<s>
A	O
topological	B-Device
quantum	I-Device
computer	I-Device
is	O
a	O
theoretical	O
quantum	B-Architecture
computer	I-Architecture
proposed	O
by	O
Russian-American	O
physicist	O
Alexei	O
Kitaev	O
in	O
1997	O
.	O
</s>
<s>
It	O
employs	O
quasiparticles	O
in	O
two-dimensional	O
systems	O
,	O
called	O
anyons	O
,	O
whose	O
world	O
lines	O
pass	O
around	O
one	O
another	O
to	O
form	O
braids	B-Algorithm
in	O
a	O
three-dimensional	O
spacetime	B-Protocol
(	O
i.e.	O
,	O
one	O
temporal	O
plus	O
two	O
spatial	O
dimensions	O
)	O
.	O
</s>
<s>
These	O
braids	B-Algorithm
form	O
the	O
logic	O
gates	O
that	O
make	O
up	O
the	O
computer	O
.	O
</s>
<s>
The	O
advantage	O
of	O
a	O
quantum	B-Architecture
computer	I-Architecture
based	O
on	O
quantum	O
braids	B-Algorithm
over	O
using	O
trapped	O
quantum	O
particles	O
is	O
that	O
the	O
former	O
is	O
much	O
more	O
stable	O
.	O
</s>
<s>
Small	O
,	O
cumulative	O
perturbations	O
can	O
cause	O
quantum	O
states	O
to	O
decohere	O
and	O
introduce	O
errors	O
in	O
the	O
computation	O
,	O
but	O
such	O
small	O
perturbations	O
do	O
not	O
change	O
the	O
braids	B-Algorithm
 '	O
topological	B-Algorithm
properties	I-Algorithm
.	O
</s>
<s>
This	O
is	O
like	O
the	O
effort	O
required	O
to	O
cut	O
a	O
string	O
and	O
reattach	O
the	O
ends	O
to	O
form	O
a	O
different	O
braid	O
,	O
as	O
opposed	O
to	O
a	O
ball	O
(	O
representing	O
an	O
ordinary	O
quantum	O
particle	O
in	O
four-dimensional	O
spacetime	B-Protocol
)	O
bumping	O
into	O
a	O
wall	O
.	O
</s>
<s>
While	O
the	O
elements	O
of	O
a	O
topological	B-Device
quantum	I-Device
computer	I-Device
originate	O
in	O
a	O
purely	O
mathematical	O
realm	O
,	O
experiments	O
in	O
fractional	O
quantum	O
Hall	O
systems	O
indicate	O
these	O
elements	O
may	O
be	O
created	O
in	O
the	O
real	O
world	O
using	O
semiconductors	O
made	O
of	O
gallium	O
arsenide	O
at	O
a	O
temperature	O
of	O
near	O
absolute	B-Application
zero	I-Application
and	O
subjected	O
to	O
strong	O
magnetic	O
fields	O
.	O
</s>
<s>
Thus	O
,	O
the	O
world	O
lines	O
of	O
two	O
anyons	O
cannot	O
intersect	O
or	O
merge	O
,	O
which	O
allows	O
their	O
paths	O
to	O
form	O
stable	O
braids	B-Algorithm
in	O
space-time	O
.	O
</s>
<s>
When	O
anyons	O
are	O
braided	O
,	O
the	O
transformation	O
of	O
the	O
quantum	O
state	O
of	O
the	O
system	O
depends	O
only	O
on	O
the	O
topological	O
class	O
of	O
the	O
anyons	O
 '	O
trajectories	O
(	O
which	O
are	O
classified	O
according	O
to	O
the	O
braid	B-Application
group	I-Application
)	O
.	O
</s>
<s>
In	O
2005	O
,	O
Sankar	O
Das	O
Sarma	O
,	O
Michael	O
Freedman	O
,	O
and	O
Chetan	O
Nayak	O
proposed	O
a	O
quantum	O
Hall	O
device	O
that	O
would	O
realize	O
a	O
topological	B-Device
qubit	I-Device
.	O
</s>
<s>
In	O
a	O
key	O
development	O
for	O
topological	B-Device
quantum	I-Device
computers	I-Device
,	O
in	O
2005	O
Vladimir	O
J	O
.	O
Goldman	O
,	O
Fernando	O
E	O
.	O
Camino	O
,	O
and	O
Wei	O
Zhou	O
claimed	O
to	O
have	O
created	O
and	O
observed	O
the	O
first	O
experimental	O
evidence	O
for	O
using	O
a	O
fractional	O
quantum	O
Hall	O
effect	O
to	O
create	O
actual	O
anyons	O
,	O
although	O
others	O
have	O
suggested	O
their	O
results	O
could	O
be	O
the	O
product	O
of	O
phenomena	O
not	O
involving	O
anyons	O
.	O
</s>
<s>
Non-abelian	O
anyons	O
,	O
a	O
species	O
required	O
for	O
topological	B-Device
quantum	I-Device
computers	I-Device
,	O
have	O
yet	O
to	O
be	O
experimentally	O
confirmed	O
.	O
</s>
<s>
Topological	B-Device
quantum	I-Device
computers	I-Device
are	O
equivalent	O
in	O
computational	O
power	O
to	O
other	O
standard	O
models	O
of	O
