Anyons and topological quantum computing · PDF fileAnyons and topological quantum computing...

Anyons and topological quantum computing

Francesca Pietracaprina

Statistical Physics PhD Course

Quantum statistical physics and Field theory

05/10/2012

Francesca Pietracaprina Topological quantum computation

Plan of the seminar

Why anyons?

Anyons:

definitions

fusion rules, F and R matrices

Summary of quantum computation

Topological quantum computation and example models:

Toric code

Honeycomb model

Universal topological quantum computation: compiling with

Fibonacci anyons

Conclusions


Why anyons and topology?

For quantum computing we need: qbits and a

way to operate.

|ξ〉 = a|0〉+ b|1〉 → U|ξ〉

Qbits any sistem which has two degenerate states;

Gates done by applying unitary operators on states.

Errors come from decoherence and inexact application of U.

Error correction can in theory be done, but not in practice.


Why anyons and topology?

Another approach: topology protects the qbits.

For Topological Quantum Computation:

qbit → anyons (excitations of the system)

unitary operator → braiding of the anyons

measurement → fusion of the anyons


Anyons

Take two identical particles and exchange them two times:

in 3D it is equivalent to the identity → ± phase on exchange;

action of the permutation group (finite, abelian);

Two types of particles: bosons and fermions.

in 2D γ1 and γ2 involve a passing through → nontrivial phase;

action of the braid group (infinite, non abelian);

Anyons.


Anyons: some properties

Fusion:

A “new type”of anyon is formed by putting together two anyons:

φa × φb =∑c

Ncabφc , Ncab = 0, 1

Abelian anyons: only one fusion channel is available.

Connection with conformal field theories:

This is the purely algebric part of an OPE.

Example: Ising CFT

12

116 0

0 116

12

σ × σ = 1 + ε, ε× σ = σ, ε× ε = 1

→ Ising anyons



Example: Tricritical Ising CFT

32

610

110 0

716

380

380

716

0 110

610

32

Notation:

even: φ21 = ε, φ31 = t, φ41 = ε′′

odd: φ12 = σ′, φ22 = σ

t × t = 1 + tε× ε = 1 + tε× t = ε+ ε′′

ε× σ′ = σ

ε× σ = σ′ + σt × σ′ = σ

t × σ = σ′ + σ

σ′ × σ′ = 1 + ε′′

σ′ × ε′′ = σ′

ε′′ × ε′′ = 1

ε′′ × ε = tε′′ × t = ε

ε′′ × σ = σ

σ′ × σ = ε+ t

σ × σ = 1 + ε+ t + ε′′

Subalgebra of Fibonacci anyons

Ising anyons



R–matrices and F–matrices:

Non abelian anyons→ higher–dim representations of Bn: R–matrix

F–matrix: relates the ways three anyons can fuse into a fourth.

Consistency conditions: pentagon and hexagon equations



Action of braid group on an “ordered basis” is specified by:

- R–matrix

- B–matrix: B = F−1RF

Example: Fibonacci anyons: rich enough for universal computing

Anyons: 1, σ

Fusion rules: 1× 1 = 1, σ × 1 = σ,

σ × σ = 1 + σ

F–matrices: F 1σbσσa = δbσδσa

Fσσσσ =

(φ−1 φ−

12

φ−12 −φ−1

)

R–matrix: R =

(e4πı5 0

0 e−2πı5

)


Quantum computation

Steps of a quantum computation:

- initialization with a state |ψ〉 of one or more qbits

- unitary operation U|ψ〉- measurement of the state

Program ↔ unitary operator = building the quantum circuit.

We can build any U with few 1- and 2-qbit gates.

- CNOT =

1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

- Hadamard = 1√

2

(1 1

1 −1

)

- π8 gate =

(1 0

0 eıπ4

)


An example: the adder

Quantum adder:


Topological Quantum computation: a model

Let us consider a simple model: Toric code

Lattice of 12 spins.

H = −Je∑s

As − Jm∑p

Bp

As =∏

j∈star(s)

σxj

Bp =∏

j∈plaquette(p)

σzj

As ,Bp commute.



- on a torus, 4–times degenerate ground state

- two kinds of excitations, with gap:

two Je contributions at the extremes of l

W(e)l =

∏j∈l σ

zj

two Jm contributions at the extremes of l∗ (dual lattice)

W(m)l∗ =

∏j∈l∗ σ

xj

- W (e) and W (m) create no excitations for closed loops



Excitations in this model are (abelian) anyons:

braiding e particles → bosons (no phase)

braiding m particles → bosons (no phase)

braiding couples ε = e–m → fermions (−1 phase)

Fusion rules: e × e = 1 m ×m = 1 ε× ε = 1

e ×m = ε e × ε = m m × ε = e


Topological Quantum computation

The goal: universal topological quantum computation.

- Degenerate ground states with quasiparticles at fixed points

- A qbit is made from quasiparticles → non–local

- Discrete operations: result depends only from topology of

trajectory of braid

Input Initialize the qbit by creating anyons in a specific way

(e.g. |0〉 by pulling from vacuum)

Operation Physically moving and braiding anyons

Output Measurement of the state of the qbit


Topological Quantum computation: another model

H = −Jx∑x links

σxj σxk − Jy

∑y links

σyj σyk − Jz

∑z links

σzj σzk

Honeycomb model: two phases

- Jx , Jy , Jz satisfy some conditions plus a

perturbation: non abelian anyons

- otherwise: abelian anyons, same as toric code


The honeycomb model: sketch of the solution

Introduce four Majorana fermions for each lattice site c , bα

H → H =ı

4

∑〈j ,k〉

Ajkcjck

Ajk = 2Jαjkujk , ujk = ıbαjkj b

αjkk

having substituted σα → ıbαc .

