<s>
In	O
quantum	B-Architecture
computing	I-Architecture
,	O
a	O
quantum	B-Device
algorithm	I-Device
is	O
an	O
algorithm	B-Algorithm
which	O
runs	O
on	O
a	O
realistic	O
model	O
of	O
quantum	B-Architecture
computation	I-Architecture
,	O
the	O
most	O
commonly	O
used	O
model	O
being	O
the	O
quantum	B-Application
circuit	I-Application
model	O
of	O
computation	O
.	O
</s>
<s>
A	O
classical	O
(	O
or	O
non-quantum	O
)	O
algorithm	B-Algorithm
is	O
a	O
finite	O
sequence	O
of	O
instructions	O
,	O
or	O
a	O
step-by-step	O
procedure	O
for	O
solving	O
a	O
problem	O
,	O
where	O
each	O
step	O
or	O
instruction	O
can	O
be	O
performed	O
on	O
a	O
classical	O
computer	O
.	O
</s>
<s>
Similarly	O
,	O
a	O
quantum	B-Device
algorithm	I-Device
is	O
a	O
step-by-step	O
procedure	O
,	O
where	O
each	O
of	O
the	O
steps	O
can	O
be	O
performed	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
Although	O
all	O
classical	O
algorithms	O
can	O
also	O
be	O
performed	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
,	O
the	O
term	O
quantum	B-Device
algorithm	I-Device
is	O
usually	O
used	O
for	O
those	O
algorithms	O
which	O
seem	O
inherently	O
quantum	O
,	O
or	O
use	O
some	O
essential	O
feature	O
of	O
quantum	B-Architecture
computation	I-Architecture
such	O
as	O
quantum	O
superposition	O
or	O
quantum	O
entanglement	O
.	O
</s>
<s>
Problems	O
which	O
are	O
undecidable	O
using	O
classical	O
computers	O
remain	O
undecidable	O
using	O
quantum	B-Architecture
computers	I-Architecture
.	O
</s>
<s>
What	O
makes	O
quantum	B-Device
algorithms	I-Device
interesting	O
is	O
that	O
they	O
might	O
be	O
able	O
to	O
solve	O
some	O
problems	O
faster	O
than	O
classical	O
algorithms	O
because	O
the	O
quantum	O
superposition	O
and	O
quantum	O
entanglement	O
that	O
quantum	B-Device
algorithms	I-Device
exploit	O
probably	O
cannot	O
be	O
efficiently	O
simulated	O
on	O
classical	O
computers	O
(	O
see	O
Quantum	B-Device
supremacy	I-Device
)	O
.	O
</s>
<s>
The	O
best-known	O
algorithms	O
are	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
factoring	O
and	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
searching	O
an	O
unstructured	O
database	O
or	O
an	O
unordered	O
list	O
.	O
</s>
<s>
Shor	B-Algorithm
's	I-Algorithm
algorithms	I-Algorithm
runs	O
much	O
(	O
almost	O
exponentially	O
)	O
faster	O
than	O
the	O
best-known	O
classical	O
algorithm	B-Algorithm
for	O
factoring	O
,	O
the	O
general	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
runs	O
quadratically	O
faster	O
than	O
the	O
best	O
possible	O
classical	O
algorithm	B-Algorithm
for	O
the	O
same	O
task	O
,	O
a	O
linear	B-Algorithm
search	I-Algorithm
.	O
</s>
<s>
Quantum	B-Device
algorithms	I-Device
are	O
usually	O
described	O
,	O
in	O
the	O
commonly	O
used	O
circuit	O
model	O
of	O
quantum	B-Architecture
computation	I-Architecture
,	O
by	O
a	O
quantum	B-Application
circuit	I-Application
which	O
acts	O
on	O
some	O
input	O
qubits	O
and	O
terminates	O
with	O
a	O
measurement	O
.	O
</s>
<s>
A	O
quantum	B-Application
circuit	I-Application
consists	O
of	O
simple	O
quantum	B-Application
gates	I-Application