quantum	B-Architecture
computation	I-Architecture
,	O
in	O
particular	O
to	O
the	O
quantum	B-Application
circuit	I-Application
model	O
and	O
to	O
the	O
quantum	O
Turing	O
machine	O
model	O
.	O
</s>
<s>
Nonetheless	O
,	O
certain	O
algorithms	O
may	O
be	O
a	O
more	O
natural	O
fit	O
to	O
the	O
topological	B-Device
quantum	I-Device
computer	I-Device
model	O
.	O
</s>
<s>
For	O
example	O
,	O
algorithms	O
for	O
evaluating	O
the	O
Jones	O
polynomial	O
were	O
first	O
developed	O
in	O
the	O
topological	O
model	O
,	O
and	O
only	O
later	O
converted	O
and	O
extended	O
in	O
the	O
standard	O
quantum	B-Application
circuit	I-Application
model	O
.	O
</s>
<s>
To	O
live	O
up	O
to	O
its	O
name	O
,	O
a	O
topological	B-Device
quantum	I-Device
computer	I-Device
must	O
provide	O
the	O
unique	O
computation	O
properties	O
promised	O
by	O
a	O
conventional	O
quantum	B-Architecture
computer	I-Architecture
design	O
,	O
which	O
uses	O
trapped	O
quantum	O
particles	O
.	O
</s>
<s>
In	O
2000	O
,	O
Michael	O
H	O
.	O
Freedman	O
,	O
Alexei	O
Kitaev	O
,	O
Michael	O
J	O
.	O
Larsen	O
,	O
and	O
Zhenghan	O
Wang	O
proved	O
that	O
a	O
topological	B-Device
quantum	I-Device
computer	I-Device
can	O
,	O
in	O
principle	O
,	O
perform	O
any	O
computation	O
that	O
a	O
conventional	O
quantum	B-Architecture
computer	I-Architecture
can	O
do	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
They	O
found	O
that	O
a	O
conventional	O
quantum	B-Architecture
computer	I-Architecture
device	O
,	O
given	O
an	O
error-free	O
operation	O
of	O
its	O
logic	O
circuits	O
,	O
will	O
give	O
a	O
solution	O
with	O
an	O
absolute	O
level	O
of	O
accuracy	O
,	O
whereas	O
a	O
topological	B-Device
quantum	I-Device
computing	I-Device
device	O
with	O
flawless	O
operation	O
will	O
give	O
the	O
solution	O
with	O
only	O
a	O
finite	O
level	O
of	O
accuracy	O
.	O
</s>
<s>
However	O
,	O
any	O
level	O
of	O
precision	O
for	O
the	O
answer	O
can	O
be	O
obtained	O
by	O
adding	O
more	O
braid	O
twists	O
(	O
logic	O
circuits	O
)	O
to	O
the	O
topological	B-Device
quantum	I-Device
computer	I-Device
,	O
in	O
a	O
simple	O
linear	O
relationship	O
.	O
</s>
<s>
Even	O
though	O
quantum	O
braids	B-Algorithm
are	O
inherently	O
more	O
stable	O
than	O
trapped	O
quantum	O
particles	O
,	O
there	O
is	O
still	O
a	O
need	O
to	O
control	O
for	O
error	O
inducing	O
thermal	O
fluctuations	O
,	O
which	O
produce	O
random	O
stray	O
pairs	O
of	O
anyons	O
which	O
interfere	O
with	O
adjoining	O
braids	B-Algorithm
.	O
</s>
<s>
Simulating	O
the	O
dynamics	O
of	O
a	O
topological	B-Device
quantum	I-Device
computer	I-Device
may	O
be	O
a	O
promising	O
method	O
of	O
implementing	O
fault-tolerant	O
quantum	B-Architecture
computation	I-Architecture
even	O
with	O
a	O
standard	O
quantum	O
information	O
processing	O
scheme	O
.	O
</s>
<s>
One	O
of	O
the	O
prominent	O
examples	O
in	O
topological	B-Device
quantum	I-Device
computing	I-Device
is	O
with	O
a	O
system	O
of	O
Fibonacci	O
anyons	O
.	O
</s>
<s>
These	O
anyons	O
can	O
be	O
used	O
to	O
create	O
generic	O
gates	O
for	O
topological	B-Device
quantum	I-Device
computing	I-Device
.	O
</s>
<s>
These	O
axioms	O
are	O
called	O
the	O
pentagonal	O
and	O
hexagonal	O
axioms	O
respectively	O
as	O
performing	O
the	O
operation	O
can	O
be	O
visualized	O
with	O
a	O
pentagon/hexagon	O
of	O
state	O
transformations	O
.	O
</s>
<s>
With	O
these	O
braid	O
operators	O
,	O
we	O
can	O
finally	O
formalize	O
the	O
notion	O
of	O
braids	B-Algorithm
in	O
terms	O
of	O
how	O
they	O
act	O
on	O
our	O
Hilbert	O
space	O
and	O
construct	O
arbitrary	O
universal	O
quantum	O
gates	O
.	O
</s>
