<s>
The	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
(	O
HSP	O
)	O
is	O
a	O
topic	O
of	O
research	O
in	O
mathematics	O
and	O
theoretical	O
computer	O
science	O
.	O
</s>
<s>
This	O
makes	O
it	O
especially	O
important	O
in	O
the	O
theory	O
of	O
quantum	B-Architecture
computing	I-Architecture
because	O
Shor	B-Algorithm
's	I-Algorithm
quantum	I-Algorithm
algorithm	I-Algorithm
for	O
factoring	O
is	O
an	O
instance	O
of	O
the	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
for	O
finite	O
Abelian	O
groups	O
,	O
while	O
the	O
other	O
problems	O
correspond	O
to	O
finite	O
groups	O
that	O
are	O
not	O
Abelian	O
.	O
</s>
<s>
Hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
:	O
Let	O
be	O
a	O
group	O
,	O
a	O
finite	O
set	O
,	O
and	O
a	O
function	O
that	O
hides	O
a	O
subgroup	O
.	O
</s>
<s>
The	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
is	O
especially	O
important	O
in	O
the	O
theory	O
of	O
quantum	B-Architecture
computing	I-Architecture
for	O
the	O
following	O
reasons	O
.	O
</s>
<s>
Shor	B-Algorithm
's	I-Algorithm
quantum	I-Algorithm
algorithm	I-Algorithm
for	O
factoring	O
and	O
discrete	O
logarithm	O
(	O
as	O
well	O
as	O
several	O
of	O
its	O
extensions	O
)	O
relies	O
on	O
the	O
ability	O
of	O
quantum	B-Architecture
computers	I-Architecture
to	O
solve	O
the	O
HSP	O
for	O
finite	O
Abelian	O
groups	O
.	O
</s>
<s>
The	O
existence	O
of	O
efficient	O
quantum	B-Device
algorithms	I-Device
for	O
HSPs	O
for	O
certain	O
non-Abelian	O
groups	O
would	O
imply	O
efficient	O
quantum	B-Device
algorithms	I-Device
for	O
two	O
major	O
problems	O
:	O
the	O
graph	O
isomorphism	O
problem	O
and	O
certain	O
shortest	O
vector	O
problems	O
(	O
SVPs	O
)	O
in	O
lattices	O
.	O
</s>
<s>
More	O
precisely	O
,	O
an	O
efficient	O
quantum	B-Device
algorithm	I-Device
for	O
the	O
HSP	O
for	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
would	O
give	O
a	O
quantum	B-Device
algorithm	I-Device
for	O
the	O
graph	O
isomorphism	O
.	O
</s>
<s>
An	O
efficient	O
quantum	B-Device
algorithm	I-Device
for	O
the	O
HSP	O
for	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
would	O
give	O
a	O
quantum	B-Device
algorithm	I-Device
for	O
the	O
unique	O
SVP	O
.	O
</s>
<s>
There	O
is	O
a	O
efficient	O
quantum	B-Device
algorithm	I-Device
for	O
solving	O
HSP	O
over	O
finite	O
Abelian	O
groups	O
in	O
time	O
polynomial	O
in	O
.	O
</s>
<s>
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
applies	O
a	O
particular	O
case	O
of	O
this	O
quantum	B-Device
algorithm	I-Device
.	O
</s>
<s>
For	O
arbitrary	O
groups	O
,	O
it	O
is	O
known	O
that	O
the	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
is	O
solvable	O
using	O
a	O
polynomial	O
number	O
of	O
evaluations	O
of	O
the	O
oracle	O
.	O
</s>
<s>
To	O
solve	O
the	O
problem	O
for	O
finite	O
Abelian	O
groups	O
,	O
the	O
'	O
standard	O
 '	O
approach	O
to	O
this	O
problem	O
involves	O
:	O
the	O
creation	O
of	O
the	O
quantum	O
state	O
,	O
a	O
subsequent	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
to	O
the	O
left	O
register	O
,	O
after	O
which	O
this	O
register	O
gets	O
sampled	O
.	O
</s>
<s>
This	O
approach	O
has	O
been	O
shown	O
to	O
be	O
insufficient	O
for	O
the	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
for	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
