<s>
In	O
quantum	B-Architecture
computing	I-Architecture
,	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
(	O
QFT	O
)	O
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
on	O
quantum	O
bits	O
,	O
and	O
is	O
the	O
quantum	O
analogue	O
of	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
a	O
part	O
of	O
many	O
quantum	B-Device
algorithms	I-Device
,	O
notably	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
factoring	O
and	O
computing	O
the	O
discrete	O
logarithm	O
,	O
the	O
quantum	B-Algorithm
phase	I-Algorithm
estimation	I-Algorithm
algorithm	I-Algorithm
for	O
estimating	O
the	O
eigenvalues	O
of	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
,	O
and	O
algorithms	O
for	O
the	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
was	O
discovered	O
by	O
Don	O
Coppersmith	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
performed	O
efficiently	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
with	O
a	O
decomposition	O
into	O
the	O
product	O
of	O
simpler	O
unitary	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
on	O
amplitudes	O
can	O
be	O
implemented	O
as	O
a	O
quantum	B-Application
circuit	I-Application
consisting	O
of	O
only	O
Hadamard	O
gates	O
and	O
controlled	O
phase	O
shift	O
gates	O
,	O
where	O
is	O
the	O
number	O
of	O
qubits	O
.	O
</s>
<s>
This	O
can	O
be	O
compared	O
with	O
the	O
classical	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
which	O
takes	O
gates	O
(	O
where	O
is	O
the	O
number	O
of	O
bits	O
)	O
,	O
which	O
is	O
exponentially	O
more	O
than	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
acts	O
on	O
a	O
quantum	O
state	O
vector	O
(	O
a	O
quantum	O
register	O
)	O
,	O
and	O
the	O
classical	O
Fourier	O
transform	O
acts	O
on	O
a	O
vector	O
.	O
</s>
<s>
Because	O
measurement	O
collapses	O
the	O
quantum	O
state	O
to	O
a	O
single	O
basis	O
state	O
,	O
not	O
every	O
task	O
that	O
uses	O
the	O
classical	O
Fourier	O
transform	O
can	O
take	O
advantage	O
of	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
's	O
exponential	O
speedup	O
.	O
</s>
<s>
The	O
best	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
algorithms	O
known	O
(	O
as	O
of	O
late	O
2000	O
)	O
require	O
only	O
gates	O
to	O
achieve	O
an	O
efficient	O
approximation	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
the	O
classical	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
applied	O
to	O
the	O
vector	O
of	O
amplitudes	O
of	O
a	O
quantum	O
state	O
,	O
where	O
we	O
usually	O
consider	O
vectors	O
of	O
length	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
acts	O
on	O
a	O
quantum	O
state	O
and	O
maps	O
it	O
to	O
a	O
quantum	O
state	O
according	O
to	O
the	O
formula	O
:	O
</s>
<s>
(	O
Conventions	O
for	O
the	O
sign	O
of	O
the	O
phase	O
factor	O
exponent	O
vary	O
;	O
here	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
has	O
the	O
same	O
effect	O
as	O
the	O
inverse	B-Algorithm
discrete	I-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
and	O
vice	O
versa	O
.	O
)	O
</s>
<s>
Since	O
is	O
a	O
rotation	O
,	O
the	O
inverse	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
acts	O
similarly	O
but	O
with	O
:	O
</s>
<s>
Most	O
of	O
the	O
properties	O
of	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
follow	O
from	O
the	O
fact	O
that	O
it	O
is	O
a	O
unitary	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
From	O
the	O
unitary	O
property	O
it	O
follows	O
that	O
the	O
inverse	O
of	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
the	O
Hermitian	O
adjoint	O
of	O
the	O
Fourier	O
matrix	O
,	O
thus	O
.	O
</s>
<s>
Since	O
there	O
is	O
an	O
efficient	O
quantum	B-Application
circuit	I-Application
implementing	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
the	O
circuit	O
can	O
be	O
run	O
in	O
reverse	O
to	O
perform	O
the	O
inverse	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Thus	O
both	O
transforms	O
can	O
be	O
efficiently	O
performed	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
written	O
as	O
the	O
tensor	O
product	O
of	O
a	O
series	O
of	O
terms	O
:	O
</s>
<s>
the	O
action	O
of	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
expressed	O
in	O
a	O
compact	O
manner	O
:	O
</s>
<s>
Because	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
an	O
operation	O
on	O
n	O
qubits	O
,	O
can	O
be	O
factored	O
into	O
the	O
tensor	O
product	O
of	O
n	O
single-qubit	O
operations	O
,	O
it	O
is	O
easily	O
represented	O
as	O
a	O
quantum	B-Application
circuit	I-Application
(	O
up	O
to	O
an	O
order	O
reversal	O
of	O
the	O
output	O
)	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
on	O
three	O
qubits	O
,	O
with	O
,	O
is	O
represented	O
by	O
the	O
following	O
transformation	O
:	O
</s>
<s>
The	O
3-qubit	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
rewritten	O
as	O
:	O
</s>
<s>
Using	O
the	O
generalized	O
Fourier	O
transform	O
on	O
finite	O
(	O
abelian	O
)	O
groups	O
,	O
there	O
are	O
actually	O
two	O
natural	O
ways	O
to	O
define	O
a	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
on	O
an	O
n-qubit	O
quantum	O
register	O
.	O
</s>
<s>
However	O
,	O
it	O
also	O
makes	O
sense	O
to	O
consider	O
the	O
qubits	O
as	O
indexed	O
by	O
the	O
Boolean	O
group	O
,	O
and	O
in	O
this	O
case	O
the	O
Fourier	O
transform	O
is	O
the	O
Hadamard	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
uses	O
both	O
types	O
of	O
Fourier	O
transforms	O
,	O
an	O
initial	O
Hadamard	B-Algorithm
transform	I-Algorithm
as	O
well	O
as	O
a	O
QFT	O
.	O
</s>
