FQH-Based Topological Quantum Computer: Materials, Devices ...

IAS-ICTP School on Quantum Information P

FQH-Based Topologica
Materials, Devic

Xin

Zhejiang

xinwan@
http://zimp.zju.edu.cn/

Processing, Singapore, January 18-29, 2016

al Quantum Computer:
ces & Algorithms

Wan

University

@zju.edu.cn
/~xinwan/Singapore2016/

Model of

• A model of anyons is a theory of a
two-dimensional medium with a m
gap, where the particles carry local
conserved charges. One defines

– A finite label set {a,b,c,…};
– The fusion rules a  b = c Nabc c;
– The F-matrix (expressing associativi
– The R-matrix (braiding rules).

F & R satisfy self-consistency equa
known as the pentagon and hexago
equations.

f Anyons

mass Ising anyon model:
lly
{1, s, y}
ity of fusion);
sxs=1+y
ations, yxy=1
on yxs=sxy=s

( )F=1 1 1
1 −1
√2

( )R=e−i π / 8 0
0 e3iπ /8

Diagr

grams

Spin and

Statistics

Initialize

Das Sarma, Freedm

e Anyons

 = 5/2

man & Nayak (2005)

Four Ising Any

● Even when one fixes the location
quasiholes, there are more than on

(Ψ(12)(34)= Pf ( zi−ξ1)( zi−ξ2)( z j−ξ3)
zi−z j

(Ψ(13)(24)= Pf ( zi−ξ1)( zi−ξ3)( z j−ξ2)
zi− z j

(Ψ(14)(23)= Pf ( zi−ξ1)( zi−ξ4)( z j−ξ2)
zi−z j

● But they are not linearly independ

Ψ(12)(34)−Ψ(13)(24)=(1− x ) ( Ψ(12)(34)−

yons as a Qubit

of all 12 34
ne states

) ∏( z j−ξ4)+i ⇔ j ( zi− z j )2
1⩽i< j⩽ N

) ∏( z j−ξ4)+i⇔ j ( zi− z j )2
1⩽i < j⩽ N

) ∏)( z j−ξ3)+i⇔ j ( zi− z j )2
1⩽i< j⩽ N

dent! x = ( ξ1− ξ2)( ξ3− ξ4 )
( ξ1− ξ3)( ξ2− ξ4 )
− Ψ (14 )(23 ) )

Four Ising Any

● Ansatz wavefunction (decompositio
wavefunctions)
Ψ(0,1)( ξ1, ξ2, ξ3, ξ4 ; z1, ... , z N ) =
+

C. Nayak and F. Wilczek, Nucl. Ph
E. Ardonne and K. Schoutens, Ann

|0⟩=|(⋅⋅)0(⋅⋅)0⟩0
|1⟩=|(⋅⋅)1(⋅⋅)1⟩0

Ising: • = s, 0 = 1, 1 = y

yons as a Qubit

on into two quasihole-paring

A(0,1)({ξ}) Ψ(12)(34)({ξ}, {z })
B(0,1)({ξ}) Ψ(13)(24)({ξ}, {z })

hys. B 479 (1996) 529 s
n. Phys. 322 (2007) 201

s ss

I/y I/y
I

Identify the Two

● The two linearly independent wav

[( ξ1 −ξ3 )( ξ2 −ξ 4 ) ]1 / 4

(1±√ 1− x)1/2
(Ψ±= √Ψ(13)(24)± 1

● Exchanging x1 and x2, we have

1− x → 1
1− x

(ξ1−ξ3)(ξ2−ξ4) → (ξ2−ξ3)(ξ1−ξ4
=( ξ1− ξ3 ) ( ξ 2− ξ4

√Φ(13)(24)± 1− x Φ(14)(23) → √Φ(23)(14)± 1
1−
√ [=
1 ±Φ(13 )( 2
1− x

Fusion Channels

ve function can be written as

)1− x Ψ(14)(23) Ψ=a+ Ψ+ + a− Ψ−

ss ss

∑= ( Rσ σ )ab
b

a=1/y b=1/y

4) 1− x= ( ξ1− ξ4 ) ( ξ2 − ξ3 )
4)(1− x ) ( ξ1− ξ3 )(ξ2−ξ4 )

x Φ(24)(13) ( )( ) ( )ΨΨ−+→1 0 ΨΨ−+
0 −1
√ ]24)+ 1− x Φ(14)(23)

R-matrix (Ising x U(1))

A Simple Quantu

I

〈 Ψ∣

Mn G ∝∣t 1U 1+t 2 U

∣Ψ 〉 〈 Ψ∣M

I

um Computation

{ }〈 〉U ∣2
2∣Ψ 〉∣2=∣t ∣2 +∣t + 2 ℜ t ∗ t e i ϕ Ψ∣M n∣Ψ
2 1
1 2

Mn∣Ψ 〉

Calculating w

with F-Matrix

NOT

T Gate

Braiding Examp

ple: CNOT Gate

Generates representation
of the braid group B6

Measuring

Das Sarma, Freedman & Nayak,

ng Anyons

, PRL 94, 166802 (2005)

