<s>
In	O
quantum	B-Architecture
computing	I-Architecture
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
also	O
known	O
as	O
the	O
quantum	O
search	O
algorithm	O
,	O
refers	O
to	O
a	O
quantum	B-Device
algorithm	I-Device
for	O
unstructured	O
search	O
that	O
finds	O
with	O
high	O
probability	O
the	O
unique	O
input	O
to	O
a	O
black	B-Device
box	I-Device
function	O
that	O
produces	O
a	O
particular	O
output	O
value	O
,	O
using	O
just	O
evaluations	O
of	O
the	O
function	O
,	O
where	O
is	O
the	O
size	O
of	O
the	O
function	O
's	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
analogous	O
problem	O
in	O
classical	O
computation	O
cannot	O
be	O
solved	O
in	O
fewer	O
than	O
evaluations	O
(	O
because	O
,	O
on	O
average	O
,	O
one	O
has	O
to	O
check	O
half	O
of	O
the	O
domain	B-Algorithm
to	O
get	O
a	O
50%	O
chance	O
of	O
finding	O
the	O
right	O
input	O
)	O
.	O
</s>
<s>
Charles	O
H	O
.	O
Bennett	O
,	O
Ethan	O
Bernstein	O
,	O
Gilles	O
Brassard	O
,	O
and	O
Umesh	O
Vazirani	O
proved	O
that	O
any	O
quantum	O
solution	O
to	O
the	O
problem	O
needs	O
to	O
evaluate	O
the	O
function	O
times	O
,	O
so	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
asymptotically	B-General_Concept
optimal	I-General_Concept
.	O
</s>
<s>
Since	O
classical	O
algorithms	O
for	O
NP-complete	O
problems	O
require	O
exponentially	O
many	O
steps	O
,	O
and	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
provides	O
at	O
most	O
a	O
quadratic	O
speedup	O
over	O
the	O
classical	O
solution	O
for	O
unstructured	O
search	O
,	O
this	O
suggests	O
that	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
by	O
itself	O
will	O
not	O
provide	O
polynomial-time	O
solutions	O
for	O
NP-complete	O
problems	O
(	O
as	O
the	O
square	O
root	O
of	O
an	O
exponential	O
function	O
is	O
an	O
exponential	O
,	O
not	O
polynomial	O
,	O
function	O
)	O
.	O
</s>
<s>
Unlike	O
other	O
quantum	B-Device
algorithms	I-Device
,	O
which	O
may	O
provide	O
exponential	O
speedup	O
over	O
their	O
classical	O
counterparts	O
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
provides	O
only	O
a	O
quadratic	O
speedup	O
.	O
</s>
<s>
However	O
,	O
even	O
quadratic	O
speedup	O
is	O
considerable	O
when	O
is	O
large	O
,	O
and	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
applied	O
to	O
speed	O
up	O
broad	O
classes	O
of	O
algorithms	O
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
could	O
brute-force	O
a	O
128-bit	O
symmetric	O
cryptographic	O
key	O
in	O
roughly	O
264	O
iterations	O
,	O
or	O
a	O
256-bit	O
key	O
in	O
roughly	O
2128	O
iterations	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
it	O
is	O
sometimes	O
suggested	O
that	O
symmetric	B-Algorithm
key	I-Algorithm
lengths	O
be	O
doubled	O
to	O
protect	O
against	O
future	O
quantum	O
attacks	O
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
along	O
with	O
variants	O
like	O
amplitude	B-Algorithm
amplification	I-Algorithm
,	O
can	O
be	O
used	O
to	O
speed	O
up	O
a	O
broad	O
range	O
of	O
algorithms	O
.	O
</s>
<s>
In	O
particular	O
,	O
algorithms	O
for	O
NP-complete	O
problems	O
generally	O
contain	O
exhaustive	O
search	O
as	O
a	O
subroutine	O
,	O
which	O
can	O
be	O
sped	O
up	O
by	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Generic	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