which	O
act	O
on	O
at	O
most	O
a	O
fixed	O
number	O
of	O
qubits	O
.	O
</s>
<s>
Quantum	B-Device
algorithms	I-Device
may	O
also	O
be	O
stated	O
in	O
other	O
models	O
of	O
quantum	B-Architecture
computation	I-Architecture
,	O
such	O
as	O
the	O
Hamiltonian	O
oracle	O
model	O
.	O
</s>
<s>
Quantum	B-Device
algorithms	I-Device
can	O
be	O
categorized	O
by	O
the	O
main	O
techniques	O
used	O
by	O
the	O
algorithm	B-Algorithm
.	O
</s>
<s>
Some	O
commonly	O
used	O
techniques/ideas	O
in	O
quantum	B-Device
algorithms	I-Device
include	O
phase	O
kick-back	O
,	O
phase	B-Algorithm
estimation	I-Algorithm
,	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
quantum	B-Algorithm
walks	I-Algorithm
,	O
amplitude	B-Algorithm
amplification	I-Algorithm
and	O
topological	O
quantum	O
field	O
theory	O
.	O
</s>
<s>
Quantum	B-Device
algorithms	I-Device
may	O
also	O
be	O
grouped	O
by	O
the	O
type	O
of	O
problem	O
solved	O
,	O
for	O
instance	O
see	O
the	O
survey	O
on	O
quantum	B-Device
algorithms	I-Device
for	O
algebraic	O
problems	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
is	O
the	O
quantum	O
analogue	O
of	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
and	O
is	O
used	O
in	O
several	O
quantum	B-Device
algorithms	I-Device
.	O
</s>
<s>
The	O
Hadamard	B-Algorithm
transform	I-Algorithm
is	O
also	O
an	O
example	O
of	O
a	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
over	O
an	O
n-dimensional	O
vector	O
space	O
over	O
the	O
field	O
F2	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
can	O
be	O
efficiently	O
implemented	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
using	O
only	O
a	O
polynomial	O
number	O
of	O
quantum	B-Application
gates	I-Application
.	O
</s>
<s>
The	O
Deutsch	O
–	O
Jozsa	O
algorithm	B-Algorithm
solves	O
a	O
black-box	O
problem	O
which	O
probably	O
requires	O
exponentially	O
many	O
queries	O
to	O
the	O
black	O
box	O
for	O
any	O
deterministic	O
classical	O
computer	O
,	O
but	O
can	O
be	O
done	O
with	O
one	O
query	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
However	O
,	O
when	O
comparing	O
bounded-error	O
classical	O
and	O
quantum	B-Device
algorithms	I-Device
,	O
there	O
is	O
no	O
speedup	O
since	O
a	O
classical	O
probabilistic	O
algorithm	B-Algorithm
can	O
solve	O
the	O
problem	O
with	O
a	O
constant	O
number	O
of	O
queries	O
with	O
small	O
probability	O
of	O
error	O
.	O
</s>
<s>
The	O
algorithm	B-Algorithm
determines	O
whether	O
a	O
function	O
f	O
is	O
either	O
constant	O
(	O
0	O
on	O
all	O
inputs	O
or	O
1	O
on	O
all	O
inputs	O
)	O
or	O
balanced	O
(	O
returns	O
1	O
for	O
half	O
of	O
the	O
input	O
domain	O
and	O
0	O
for	O
the	O
other	O
half	O
)	O
.	O
</s>
<s>
The	O
Bernstein	O
–	O
Vazirani	O
algorithm	B-Algorithm
is	O
the	O
first	O
quantum	B-Device
algorithm	I-Device
that	O
solves	O
a	O
problem	O
more	O
efficiently	O
than	O
the	O
best	O
known	O
classical	O
algorithm	B-Algorithm
.	O
</s>
<s>
Simon	O
's	O
algorithm	B-Algorithm
solves	O
a	O
black-box	O
problem	O
exponentially	O
faster	O
than	O
any	O
classical	O
algorithm	B-Algorithm
,	O
including	O
bounded-error	O
probabilistic	O
