milestones toward majorana-based quantum...

Milestones toward Majorana-based quantum computing

Jason Alicea (Caltech, IQIM, & Walter Burke Institute)

Majorana-based quantum computing

“Majorana mountain”Device fabrication

Majorana signatures

Exponential degeneracy

Albrecht et al., Nature 531, 206 (2016)

Dave Aasen (current KITP grad fellow)

Michael Hell Ryan Mishmash Andrew Higginbotham

Jeroen Danon Martin Leijnse Thomas Jespersen Josh Folk Charlie Marcus Karsten Flensberg

Physical Review X 6, 031016 (2016)

(Ising) non-Abelian anyonsExcited states

Gap

Ground state with no anyons

Ground states w/anyons present

Excited states

Gap

(Ising) non-Abelian anyons


Excited states

Gap

But what about, say, free spin-1/2’s?

Gap



Excited states

Gap


Gap

???



Excited states

Gap


B



Excited states

Gap


B

B


Excited states

GapLocally indistinguishable ground states

Majorana zero modes encode degeneracy


Locally indistinguishable ground states

Excited states

Gap


�i ! Uij�j


Excited states

Gap


�i ! Uij�j�i ! Uij�j

Non-Abelian statistics


Excited states

Gap

Non-Abelian statistics Nontrivial fusion rules


� ⇥ � = I + �i ! Uij�j�i ! Uij�j

I


Excited states

Gap



� ⇥ � = I + �i ! Uij�j�i ! Uij�j

I

Barkeshli, Bonderson, Cheng, and Wang, arXiv:1410.4540; Rowell and Wang, PRA 93, 030102(R)


Excited states

Gap

‘$3 million idea’: fault-tolerant topological quantum computation!

Alexei Kitaev

|000i, |001i, . . .

Quantum gates Readout

Qubits




� ⇥ � = I + I


Excited states

Gap

‘$3 million idea’: fault-tolerant topological quantum computation!

Billion $ question: how to build the

hardware?Alexei Kitaev

|000i, |001i, . . .Qubits


Quantum gates Readout



� ⇥ � = I + I

Inception (late 80’s) Moore & Seiberg, Witten, …


Realizations(1991-2000’s)

γ1 γ2

Kitaev (2001)

1D p-wave SC

e/4

�

Ising anyon

Moore-Read state

�

h/2e vortex

Moore & Read (1991)

2D p+ip SC

Read & Green (2000)



Design era(2008-2012)

γ1 γ2

Kitaev (2001)

1D p-wave SC

e/4

�

Ising anyon

Moore-Read state

�

h/2e vortex

Moore & Read (1991)

2D p+ip SC

Read & Green (2000)

3D TI

s-wave SC

Fu & Kane (2008)Lutchyn et al., Oreg et al. (2010)

(many others)




Fabrication & characterization era (2012-present)

γ1 γ2

Kitaev (2001)

1D p-wave SC

e/4

�

Ising anyon

Moore-Read state

�

h/2e vortex

Moore & Read (1991)

2D p+ip SC

Read & Green (2000)

3D TI

s-wave SC


(many others)

Mourik et al. (2012)(many others)


Marcus et al. Nadj-Perge et al.




Fabrication & characterization era (2012-present)

Majorana control era (coming soon?)

γ1 γ2

Kitaev (2001)

1D p-wave SC

e/4

�

Ising anyon

Moore-Read state

�

h/2e vortex

Moore & Read (1991)

2D p+ip SC

Read & Green (2000)

3D TI

s-wave SC


(many others)

Mourik et al. (2012)(many others)


Marcus et al. Nadj-Perge et al.

1. New all-electrical Majorana control scheme

1I. Majorana-control milestones

Fusion-rule detection

Topological qubit validation

Braiding

�1 �2

� ⇥ � = I +


Goal: leverage quantum dot/spin qubit tools for experiments below

Related to T. W. Larsen, K. D. Petersson, F. Kuemmeth, T. S. Jespersen, P. Krogstrup, J. Nygard, and C. M. Marcus, PRL 115, 127001 (2015); G. de Lange, B. van Heck, A. Bruno, D. J. van Woerkom, A. Geresdi, S. R. Plissard, E. P. A. M. Bakkers, A. R. Akhmerov, and L. DiCarlo, PRL 115, 127002 (2015); B. van Heck, A. R. Akhmerov, F. Hassler, M. Burrello, and C. W. J. Beenakker, New Journal of Physics 14, 035019 (2012); T. Hyart, B. van Heck, I. C. Fulga, M. Burrello, A. R. Akhmerov, and C. W. J. Beenakker, Phys. Rev. B 88, 035121 (2013).




Braiding

�1 �2

� ⇥ � = I +

Majorana control scheme: setup

Vg

nanowire (e.g, InAs)

Vg


mesoscopic superconducting island (e.g., Al)

island gate


Vg



island gate

Magnetic field needed for Majoranas

B


Lutchyn, Sau, Das Sarma PRL 105, 077001 (2010); Oreg, Refael, von Oppen PRL 105, 177002 (2010)

Vg



island gate

Magnetic field needed for Majoranas

B

trivial bulk superconductor


Vg

BEC

H = HJ [�̂] + EC(n̂� no↵set

)2

EJ


Vg

BEC

EJ


)2

gate-tunable ‘valve’


Vg

BEC

EJ


)2

gate-tunable ‘valve’

�2�1


Vg Vg Vg

�2�1

Initialization

Readout

Parity-to-charge conversion

Charge eigenstates

Parity eigenstates

Multi-island extension

SC island SC islandBulkSC

BulkSC

Double dot



BulkSC

Double dot

Single dot



BulkSC

Double dot

Single dot

Topological superconductor

�1 �4



BulkSC

Double dot

Single dot


Double topological superconductor

�1 �4�2 �3



BulkSC

Double dot

Single dot



�1 �4�2 �3

Suffice for fusion rules, topological qubit validation.