also	O
see	O
quadratic	O
speedups	O
with	O
Grover	O
.	O
</s>
<s>
These	O
algorithms	O
do	O
not	O
require	O
that	O
the	O
input	O
be	O
given	O
in	O
the	O
form	O
of	O
an	O
oracle	O
,	O
since	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
being	O
applied	O
with	O
an	O
explicit	O
function	O
,	O
e.g.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
also	O
give	O
provable	O
speedups	O
for	O
black-box	O
problems	O
in	O
quantum	O
query	O
complexity	O
,	O
including	O
element	O
distinctness	O
and	O
the	O
collision	B-Algorithm
problem	I-Algorithm
(	O
solved	O
with	O
the	O
Brassard	B-Algorithm
–	I-Algorithm
Høyer	I-Algorithm
–	I-Algorithm
Tapp	I-Algorithm
algorithm	I-Algorithm
)	O
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
essentially	O
solves	O
the	O
task	O
of	O
function	O
inversion	O
.	O
</s>
<s>
Roughly	O
speaking	O
,	O
if	O
we	O
have	O
a	O
function	O
that	O
can	O
be	O
evaluated	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
allows	O
us	O
to	O
calculate	O
when	O
given	O
.	O
</s>
<s>
Consequently	O
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
gives	O
broad	O
asymptotic	O
speed-ups	O
to	O
many	O
kinds	O
of	O
brute-force	O
attacks	O
on	O
symmetric-key	B-Algorithm
cryptography	I-Algorithm
,	O
including	O
collision	B-Algorithm
attacks	I-Algorithm
and	O
pre-image	O
attacks	O
.	O
</s>
<s>
However	O
,	O
this	O
may	O
not	O
necessarily	O
be	O
the	O
most	O
efficient	O
algorithm	O
since	O
,	O
for	O
example	O
,	O
the	O
parallel	O
rho	B-Algorithm
algorithm	I-Algorithm
is	O
able	O
to	O
find	O
a	O
collision	O
in	O
SHA2	O
more	O
efficiently	O
than	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
To	O
account	O
for	O
such	O
effects	O
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
viewed	O
as	O
solving	O
an	O
equation	O
or	O
satisfying	B-Application
a	I-Application
constraint	I-Application
.	O
</s>
<s>
The	O
major	O
barrier	O
to	O
instantiating	O
a	O
speedup	O
from	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
that	O
the	O
quadratic	O
speedup	O
achieved	O
is	O
too	O
modest	O
to	O
overcome	O
the	O
large	O
overhead	O
of	O
near-term	O
quantum	B-Architecture
computers	I-Architecture
.	O
</s>
<s>
However	O
,	O
later	O
generations	O
of	O
fault-tolerant	O
quantum	B-Architecture
computers	I-Architecture
with	O
better	O
hardware	O
performance	O
may	O
be	O
able	O
to	O
realize	O
these	O
speedups	O
for	O
practical	O
instances	O
of	O
data	O
.	O
</s>
<s>
As	O
input	O
for	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
suppose	O
we	O
have	O
a	O
function	O
.	O
</s>
<s>
In	O
the	O
"	O
unstructured	O
database	O
"	O
analogy	O
,	O
the	O
domain	B-Algorithm
represent	O
indices	O
to	O
a	O
database	O
,	O
and	O
if	O
and	O
only	O
if	O
the	O
data	O
that	O
x	O
points	O
to	O
satisfies	O
the	O
search	O
criterion	O
.	O
</s>
<s>
We	O
can	O
access	O
f	O
with	O
a	O
subroutine	O
(	O
sometimes	O
called	O
an	O
oracle	O
)	O
in	O
the	O
form	O
of	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
Uω	O
that	O
acts	O
as	O
follows	O
:	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
outputs	O
ω	O
with	O
probability	O
at	O
least	O
1/2	O
using	O
applications	O
of	O
Uω	O
.	O
</s>
<s>
This	O
probability	O
can	O
be	O
made	O
arbitrarily	O
large	O
by	O
running	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