algorithms	O
.	O
</s>
<s>
This	O
algorithm	B-Algorithm
,	O
which	O
achieves	O
an	O
exponential	O
speedup	O
over	O
all	O
classical	O
algorithms	O
that	O
we	O
consider	O
efficient	O
,	O
was	O
the	O
motivation	O
for	O
Shor	O
's	O
factoring	O
algorithm	B-Algorithm
.	O
</s>
<s>
The	O
quantum	B-Algorithm
phase	I-Algorithm
estimation	I-Algorithm
algorithm	I-Algorithm
is	O
used	O
to	O
determine	O
the	O
eigenphase	O
of	O
an	O
eigenvector	O
of	O
a	O
unitary	O
gate	O
given	O
a	O
quantum	O
state	O
proportional	O
to	O
the	O
eigenvector	O
and	O
access	O
to	O
the	O
gate	O
.	O
</s>
<s>
The	O
algorithm	B-Algorithm
is	O
frequently	O
used	O
as	O
a	O
subroutine	O
in	O
other	O
algorithms	O
.	O
</s>
<s>
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
solves	O
the	O
discrete	O
logarithm	O
problem	O
and	O
the	O
integer	O
factorization	O
problem	O
in	O
polynomial	O
time	O
,	O
whereas	O
the	O
best	O
known	O
classical	O
algorithms	O
take	O
super-polynomial	O
time	O
.	O
</s>
<s>
It	O
is	O
also	O
one	O
of	O
the	O
few	O
quantum	B-Device
algorithms	I-Device
that	O
solves	O
a	O
nonblack-box	O
problem	O
in	O
polynomial	O
time	O
where	O
the	O
best	O
known	O
classical	O
algorithms	O
run	O
in	O
super-polynomial	O
time	O
.	O
</s>
<s>
The	O
abelian	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
is	O
a	O
generalization	O
of	O
many	O
problems	O
that	O
can	O
be	O
solved	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
,	O
such	O
as	O
Simon	O
's	O
problem	O
,	O
solving	O
Pell	O
's	O
equation	O
,	O
testing	O
the	O
principal	O
ideal	O
of	O
a	O
ring	O
R	O
and	O
factoring	O
.	O
</s>
<s>
There	O
are	O
efficient	O
quantum	B-Device
algorithms	I-Device
known	O
for	O
the	O
Abelian	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
more	O
general	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
,	O
where	O
the	O
group	O
is	O
n't	O
necessarily	O
abelian	O
,	O
is	O
a	O
generalization	O
of	O
the	O
previously	O
mentioned	O
problems	O
and	O
graph	O
isomorphism	O
and	O
certain	O
lattice	O
problems	O
.	O
</s>
<s>
Efficient	O
quantum	B-Device
algorithms	I-Device
are	O
known	O
for	O
certain	O
non-abelian	O
groups	O
.	O
</s>
<s>
However	O
,	O
no	O
efficient	O
algorithms	O
are	O
known	O
for	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
,	O
which	O
would	O
give	O
an	O
efficient	O
algorithm	B-Algorithm
for	O
graph	O
isomorphism	O
and	O
the	O
dihedral	B-Algorithm
group	I-Algorithm
,	O
which	O
would	O
solve	O
certain	O
lattice	O
problems	O
.	O
</s>
<s>
Solving	O
this	O
problem	O
with	O
a	O
classical	O
computer	O
algorithm	B-Algorithm
requires	O
computing	O
the	O
permanent	O
of	O
the	O
unitary	O
transform	O
matrix	O
,	O
which	O
may	O
be	O
either	O
impossible	O
or	O
take	O
a	O
prohibitively	O
long	O
time	O
.	O
</s>
<s>
In	O
2014	O
,	O
it	O
was	O
proposed	O
that	O
existing	O
technology	O
and	O
standard	O
probabilistic	O
methods	O
of	O
generating	O
single	O
photon	O
states	O
could	O
be	O
used	O
as	O
input	O
into	O
a	O
suitable	O
quantum	O
computable	O
linear	O
optical	O
network	O
and	O
that	O
sampling	O
of	O
the	O
output	O
probability	O
distribution	O
