Topological quantum computation

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Quantum computation in lab Topological quantum computation Search for Ising anyons Experimental realizations

Topological quantum computation with Non Abelian Anyons

Pavithran Iyer, Matrise En Physique,

Superviseur: David Poulin, Universite de Sherbrooke

Term project for PHY889, December 13th, 2013

Pavithran Iyer TQC, Majorana Fermions


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Contents of this talk

1 Quantum computation in lab

2 Topological quantum computation

3 Search for Ising anyons

4 Experimental realizations

5 Conclusions



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What is quantum computation ?

Quantum Computation: Information: quantum states, manipulations: unitary operato

nqubitQuantum state: A state | Hn | =|000 . . . 0 + |11 1

Operations Quantum gates: All matrices in SU(n). (Unitary evolutions of stored



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Example: n= 1 | = |0 + |1 and gates are

X=

0 1

1 0

, Y =

0 i

i 0

, Z=

1 0

0 1

Actions on a qubit state: X(|0 + |1) =|1 + |0 , Z(|0 + |1) = (|0


Q i i l b T l i l i S h f I i E i l li i


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Example: n= 1 | = |0 + |1 and gates are

X=

0 1

1 0

, Y =

0 i

i 0

, Z=

1 0

0 1

Actions on a qubit state: X(|0 + |1) =|1 + |0 , Z(|0 + |1) = (|0

Until Now: Many body systems, Quantum Dots, Ion Traps, Neutral Atoms, etc.

Concerns: No quantum system is isolated decoherence time computation time

Perfect quantum gates Error correction, concatenation, thresholds,etc.


Q t t ti i l b T l i l t t ti S h f I i E i t l li ti


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5 Conclusions




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Encoding information in non local properties

Encode information in a pair of spin1/2 systems two spin 1/2 havetotal spin 1 or

Fusingtwo spin half: 12 12 = 1 + 0 Fused state is| =|

12

12 = 0 + |

12

12




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12

12 = 0 + |

12

12

Locally 12(X1l 1lX), then a total spin 1 0. (data destroyed, irrespective of distan




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12

12 = 0 + |

12

12

Locally 12(X1l 1lX), then a total spin 1 0. (data destroyed, irrespective of distan

Separation adds stability of states . . . Nontrivial operation must be only exchanges. . .

But fermions, bosons arent good candidates trivial exchange need richer statisti




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A richer statistics

Total spin is not affected by exchanging general degree of freedom: Topological Cha




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Q p p g q p g y p

A richer statistics


Take an analogous fusion rule: = 1 + f. (, 1vac, f are some topological charg

Total Topological charge of a two-anyon state is onlyaltered by a physical exchang




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p p g q p g y p

A richer statistics




Particles with this kind of a freedom are called Ising Anyons. (Non-Abelian any

(Fusion state) of two anyons: | =| = 1vac + | =f. (exactly a qu




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g g

A richer statistics




Particles with this kind of a freedom are called Ising Anyons. (Non-Abelian any

(Fusion state) of two anyons: | =| = 1vac + | =f. (exactly a qu

How does charge of two anyons 1, 2 change ? (Total spin S1 S2, Total Charge

On exchanging12 we will affect the total charge of the system: 1 2 and2 Analogue in the spin language: exchange two spin half total spingoesfrom0to




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Topological quantum computation using Ising Anyons

Specific example: 2n anyon (a1, a2), . . . , (a2n1, a2n) where a a= 1vac+f (f: ferm

Fuseall anyons a state with 0 or1 or2 or . . . or n fermions ! (nqubit fusion sFusion state superposition of2n states |00 0 = 1vac, |11 11 =f1 fn.




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Transformation on states on exchange second quantization: (a1, a2) =c1c21vac.

Exchange (1, 2) : c1 c2, c2 c1. Ui=1

2(1l cici+1).




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Transformation on states on exchange second quantization: (a1, a2) =c1c21vac.

Exchange (1, 2) : c1 c2, c2 c1. Ui=1

2(1l cici+1).

1qubit gates: X, Y , Zconstructed by performing U1, U2, U3, . . .0 1

1 0

0 i

i 0

1 0

0 1

i 1

i 1

Ising anyons: model simulates all 1qubit clifford gates by exchanges ! (not others)




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5 Conclusions




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Kitaevs toy hamiltonian

Are Ising anyons real particles ? What hamiltonians support it ?




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H= i

aiai

1

2

t(aiai+1+ a

i+1ai) + aiai+1+

ai+1ai

{ai , ai} fermion creation/annhilation, t: hopping, = ||ei : superconducting pa




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H= i

aiai

1

2

t(aiai+1+ a

i+1ai) + aiai+1+

ai+1ai


Remove the Xtype errors spinless, small hopping, Ztype write aiai cjcj+




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H= i

aiai

1

2

t(aiai+1+ a

i+1ai) + aiai+1+

ai+1ai



Recall Bogoliubouv transformations: phase errors hoppings make hopping small:

aj =c2j1+ ic2j and ai =c2j1 ic2j , where (c

j =cj , {ck, cj} kj)

H= i

2 jc

2j1c2j+ (t + )c2jc2j+1+ (t + )c2j1c2j+2




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H= i

aiai

1

2

t(aiai+1+ a

i+1ai) + aiai+1+

ai+1ai



Recall Bogoliubouv transformations: phase errors hoppings make hopping small:

aj =c2j1+ ic2j and ai =c2j1 ic2j , where (c

j =cj , {ck, cj} kj)

H= i

2 jc

2j1c2j+ (t + )c2jc2j+1+ (t + )c2j1c2j+2

What have we done physically with the transformation ? Where are Ising anyons ?