multiple	O
times	O
.	O
</s>
<s>
If	O
one	O
runs	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
until	O
ω	O
is	O
found	O
,	O
the	O
expected	O
number	O
of	O
applications	O
is	O
still	O
,	O
since	O
it	O
will	O
only	O
be	O
run	O
twice	O
on	O
average	O
.	O
</s>
<s>
So	O
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
run	O
regardless	O
of	O
which	O
oracle	O
is	O
given	O
.	O
</s>
<s>
The	O
steps	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
are	O
given	O
as	O
follows	O
:	O
</s>
<s>
There	O
is	O
a	O
geometric	O
interpretation	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
following	O
from	O
the	O
observation	O
that	O
the	O
quantum	O
state	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
stays	O
in	O
a	O
two-dimensional	O
subspace	O
after	O
each	O
step	O
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
begins	O
with	O
the	O
initial	O
ket	O
,	O
which	O
lies	O
in	O
the	O
subspace	O
.	O
</s>
<s>
This	O
can	O
be	O
seen	O
by	O
writing	O
in	O
the	O
form	O
of	O
a	O
Householder	B-Algorithm
reflection	I-Algorithm
:	O
</s>
<s>
Therefore	O
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
stays	O
in	O
this	O
plane	O
for	O
the	O
entire	O
algorithm	O
.	O
</s>
<s>
A	O
simple	O
solution	O
performs	O
optimally	O
up	O
to	O
a	O
constant	O
factor	O
:	O
run	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
repeatedly	O
for	O
increasingly	O
small	O
values	O
of	O
k	O
,	O
e.g.	O
,	O
taking	O
k	O
=	O
N	O
,	O
N/2	O
,	O
N/4	O
,	O
...	O
,	O
and	O
so	O
on	O
,	O
taking	O
for	O
iteration	O
t	O
until	O
a	O
matching	O
entry	O
is	O
found	O
.	O
</s>
<s>
A	O
version	O
of	O
this	O
algorithm	O
is	O
used	O
in	O
order	O
to	O
solve	O
the	O
collision	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
A	O
modification	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
called	O
quantum	O
partial	O
search	O
was	O
described	O
by	O
Grover	O
and	O
Radhakrishnan	O
in	O
2004	O
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
requires	O
iterations	O
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
optimal	O
up	O
to	O
sub-constant	O
factors	O
.	O
</s>
<s>
That	O
is	O
,	O
any	O
algorithm	O
that	O
accesses	O
the	O
database	O
only	O
by	O
using	O
the	O
operator	O
Uω	O
must	O
apply	O
Uω	O
at	O
least	O
a	O
fraction	O
as	O
many	O
times	O
as	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
The	O
extension	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
k	O
matching	O
entries	O
,	O
(	O
N/k	O
)	O
1/2/4	O
,	O
is	O
also	O
optimal	O
.	O
</s>
<s>
This	O
result	O
is	O
important	O
in	O
understanding	O
the	O
limits	O
of	O
quantum	B-Architecture
computation	I-Architecture
.	O
</s>
<s>
The	O
optimality	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
suggests	O
that	O
quantum	B-Architecture
computers	I-Architecture
cannot	O
solve	O
NP-Complete	O
problems	O
in	O
polynomial	O
time	O
,	O
and	O
thus	O
NP	O
is	O
not	O
contained	O
in	O
BQP	O
.	O
</s>
<s>
It	O
has	O
been	O
shown	O
that	O
a	O
class	O
of	O
non-local	O
hidden	O
variable	O
quantum	B-Architecture
computers	I-Architecture
could	O
implement	O
a	O
search	O
of	O
an	O
-item	O
database	O
in	O
at	O
most	O
steps	O
.	O
</s>
<s>
This	O
is	O
faster	O
than	O
the	O
steps	O
taken	O
by	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