BulkSC

Double dot

Single dot



�1 �4�2 �3

Readily extends to networks (braiding).

Suffice for fusion rules, topological qubit validation.





Braiding

�1 �2

� ⇥ � = I +

Related to W. Bishara, P. Bonderson, C. Nayak, K. Shtengel, and J. K. Slingerland, PRB 80, 155303 (2009); JA, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat. Phys. 7, 412 (2011); J. Ruhman, E. Berg, and E. Altman, PRL 114, 100401 (2015).

Fusion-rule protocol

SC islandBulkSC

BulkSCSC island

Control (boring)


SC islandBulkSC

BulkSCSC island

|QL, QRi

Control (boring)


SC islandBulkSC

BulkSCSC island

�1 �4�2 �3

|QL, QRi

Control (boring)


SC islandBulkSC

BulkSCSC island

�1 �4�2 �3

� ⇥ � = I +

|QL, QRi

|QL, QRi

Deterministic charge measurement

Control (boring)

Fuse


SC islandBulkSC

BulkSCSC island

�1 �4�2 �3

� ⇥ � = I +

�1 �4|QL, QRi

|QL, QRi


Control (boring) Nontrivial

Fuse


SC islandBulkSC

BulkSCSC island

�1 �4�2 �3

� ⇥ � = I +

�1 �4�2 �3

�1 �4|QL, QRi

|QL, QRi

Islands entangled!



Fuse


SC islandBulkSC

BulkSCSC island

�1 �4�2 �3

� ⇥ � = I +

�1 �4�2 �3

�1 �4|QL, QRi

|QL, QRi 1p2(|QL, QRi+ |QL � 1, QR + 1i)

Islands entangled!


� ⇥ � = I + Probabilistic charge measurement!


Fuse Fuse


SC islandBulkSC

BulkSCSC island

�1 �4�2 �3

� ⇥ � = I +

�1 �4�2 �3

�1 �4|QL, QRi

|QL, QRi 1p2(|QL, QRi+ |QL � 1, QR + 1i)

Islands entangled!



Fuse Fuse

Comments1. Experiment is “non-Abelian”2. Control distinguishes from probabilistic outcome due to noise3. Charge sensing unnecessary—can use “Majorana-mediated pump”!

� ⇥ � = I + Probabilistic charge measurement!





Braiding

�1 �2

� ⇥ � = I +

Prototype topological qubit

�1 �4�2 �3 Locally indistinguishable ground states

Excited states

Gap

|0i |1i



Excited states

Gap

|0i |1i

Topological protection requires 1. Time scales << poisoning time2. System confined to ground states w/high probability (i.e., low T, low frequency noise)



Excited states

Gap

|0i |1i


Fundamental question: Assuming these hold, how to verify topological protection in above device??



Excited states

Gap

|0i |1i



“Accidental” degeneracies

Excited states

Gap

|0i |1iConventional zero-energy bound states



Excited states

Gap

|0i |1i


“Accidental” degeneracies

Excited states

Gap

|0i |1iConventional zero-energy bound states

Answer must sharply distinguish these qubits!

Peculiar noise sensitivity

�1 �4�2 �3 |0i|1i

!0 / cos(L)e�L/⇠

!0

µ

e�L/⇠(µ)

chem pot

�1 �4�2 �3 |0i|1i

!0 / cos(L)e�L/⇠

Time-averaged quantities & fluctuations deeply linked for topological qubit!

!0

µ

e�L/⇠(µ)

chem pot


�1 �4�2 �3 |0i|1i

!0 / cos(L)e�L/⇠

!0

µ

e�L/⇠(µ)

chem pot

!0 ⇠ �Etypical

T2 ⇠ ~/�Etypical

1/T2 ⇠ !0



�1 �4�2 �3 |0i|1i

!0 / cos(L)e�L/⇠

!0

µ

e�L/⇠(µ)

chem pot

µchem pot

1/T2

!0 ⇠ �Etypical

T2 ⇠ ~/�Etypical

1/T2 ⇠ !0



Measuring oscillation frequency & T2

!0

µ

e�L/⇠(µ)

chem pot

µchem pot

1/T2

SC SCSC SC

Initialize double dot

Evolution on Bloch sphere:

Unitary evolution for time t

Readout

Initialize qubit into |0i = |012, 034i

�1 �2 �3 �4

Apply pulse⇡/2

Apply pulse⇡/2

|0i

1p2(|0i+ i|1i)

1p2

⇣|0i+ iei��(t)|1i

⌘

↵|0i+ �|1i

P (t) = |�|2






Braiding

�1 �2

� ⇥ � = I +

Braiding protocol

|QL, QRi

Initialize

Braiding protocol

�1 �4�2 �3

|QL, QRi |0i

Initialize

Braiding protocol

�1 �4�3

�2

|QL, QRi |0i

Initialize

Braiding protocol

�1 �4

�2

�3

|QL, QRi |0i

Initialize

Braiding protocol

�1 �4�3 �2

|QL, QRi |0i

Initialize

|0i+ i|1ip2



Braiding

�1 �2

� ⇥ � = I +

Milestones for the Majorana control era



Braiding

�1 �2

� ⇥ � = I +

Realizable with single-wire devices! (May generalize to other platforms/schemes)




Braiding

�1 �2

� ⇥ � = I +


Reveals defining property of non-Abelian anyons;

natural braiding precursor




Braiding

�1 �2

� ⇥ � = I +


Verify basic tenets of topological quantum computing

All protocols very difficult to mimic in non-topological setups.


Reveals defining property of non-Abelian anyons;

natural braiding precursor

milestones toward majorana-based quantum...

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