would	O
be	O
demonstrably	O
superior	O
using	O
quantum	B-Device
algorithms	I-Device
.	O
</s>
<s>
The	O
best	O
known	O
classical	O
algorithm	B-Algorithm
for	O
estimating	O
these	O
sums	O
takes	O
exponential	O
time	O
.	O
</s>
<s>
Since	O
the	O
discrete	O
logarithm	O
problem	O
reduces	O
to	O
Gauss	O
sum	O
estimation	O
,	O
an	O
efficient	O
classical	O
algorithm	B-Algorithm
for	O
estimating	O
Gauss	O
sums	O
would	O
imply	O
an	O
efficient	O
classical	O
algorithm	B-Algorithm
for	O
computing	O
discrete	O
logarithms	O
,	O
which	O
is	O
considered	O
unlikely	O
.	O
</s>
<s>
However	O
,	O
quantum	B-Architecture
computers	I-Architecture
can	O
estimate	O
Gauss	O
sums	O
to	O
polynomial	O
precision	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
Amplitude	B-Algorithm
amplification	I-Algorithm
is	O
a	O
technique	O
that	O
allows	O
the	O
amplification	O
of	O
a	O
chosen	O
subspace	O
of	O
a	O
quantum	O
state	O
.	O
</s>
<s>
Applications	O
of	O
amplitude	B-Algorithm
amplification	I-Algorithm
usually	O
lead	O
to	O
quadratic	O
speedups	O
over	O
the	O
corresponding	O
classical	O
algorithms	O
.	O
</s>
<s>
It	O
can	O
be	O
considered	O
to	O
be	O
a	O
generalization	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
searches	O
an	O
unstructured	O
database	O
(	O
or	O
an	O
unordered	O
list	O
)	O
with	O
N	O
entries	O
,	O
for	O
a	O
marked	O
entry	O
,	O
using	O
only	O
queries	O
instead	O
of	O
the	O
queries	O
required	O
classically	O
.	O
</s>
<s>
Theorists	O
have	O
considered	O
a	O
hypothetical	O
generalization	O
of	O
a	O
standard	O
quantum	B-Architecture
computer	I-Architecture
that	O
could	O
access	O
the	O
histories	O
of	O
the	O
hidden	O
variables	O
in	O
Bohmian	O
mechanics	O
.	O
</s>
<s>
(	O
Such	O
a	O
computer	O
is	O
completely	O
hypothetical	O
and	O
would	O
not	O
be	O
a	O
standard	O
quantum	B-Architecture
computer	I-Architecture
,	O
or	O
even	O
possible	O
under	O
the	O
standard	O
theory	O
of	O
quantum	O
mechanics	O
.	O
)	O
</s>
<s>
This	O
is	O
slightly	O
faster	O
than	O
the	O
steps	O
taken	O
by	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Neither	O
search	O
method	O
would	O
allow	O
either	O
model	O
of	O
quantum	B-Architecture
computer	I-Architecture
to	O
solve	O
NP-complete	O
problems	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
Quantum	B-Algorithm
counting	I-Algorithm
solves	O
a	O
generalization	O
of	O
the	O
search	O
problem	O
.	O
</s>
<s>
More	O
precisely	O
,	O
the	O
algorithm	B-Algorithm
outputs	O
an	O
estimate	O
for	O
,	O
the	O
number	O
of	O
marked	O
entries	O
,	O
with	O
the	O
following	O
accuracy	O
:	O
.	O
</s>
<s>
A	O
quantum	B-Algorithm
walk	I-Algorithm
is	O
the	O
quantum	O
analogue	O
of	O
a	O
classical	O
random	O
walk	O
,	O
which	O
can	O
be	O
described	O
by	O
a	O
probability	O
distribution	O
over	O
some	O
states	O
.	O
</s>
<s>
A	O
quantum	B-Algorithm
walk	I-Algorithm
can	O
be	O
described	O
by	O
a	O
quantum	O
superposition	O
over	O
states	O
.	O
</s>
<s>
Quantum	B-Algorithm
walks	I-Algorithm
are	O
known	O
to	O
give	O
exponential	O
speedups	O
for	O
some	O
black-box	O
problems	O
.	O
</s>
<s>
A	O
framework	O