#4: Pictoria

a

Planer graph with punctures <==>

g

initialization/
measurement (inverse proces

Advantages: GS degeneracy and braidin

al Messages

 M ab  b Excited
states

Gap D

ground state
manifold

> Condensate with quasiparticles

braiding = computing
ss)

ng operation robust against local perturbation

Universal Qua

● A set of universal quantum gates is
operation possible on a quantum c
other unitary operation can be exp
from the set. We only require that
approximated by a sequence of ga
the specific case of single qubit ga
guarantees that this can be done ef

● From a more mathematical point o
a remarkable general statement ab
“filled in” by a universal set of gat

● One simple set of universal quantu
p/8-gate R(p/4), and the controlled

antum Gate Set

s any set of gates to which any
computer can be reduced, that is, any
pressed as a finite sequence of gates
any quantum operation can be
ates from this finite set. Moreover, for
ates, the Solovay-Kitaev theorem
fficiently.

of view, the Solovay-Kitaev theorem is
bout how quickly the group SU(d) is
tes.

um gates is the Hadamard gate H, the
d-NOT gate.

Fibonacc

● Suppose we have only two types o

– A trivial anyon I (or 0): representing
– A non-trivial anyon t (or 1) – must b

● Anyons can be fused to a new one

Two possibilities: ×=I 
non-Abelian! Ising: σ×σ= I

k = 3 Read-Rezayi state; non-Abelian s

ci Anyons

of anyons

the ground state of the system (vacuum)
be the antiparticle of itself

e or nothing

I+ ψ

spin-singlet state (Ardonne & Schoutens)

Quantum D

×××= I  ××=×

V n1=V n−1V n

Dimension of V : 1, 1, 2, 3,
n

Dimension

× ××= I   I  

Dim(V n)∼ϕn , ϕ=(√5+1)/ 2

, 5, 8, 13, 21, 34, 55, 89, 144, ...

Single Qubit and E

● Either three or four anyons can enco

● A braid represents the worldline of a

Elementary Braids

ode one qubit of information.

anyons in the (2+1)-dim spacetime.

 = 5−1

2

Identical to s1

Braiding Matrices from

m the Hexagon Equation

Universal Qu

● Single-qubit gates (rotation)

∣ 〉 U U ∣ 〉

• At least a two-qubit gate, such as CN

∣ 〉 ∣ 〉
∣〉 ∣ ' 〉

• Any N-qubit gates can be realized by
• Freedman et al. proved TQC is as po

implemented by Bonesteel and co-w

uantum Gates

NOT U −1 = −1 2  2  4 ⋯
2 1

Goal: Efficiently find a
sequence that approximates
the target gate within a given
error e.

y the set of universal gates

owerful as conventional QC;
workers using Fibonacci anyons.

Importance o

● In a classical computer, one can bu
numbers using OR and NOT gates

● In a quantum computer, the set of
continuum, and it’s not necessarily
simulate another exactly. Instead, s
necessary.

● We explore an algorithm that guar
any quantum gate, to a very good a

– From a practical point of view, this is
algorithms (like Shor’s) into a form t

– From a more mathematical point of v
how quickly the group SU(d) is “fille

● This is also the importance of the t
Kitaev algorithm (c.f. Nielsen & C

of Algorithm

uild up a circuit, e.g., to add two
s.
possible quantum gates forms a
y possible to use one gate set to
some approximation may be

rantees the efficient construction of
approximation.

s important in compiling quantum
that can be implemented fault-tolerantly.
view, we give a general statement about
ed in” by a universal set of gates.

textbook example – the Solovay-
Chung).

Single-Qubit Gates:

 10 =1 ● We have s1
i ● Each excha
● Finding the
● Exhaustive

1 21 1 21=1

Brute-Force Search

1 (or s3), s2, and inverses s1-1, s2-1
ange has 3 possibilities (no return)
e best braid in ~3N possibilities
e search: non-polynomial time

Exchange-by-Ex

L = 44, e = 0
Distance between matrices U and V is defined
root of the highest eigenvalues of (U-V)*(U-V

xchange Distance

0.00191937

as the square rare fluctuation!
V)

Distance Distribution

Distribution of distance to
the identity for all weaves
(a subset of braids in
which only one anyon
moves) with a length 24:

How to enhanced the
sampling at small d?

d=2 sin / 4 P BF d = 4 d


assuming that the braids distributed un

n for a Fixed Length

d2 1−d2/4

niformly in the space of unitary matrices

Randomly Uniform

● Assumption: The matrix represent
randomly in the space of unitary m
correlation.

– Total number of weaves for a fixed bra

N ( L)∼αL/2 , α≈2.732< 3

 1i 0=1 1 211 21=

– Average volume per weave on the 3-sp

[ϵ( L)]3∼1/ N ( L)∼α−L/2

– Average error:   L~−L/6

or L~ln 1/ ineffic
T ∼(1/ ϵ)3∼e γ L

rm Approximation

tations of long enough braids distribute
matrices (3-sphere). There is no local

aid length L: g =ei m̂⋅σ⃗ (ϕ/2)

=1 σ σ σ σ ⋯σ σn1 n2 n3 n4nm−1nm
phere: 1 21 2 1 2

ni =±2,± 4

L=∑∣ni∣
i

N ( L)∼( 1+ √ 3) L/2

cient! Burrello et al., 2011