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

Two-Qub

● Single-qubit gate: 3 free parameters

● Two-qubit gate: too many paramete

∣ 〉
∣ 〉

bit Gates

s [SU(2)]

ers 1?

∣ 〉 leakage
∣ ' 〉
1?

Decomposing Tw

● Idea proposed by Bonesteel et al. (20
● [Xu & Wan, 08] Reduces leakage er

● Inject the control pair into the targ

● Perform a single-qubit rotation U
● Extract the control pair use the in

D U

0
0

Two-Qubit Gates

005) – leakage error ~10-3
rror significantly, ~10-9

get qubit – exchange braid D

– implemented by a weave
nverse of the inverse of the exchange braid D-1

D-1 Generic
controlled-gates
with leakage error
~10-9.

Leakag

Create a pair of anyons out of
vacuum (so fuse to 0).

Note they could also be stray
anyons thermally excited.

ge Error

Computing basis
Non-computing basis

Leakage-Err

W

ror Analysis

What kind of braids (of a , a , a ) leave
456
the left qubit in state 0, after

exchanging a and a ?
45

101 = 1
10 0
    0

    ei 01 = ei 1
0 ei  0 0

Phase

• Let us look for diagonal matrices, ra
means we introduce a phase error

• For small b, g is irrelavent. Targeting

∣1∣=∣2∣= 1−

• But how do we use it? What about th

Gates

ather than the identity matrix; this
r.

g less parameter – higher accuracy!

−r2=1.56×10−10

he phase?

Exchang

• Apply the diagonal gate (with the irr

 ∣1∣= ∣2∣2∣3∣2=∣4∣= ∣5∣2∣6

ge Braid

relevant phase) to the leakage model

Low leakage! ∣1∣=1.56×10−10

6∣2=∣1∣= 1−r2≈1.56×10−10

5-Dimensional

• One calculate the braiding matrix in
computing bases.

Representation

an enlarged space, including non-

Improvement in the Bru


≈0.76 e−L/4.3

Leakage error reduction by se

rute-Force Performance

≈1.6 e−L/7.3

Search for one point
on the bloch sphere

Search for any point
on a circle on the
Bloch sphere

everal orders of magnitude

Single-Qubit Con

● Single-qubit construction hides an S
arbitrary axis l by an angle q on a B
rotating l to another direction l', then
finally rotating l' back to l.

|ψ⟩ G1 G2

● Implementation: Instead of search f
G and G , such that G ≈ G1 G 2 G 1+
12

onstruction Again

SU(2) symmetry. A rotation around an
Bloch sphere can be carried out by first
n rotating around l' by an angle q, and

U(1) symmetry

G1+ U |ψ⟩

SU (2)/U (1) ∼ S 2

for a gate G, we search a pair of gates

Geometric Redundancy

● We first rotate the axis of rotation,
finally rotate the axis back – physi
geometric redundancy in search, d

– We can search G and G separately
12

– Both searches are achievable in lowe

– i.e., we can fix

G up to a U(1) rotation, and
1

G up to SU(2) / U(1) ~ S2
2

● Outcome: Generic single-qubit gat
braids of ~300 exchanges (length)

– Hormozi et al. (07): 4 ×10-5 for a brai
algorithm

y for Single-qubit Gates

, then rotate around the axis, and
ically, this means that we have a
due to the SU(2) rotation symmetry.

er (than 3) dimensions

tes with error (distance) ~10-10 with
) – Xu & XW (2009).

id of length 220 with Solovay-Kitaev

#5: Messages on Topol

● Three or four Fibonacci anyons ca
● Quantum gates can be achieved by

moving one or one pair of anyons
gates.
● Braids for quantum gates can be co
elementary exchanges and their in
● The construction of two-qubit gate
qubit gates. But at least one high-p
eliminate leakage errors.
● In the brute-force search for braids
explored to boost the efficiency.

logical Quantum Gates

an encode one qubit of information.
y braiding anyons; in particular,
is enough to generate all quantum

ompiled into sequences of two
nverses.
es can be mapped to that of single-
precision phase gate is needed to

s geometrical redundancy can be

Renormalization G

1. Start from a collection of braids of
2. Find the cluster of braids that appr
3. Moving on to a collection of longe

the residual error
4. Repeat 2-3, and stop when the des

Group Like Scheme

f certain length
roximates the target best
er braids (finer in distance) matching

sired error scale is reached

Icosahedr

● The following Cartesian coordinat
the vertices of an icosahedron with
edge-length 2, centered at the orig

0,±1,± ,±1,± ,0 ,± , 0,±

● The icosahedral group is the larges
subgroup of SU(2). It is composed
60 rotations around the axes of sym
of the icosahedron.