for	O
the	O
creation	O
of	O
quantum	B-Algorithm
walk	I-Algorithm
algorithms	O
exists	O
and	O
is	O
quite	O
a	O
versatile	O
tool	O
.	O
</s>
<s>
Classically	O
,	O
Ω(N )	O
queries	O
are	O
required	O
for	O
a	O
list	O
of	O
size	O
N	O
.	O
However	O
,	O
it	O
can	O
be	O
solved	O
in	O
queries	O
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
The	O
optimal	O
algorithm	B-Algorithm
is	O
by	O
Andris	O
Ambainis	O
.	O
</s>
<s>
The	O
best-known	O
lower	O
bound	O
for	O
quantum	B-Device
algorithms	I-Device
is	O
Ω(N )	O
,	O
but	O
the	O
best	O
algorithm	B-Algorithm
known	O
requires	O
O( 	O
N1.297	O
)	O
queries	O
,	O
an	O
improvement	O
over	O
the	O
previous	O
best	O
O( 	O
N1.3	O
)	O
queries	O
.	O
</s>
<s>
With	O
a	O
quantum	B-Device
algorithm	I-Device
however	O
,	O
it	O
can	O
be	O
solved	O
in	O
Θ( 	O
N0.5	O
)	O
queries	O
.	O
</s>
<s>
No	O
better	O
quantum	B-Device
algorithm	I-Device
for	O
this	O
case	O
was	O
known	O
until	O
one	O
was	O
found	O
for	O
the	O
unconventional	O
Hamiltonian	O
oracle	O
model	O
.	O
</s>
<s>
Fast	O
quantum	B-Device
algorithms	I-Device
for	O
more	O
complicated	O
formulas	O
are	O
also	O
known	O
.	O
</s>
<s>
A	O
quantum	B-Device
algorithm	I-Device
requires	O
queries	O
but	O
the	O
best	O
known	O
algorithm	B-Algorithm
uses	O
queries	O
.	O
</s>
<s>
The	O
complexity	O
class	O
BQP	O
(	O
bounded-error	O
quantum	O
polynomial	O
time	O
)	O
is	O
the	O
set	O
of	O
decision	O
problems	O
solvable	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
in	O
polynomial	O
time	O
with	O
error	O
probability	O
of	O
at	O
most	O
1/3	O
for	O
all	O
instances	O
.	O
</s>
<s>
A	O
problem	O
is	O
BQP-complete	O
if	O
it	O
is	O
in	O
BQP	O
and	O
any	O
problem	O
in	O
BQP	O
can	O
be	O
reduced	B-Algorithm
to	O
it	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
Informally	O
,	O
the	O
class	O
of	O
BQP-complete	O
problems	O
are	O
those	O
that	O
are	O
as	O
hard	O
as	O
the	O
hardest	O
problems	O
in	O
BQP	O
and	O
are	O
themselves	O
efficiently	O
solvable	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
(	O
with	O
bounded	O
error	O
)	O
.	O
</s>
<s>
A	O
quantum	B-Architecture
computer	I-Architecture
can	O
simulate	O
a	O
TQFT	O
,	O
and	O
thereby	O
approximate	O
the	O
Jones	O
polynomial	O
,	O
which	O
as	O
far	O
as	O
we	O
know	O
,	O
is	O
hard	O
to	O
compute	O
classically	O
in	O
the	O
worst-case	O
scenario	O
.	O
</s>
<s>
The	O
idea	O
that	O
quantum	B-Architecture
computers	I-Architecture
might	O
be	O
more	O
powerful	O
than	O
classical	O
computers	O
originated	O
in	O
Richard	O
Feynman	O
's	O
observation	O
that	O
classical	O
computers	O
seem	O
to	O
require	O
exponential	O
time	O
to	O
simulate	O
many-particle	O
quantum	O
systems	O
.	O
</s>
<s>
Since	O
then	O
,	O
the	O
idea	O
that	O
quantum	B-Architecture
computers	I-Architecture
can	O
simulate	O
quantum	O
physical	O
processes	O
exponentially	O
faster	O
than	O
classical	O
computers	O
has	O
been	O
greatly	O
fleshed	O
out	O
and	O
elaborated	O
.	O
</s>
<s>
Efficient	O
(	O
that	O
is	O
,	O
polynomial-time	O
)	O
quantum	B-Device
algorithms	I-Device
have	O
been	O
developed	O
for	O
simulating	O