Note: [H, ujk ] = 0 → L =⊕L{u}

in the subspace L{u}, H{u} is free fermions.

Introduce for each plaquette p the hexagon vortex operators

Wp = σx1σy2σz3σx4σy5σz6 →

∏〈j ,k〉∈p

ujk

[Wp,H] = 0→ L =⊕L{w}

Then L{w} = PL L{u}, PL projector on the physical space.


The honeycomb model: sketch of the solution

Results:

Minimum energy: state with wp = 1

∀p (no vortices)

The spectrum is given by

ε(q) = ± |2(Jx eıq·n1 + Jy eıq·n2 + Jz )|

Jx , Jy , Jz satisfy triangle inequalities:

gapless spectrum → phase B

For other values of Jx , Jy , Jz :

gapped spectrum → phase Aα


The honeycomb model

In the gapless phase:

Particle types: fermions (gapless), vortices (gapped).

A gap appears if we apply a perturbation: −∑

α=x ,y ,zj

hασαj

Results:

a gap (∝ hxhyhzJ2

) opens for the fermions

hexagon operators Wp are no longer conserved

→ vortices can hop between hexagons

Anyon types: 1, ε (fermion), σ (vortex)

Fusion rules: ε× ε = 1 ε× σ = σ σ × σ = 1 + ε

Ising anyons


Universal computation

With Fibonacci anyons we can build a universal quantum

computer:

- Qbit encoding: one qbit ↔ four anyons 1234 (excitation);

|0〉 → fusion of 12 gives 1, |1〉 → fusion of 12 gives σ

Equivalently: three anyons (with one inadmissible state)

- Measurement: fusing the couples of anyons → observe

vacuum or not;


Universal computation: universal gates

- Unitary operations: implement 1– and 2– qbit gates

- braid matrices: R and B = F σσσσ R (F σσσσ )−1

- on one qbit: restrict on weaves (only one anyon moves)

σ1 =

e4πı5 0

0 e−2πı5

e−2πı5

, σ2 =

−φ−1 eπı5 φ−

12 e

3πı5

φ−12 e

3πı5 −φ−1

e−3πı5

A generic weave is U({ni}) = σnm1 σ

nm−12 . . . σn22 σ

n11 , with

ni = ±2,±4 or 0 for the extremes.

Search for the weave which approximate best U:

brute force searchbest accuracy

or icosahedral group hashingbest time

.


Icosahedral group hashing

Largest finite subgroup of SU(2) → we use it to do iteratively

better approximations.

Icosahedral group: I = {g0 = 1, g1, . . . , g59}.1 brute force search (length L) for I(L) = {g0, g1, . . . , g59}2 construct 1 = gi1 . . . gingin+1 , with gn+1 = [gi1 . . . gin ]

−1;

for the approximate rotations, S(L, n) = {gi1 . . . gin gin+1}{ik} is

a mesh of fine rotations around 1

3 search for approximation of U with minimum distance:

- initial approximation: search for best U0 in I(L0)m- k−th iteration: search for best correction V ∈ S(Lk , n) so that

Uk = Uk−1V has minimum distance from U;

- iterate with bigger Lk .

Note: Lk are chosen so that error of 1 ∼ error of Uk .



- for two qbits:

controlled gates → one control qbit.

action of control pair:

{1 vacuum→ 1

σ one anyon

Let’s build a controlled gate:1 injection of the control pair of the control qbit in the

controlled qbit;

(found by searching for σ2U({ni})σ2 = 1)

2 one qbit operation. Example: CNOT → σx ;

3 inverse injection of the control pair.



It can also be done with one anyon instead of a couple:

Swap of one anyon: action of F–matrix. U({ni})σ2 = −F

Summing up: for a controlled gate

with control pair:

Injection + U + Ejection

with control anyon:

F + U + F−1

All ingredients to build any circuit up to arbitrary precision.


Conclusions

In 2D: anyons, characterised by fusion rules, braidings;

It is possible to do topologically protected quantum

computation;

For some types of anyons: universal quantum computation;

Proposed models: lattice surface codes;

It is possible to efficiently compile a quantum circuit into

braidings.


Thank you for your attention

References:

1 Preskill, Topological quantum computation (lecture notes)

2 Nielsen, Chuang Quantum computation and quantum information, Cambridge

University Press

3 Brennen, Pachos, Why should anyone care about computing with anyons?, Proc.

R. Soc. A 2008 464, 1-24

4 Das Sarma, Freedman, Nayak, Simon, Stern, Non–Abelian anyons and topological

quantum computation, ArXiv 0707.1889v1

5 Trebst, Troyer, Wang, Ludwig, A short introduction to Fibonacci anyon models,

Progress of Theoretical Physics 176, 2008

6 Kitaev, Laumann, Topological phases and quantum computation, ArXiv

0904.2771v1

7 Kitaev, Anyons in an exactly solved model and beyond, ArXiv

cond-mat/0506438v3

8 Hormozi, Zikos, Bonesteel, Simon, Topological quantum compiling, Phys. Rev. B,

75, 165310 (2007)

9 Burrello, Mussardo, Wan, Topological quantum gate construction by iterative

pseudogroup hashing, New J. Phys. 13, 025023 (2011)


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