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Interesting physics at the boudaries

Interesting limits ofH= i

2 j[c2j1c2j+ (t + )c2jc2j+1+ (t + )c2j1c2j+2 =t = 0,


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2 j[c2j1c2j+ (t + )c2jc2j+1+ (t + )c2j1c2j+2 =t = 0, 0, = 0

H= i2

n1j=1

c2jc2j+1




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2 j[c2j1c2j+ (t + )c2jc2j+1+ (t + )c2j1c2j+2 =t = 0, 0, = 0

H= i2

n1j=1

c2jc2j+1

Almost the same but c1, c2n neve

Zero energy irrespective of occupan

1, 2n sites 2 degenerate ground st

Generally, ground state: unpairedMajorana fermions if|| 2




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Encoding Majorana fermions

Majorana modes are well separated Hint enc1c2nimmune to any kind of er

Majorana fermions obey the same rules as Ising Anyons: [1]




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Majorana modes are well separated Hint en

c1c2nimmune to any kind of er


2 Majorana fermionsfuseone or no fermions:

Majorana transformation: ic1c2= (1 2n)

ic1c2 +1: no fermion, 1 : one fermion




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Majorana modes are well separated Hint en

c1c2nimmune to any kind of er


2 Majorana fermionsfuseone or no fermions:

Majorana transformation: ic1c2= (1 2n)

ic1c2 +1: no fermion, 1 : one fermion

Exchange1 and 2: c1 c2 and c2

Little tricky: superconducting phase

Equivalent to multiplying the H by

c1 c2 and c2 c1.

Quantum/Thermal fluctuations spurious spontaneous anyon-pair creation suppr

heavily: mass gap and at low temperatures.




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3

Search for Ising anyons


5 Conclusions




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Engineering real systems to see Majorana modes

Can real systems host Majorana fermions ? Any system in a lab be have HKitaev ? [2]

Basic 1D semiconducting wire: H0=

={,}

dr(r)

p2

2m wire V(r)

(r

Standard procedure1: (r) =

i W(r r0)ai , H=

i

effa

iai ta

iai+1+h.c

Semiconducting wire:

H,0 (k) =

H0(k) 00 H0(k)

H=0 (k) =

k

ck ck

eff 2t cos k 00 eff 2t cos k

ckck

Gap = 2t + eff

Bands: spin upand s

1eff=wire

i,j

drV(r)Wi(rr0)Wj(r r0), t=

i,j

drWi(rr0)(2d2r

2m )Wj(rr0).




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={,}

dr(r)

p2

2m wire V(r)

(r


i W(r r0)ai , H=

i

effa

iai ta

iai+1+h.c

Semiconducting wire: Bfield

H=(k) =k

ck ck

H0(k) + z 00 H0(k) + z

ckck

Gap Bz/2 eff

Bands: spin upand s

1eff=wire

i,j


i,j

drWi(rr0)(2d2r

2m )Wj(rr0).




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={,}

dr(r)

p2

2m wire V(r)

(r


i W(r r0)ai , H=

i

effa

iai ta

iai+1+h.c

Semiconducting wire: Bfield, swave pairing

H=(k) =k

ck ck

H0(k) + Bz2 z 1l1l H0(k) +

Bz2 z

ckck

Gap Bz/2

2

eff+ ||2

Bands: spin upand s

1eff=wire

i,j


i,j

drWi(rr0)(2d2r

2m )Wj(rr0).




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Experimental topological quantum computation

A long wire with different sections at different eff gates locally tune V(r) since[1]

eff=wire i,j

drV(r)Wi(r r0)Wj(r r0) and we need |eff|


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eff=wire i,j



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eff=wire i,j



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5 Conclusions




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What have we seen ?

In this talk we learnt:

1 What are anyons ? What is the idea and need for topological quantum computat

2 An example of a simple Hamiltonian hosting zero energy quasiparticles that are an

3 Real systems hosting anyons and outlining quantum computation operations




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What have we seen ?

In this talk we learnt:

1 What are anyons ? What is the idea and need for topological quantum computat

2 An example of a simple Hamiltonian hosting zero energy quasiparticles that are an

3 Real systems hosting anyons and outlining quantum computation operations

In this talk we did not learn:

1 Readout a qubit measurements to probe existence of zero modes on the chain

2 Universal computation, Fault tolerant (Topological quantum error correction) sch

3 Other systems host Majorana fermions 2D Hkitaev, 5/2 quantum hall state, etc

4 Different types of anyons some can be used for universal quantum computation




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References I

Jason Alicea.

New directions in the pursuit of majorana fermions in solid state systems.

Reports on Progress in Physics, 75(7):076501, 2012.

CWJ Beenakker.

Search for majorana fermions in superconductors.

arXiv preprint arXiv:1112.1950, 2011.

Sergey Bravyi.

Universal quantum computation with the = 5/ 2 fractional quantum hall state.

Physical Review A, 73(4):042313, 2006.




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References II

A Yu Kitaev.

Unpaired majorana fermions in quantum wires.

Physics-Uspekhi, 44(10S):131, 2001.

A Yu Kitaev.

Fault-tolerant quantum computation by anyons.

Annals of Physics, 303(1):230, 2003.

John Preskil.

Topological quantum computation.

Lecture Notes for Physics 219: Quantum Computation, June 2004.




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Thank you


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Topological quantum computation