both	O
Bosonic	O
and	O
Fermionic	O
systems	O
and	O
in	O
particular	O
,	O
the	O
simulation	O
of	O
chemical	O
reactions	O
beyond	O
the	O
capabilities	O
of	O
current	O
classical	O
supercomputers	O
requires	O
only	O
a	O
few	O
hundred	O
qubits	O
.	O
</s>
<s>
Quantum	B-Architecture
computers	I-Architecture
can	O
also	O
efficiently	O
simulate	O
topological	O
quantum	O
field	O
theories	O
.	O
</s>
<s>
In	O
addition	O
to	O
its	O
intrinsic	O
interest	O
,	O
this	O
result	O
has	O
led	O
to	O
efficient	O
quantum	B-Device
algorithms	I-Device
for	O
estimating	O
quantum	O
topological	O
invariants	O
such	O
as	O
Jones	O
and	O
HOMFLY	O
polynomials	O
,	O
and	O
the	O
Turaev-Viro	O
invariant	O
of	O
three-dimensional	O
manifolds	O
.	O
</s>
<s>
In	O
2009	O
Aram	O
Harrow	O
,	O
Avinatan	O
Hassidim	O
,	O
and	O
Seth	O
Lloyd	O
,	O
formulated	O
a	O
quantum	B-Device
algorithm	I-Device
for	O
solving	O
linear	O
systems	O
.	O
</s>
<s>
The	O
algorithm	B-Algorithm
estimates	O
the	O
result	O
of	O
a	O
scalar	O
measurement	O
on	O
the	O
solution	O
vector	O
to	O
a	O
given	O
linear	O
system	O
of	O
equations	O
.	O
</s>
<s>
Provided	O
the	O
linear	O
system	O
is	O
a	O
sparse	B-Algorithm
and	O
has	O
a	O
low	O
condition	B-Algorithm
number	I-Algorithm
,	O
and	O
that	O
the	O
user	O
is	O
interested	O
in	O
the	O
result	O
of	O
a	O
scalar	O
measurement	O
on	O
the	O
solution	O
vector	O
,	O
instead	O
of	O
the	O
values	O
of	O
the	O
solution	O
vector	O
itself	O
,	O
then	O
the	O
algorithm	B-Algorithm
has	O
a	O
runtime	O
of	O
,	O
where	O
is	O
the	O
number	O
of	O
variables	O
in	O
the	O
linear	O
system	O
.	O
</s>
<s>
This	O
offers	O
an	O
exponential	O
speedup	O
over	O
the	O
fastest	O
classical	O
algorithm	B-Algorithm
,	O
which	O
runs	O
in	O
(	O
or	O
for	O
positive	O
semidefinite	O
matrices	O
)	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
approximate	I-Algorithm
optimization	I-Algorithm
algorithm	I-Algorithm
is	O
a	O
toy	O
model	O
of	O
quantum	O
annealing	O
which	O
can	O
be	O
used	O
to	O
solve	O
problems	O
in	O
graph	O
theory	O
.	O
</s>
<s>
The	O
algorithm	B-Algorithm
makes	O
use	O
of	O
classical	O
optimization	O
of	O
quantum	O
operations	O
to	O
maximize	O
an	O
objective	O
function	O
.	O
</s>
<s>
The	O
variational	B-Algorithm
quantum	I-Algorithm
eigensolver	I-Algorithm
(	O
VQE	O
)	O
algorithm	B-Algorithm
applies	O
classical	O
optimization	O
to	O
minimize	O
the	O
energy	O
expectation	O
of	O
an	O
ansatz	O
state	O
to	O
find	O
the	O
ground	O
state	O
energy	O
of	O
a	O
molecule	O
.	O
</s>
<s>
The	O
CQE	O
algorithm	B-Algorithm
minimizes	O
the	O
residual	O
of	O
a	O
contraction	O
(	O
or	O
projection	O
)	O
of	O
the	O
Schrödinger	O
equation	O
onto	O
the	O
space	O
of	O
two	O
(	O
or	O
more	O
)	O
electrons	O
to	O
find	O
the	O
ground	O
-	O
or	O
excited-state	O
energy	O
and	O
two-electron	O
reduced	B-Algorithm
density	O
matrix	O
of	O
a	O
molecule	O
.	O
</s>
<s>
It	O
is	O
based	O
on	O
classical	O
methods	O
for	O
solving	O
energies	O
and	O
two-electron	O
reduced	B-Algorithm
density	O
matrices	O
directly	O
from	O
the	O
anti-Hermitian	O
contracted	O
Schrödinger	O
equation	O
.	O
</s>
