, 125105 (2016) Klinovaja and DL, PRB 90, 045118 (2014)

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III. From Majorana to Parafermions - Capri SchoolIII. From Majorana to Parafermions In collabora9on...

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