● There are 6 axes of the 5th order, 1
I 60={ g 0, g1, g 2, ⋯

● We approximate all group element

ral Group

tes define
h
gin:

±1

st finite
d by the
mmetry

= 1 5

2

10 of the 3rd, and 15 of the 2nd.

⋯, g59 } g0=e
ts by braids of various length.

Braid Representatio

● L = 8, e = 0.236068

● L = 24, e = 0.0344419

● L = 44, e = 0.00191937

● L = 68, e = 0.0000304193

The braid representations can be c
Hence no additional cost to the sea

ons for the Identity e

g̃0 ( 8 )= σ−2 2 σ 2 σ−2 2 σ 12= g e i Δ (8 )
1 0

0

Δ (8) : a Hermitian matrix
0 characterizing error

computed and stored once for all.
arch later.

Connection to Rand

● Pseudogroup of braids (for small D

gi g j=g k , g̃ i g̃ j= gi ei Δ i g j ei Δ j≈

● To approximate gi g j⋯g n+ 1=e

g̃i g̃ j ⋯ gñ+ 1= gi ei Δi g j ei Δ j ⋯ g n+ 1 ei Δ n+ 1≡ei H n

H n= gi Δ i g−i 1+ gi g j Δ j g−j 1 g−i 1+ ⋯
+ gi g j ⋯ g n Δ n g−n 1⋯ g−j 1 g−i 1

+ Δ n+ 1+ O (Δ 2)

● We conjecture H is a random mat
n
in the Wigner-Dyson Gaussian

Unitary Ensemble (s for eigenvalu

( )P(s)= 32 2
π2 s
0 s −(4 / π)(s / s0)2
s e0

A single parameter s controls the flow of
0

dom Matrix Theory

Di )

i ( g−j 1 Δi g j + Δ j)k i Δk
k
≈ g e ≠ g̃ = g ek

n

trix
ue/error)

n = 3 is large enough
f the (distribution of) error.

Understanding Erro

● First approximate by gluing 3 shor

● Reduce the error (e1) by gluing
4 (= n + 1) longer (L = 24)
segments (1 out of 603 ).

● The resulting error (e2) follows
the Wigner-Dyson distribution.

● Average error reduction:

<e > / <e > ~ f = 60n/ 3
12
√ n+ 1

Initial approximation e1 Correction: a

or Renormalization

rt (L = 8) segments (1 out of 603 ).

distribution of e
2

approximation to e
2

Scaling A

● The number of iteration for a
given final error ≪1

● Choose suitable braid segment
length to match the residual error

● Each iteration increases the
length by 4 (= n + 1) segments

● Length of braid after q iterations

● Time

Analysis

… ln(f) ln(f) ln(error)

ln(e) ~ 0

Comparison with

● Compiling with the RG-like algori
● Brute-force search
● Solovay-Kitaev

Thanks to randomness in the building bloc

Other Algorithms

ithm tpIfIsotnoetIsIpfrotnoecetrtrreacteroctearelakocineaanlskEeinacdnisEesaoc6hdisoaro6hln8beaornln85beasnion5t0sasniofrt0asaafrtp37avhartp37rvyehraGxoryeranGxocgaHr1ncegagaHa10eszgatea0−esszte4−oes.4o.r r

U i1= Ai Bi A−i 1 B−i 1 U i

cks, we save time in search exponentially.

#6: Importance

● 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.

e 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-

I Ching o

● I Ching (~1100 BC): Ancient peop
record, while people during later p

● 《易 · 系辞》载：“上古结绳而

纠缠 ↔ entanglem

of Knots

ple tied knots on cords to keep
periods replaced with writing
而治，后世圣人易之以书契”

ment

Stability in Chin

● Decimal system in China (over 30

一二三四五六七八九

百

千

nese Characters

000 years ago) Chinese characters

十 encode information,
in some sense, in

the topology of the
strokes.

Topological Quan

ntum Computation

Next: compute with Fibonacci anyons