<s>
In	O
computational	O
complexity	O
theory	O
and	O
quantum	B-Architecture
computing	I-Architecture
,	O
Simon	B-Algorithm
's	I-Algorithm
problem	I-Algorithm
is	O
a	O
computational	O
problem	O
that	O
is	O
proven	O
to	O
be	O
solved	O
exponentially	O
faster	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
than	O
on	O
a	O
classical	O
(	O
that	O
is	O
,	O
traditional	O
)	O
computer	O
.	O
</s>
<s>
The	O
quantum	B-Device
algorithm	I-Device
solving	O
Simon	B-Algorithm
's	I-Algorithm
problem	I-Algorithm
,	O
usually	O
called	O
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
served	O
as	O
the	O
inspiration	O
for	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Both	O
problems	O
are	O
special	O
cases	O
of	O
the	O
abelian	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
,	O
which	O
is	O
now	O
known	O
to	O
have	O
efficient	O
quantum	B-Device
algorithms	I-Device
.	O
</s>
<s>
The	O
problem	O
is	O
set	O
in	O
the	O
model	O
of	O
decision	B-Application
tree	I-Application
complexity	I-Application
or	O
query	B-Application
complexity	I-Application
and	O
was	O
conceived	O
by	O
Daniel	O
Simon	O
in	O
1994	O
.	O
</s>
<s>
Simon	O
exhibited	O
a	O
quantum	B-Device
algorithm	I-Device
that	O
solves	O
Simon	B-Algorithm
's	I-Algorithm
problem	I-Algorithm
exponentially	O
faster	O
and	O
with	O
exponentially	O
fewer	O
queries	O
than	O
the	O
best	O
probabilistic	B-General_Concept
(	O
or	O
deterministic	O
)	O
classical	O
algorithm	O
.	O
</s>
<s>
In	O
particular	O
,	O
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
uses	O
a	O
linear	O
number	O
of	O
queries	O
and	O
any	O
classical	O
probabilistic	B-General_Concept
algorithm	I-General_Concept
must	O
use	O
an	O
exponential	O
number	O
of	O
queries	O
.	O
</s>
<s>
This	O
problem	O
yields	O
an	O
oracle	O
separation	O
between	O
the	O
complexity	O
classes	O
BPP	O
(	O
bounded-error	O
classical	O
query	B-Application
complexity	I-Application
)	O
and	O
BQP	O
(	O
bounded-error	O
quantum	O
query	B-Application
complexity	I-Application
)	O
.	O
</s>
<s>
This	O
is	O
the	O
same	O
separation	O
that	O
the	O
Bernstein	O
–	O
Vazirani	O
algorithm	O
achieves	O
,	O
and	O
different	O
from	O
the	O
separation	O
provided	O
by	O
the	O
Deutsch	B-Algorithm
–	I-Algorithm
Jozsa	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
separates	O
P	O
and	O
EQP	O
.	O
</s>
<s>
Unlike	O
the	O
Bernstein	O
–	O
Vazirani	O
algorithm	O
,	O
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
's	O
separation	O
is	O
exponential	O
.	O
</s>
<s>
Because	O
this	O
problem	O
assumes	O
the	O
existence	O
of	O
a	O
highly-structured	O
"	O
black	B-Device
box	I-Device
"	O
oracle	O
to	O
achieve	O
its	O
speedup	O
,	O
this	O
problem	O
has	O
little	O
practical	O
value	O
.	O
</s>
<s>
Given	O
a	O
function	O
(	O
implemented	O
by	O
a	O
black	B-Device
box	I-Device
or	O
oracle	O
)	O
with	O
the	O
promise	O
that	O
,	O
for	O
some	O
unknown	O
,	O
for	O
all	O
,	O
</s>
<s>
where	O
denotes	O
bitwise	O
XOR	B-Application
.	O
</s>
<s>
The	O
associated	O
decision	O
problem	O
formulation	O
of	O
Simon	B-Algorithm
's	I-Algorithm
problem	I-Algorithm
is	O
to	O
distinguish	O
when	O
(	O
is	O
one-to-one	O
)	O
,	O
and	O
when	O
(	O
is	O
two-to-one	O
)	O
.	O
</s>
<s>
Every	O
output	O
of	O
occurs	O
twice	O
,	O
and	O
the	O
two	O
input	O
strings	O
corresponding	O
to	O
any	O
one	O
given	O
output	O
have	O
bitwise	O
XOR	B-Application
equal	O
to	O
.	O
</s>
<s>
Applying	O
XOR	B-Application
to	O
001	O
and	O
111	O
obtains	O
110	O
,	O
that	O
is	O
.	O
</s>
<s>
Since	O
,	O
classically	O
to	O
find	O
s	O
with	O
a	O
100%	O
certainty	O
it	O
would	O
require	O
checking	O
up	O
to	O
inputs	O
,	O
Simon	B-Algorithm
's	I-Algorithm
problem	I-Algorithm
seeks	O
to	O
find	O
s	O
using	O
fewer	O
queries	O
than	O
this	O
classical	O
method	O
.	O
</s>
<s>
The	O
quantum	O
circuit	O
(	O
see	O
the	O
picture	O
)	O
is	O
the	O
implementation	O
of	O
the	O
quantum	O
part	O
of	O
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
quantum	O
subroutine	O
of	O
the	O
algorithm	O
makes	O
use	O
of	O
the	O
Hadamard	O
transformwhere	O
,	O
where	O
denotes	O
XOR	B-Application
.	O
</s>
<s>
Apply	O
another	O
Hadamard	B-Algorithm
transform	I-Algorithm
to	O
the	O
first	O
register	O
.	O
</s>
<s>
We	O
can	O
repeat	O
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
a	O
constant	O
number	O
of	O
times	O
to	O
increase	O
the	O
probability	O
of	O
success	O
arbitrarily	O
,	O
while	O
still	O
having	O
the	O
same	O
time	O
complexity	O
.	O
</s>
<s>
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
requires	O
queries	O
to	O
the	O
black	B-Device
box	I-Device
,	O
whereas	O
a	O
classical	O
algorithm	O
would	O
need	O
at	O
least	O
queries	O
.	O
</s>
<s>
It	O
is	O
also	O
known	O
that	O
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
optimal	O
in	O
the	O
sense	O
that	O
any	O
quantum	B-Device
algorithm	I-Device
to	O
solve	O
this	O
problem	O
requires	O
queries	O
.	O
</s>
