<s>
Quantum	B-Algorithm
optimization	I-Algorithm
algorithms	I-Algorithm
are	O
quantum	B-Device
algorithms	I-Device
that	O
are	O
used	O
to	O
solve	O
optimization	O
problems	O
.	O
</s>
<s>
The	O
power	O
of	O
quantum	B-Architecture
computing	I-Architecture
may	O
allow	O
problems	O
which	O
are	O
not	O
practically	O
feasible	O
on	O
classical	O
computers	O
to	O
be	O
solved	O
,	O
or	O
suggest	O
a	O
considerable	O
speed	O
up	O
with	O
respect	O
to	O
the	O
best	O
known	O
classical	O
algorithm	O
.	O
</s>
<s>
Data	B-Algorithm
fitting	I-Algorithm
is	O
a	O
process	O
of	O
constructing	O
a	O
mathematical	O
function	O
that	O
best	B-Algorithm
fits	I-Algorithm
a	O
set	O
of	O
data	O
points	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
common	O
types	O
of	O
data	B-Algorithm
fitting	I-Algorithm
is	O
solving	O
the	O
least	B-Algorithm
squares	I-Algorithm
problem	I-Algorithm
,	O
minimizing	O
the	O
sum	O
of	O
the	O
squares	O
of	O
differences	O
between	O
the	O
data	O
points	O
and	O
the	O
fitted	O
function	O
.	O
</s>
<s>
The	O
quantum	O
least-squares	B-Algorithm
fitting	I-Algorithm
algorithm	O
makes	O
use	O
of	O
a	O
version	O
of	O
Harrow	O
,	O
Hassidim	O
,	O
and	O
Lloyd	O
's	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
(	O
HHL	O
)	O
,	O
and	O
outputs	O
the	O
coefficients	O
and	O
the	O
fit	O
quality	O
estimation	O
.	O
</s>
<s>
Because	O
the	O
quantum	B-Device
algorithm	I-Device
is	O
mainly	O
based	O
on	O
the	O
HHL	B-Algorithm
algorithm	I-Algorithm
,	O
it	O
suggests	O
an	O
exponential	O
improvement	O
in	O
the	O
case	O
where	O
is	O
sparse	B-Algorithm
and	O
the	O
condition	B-Algorithm
number	I-Algorithm
(	O
namely	O
,	O
the	O
ratio	O
between	O
the	O
largest	O
and	O
the	O
smallest	O
eigenvalues	O
)	O
of	O
both	O
and	O
is	O
small	O
.	O
</s>
<s>
Semidefinite	O
programming	O
(	O
SDP	O
)	O
is	O
an	O
optimization	O
subfield	O
dealing	O
with	O
the	O
optimization	O
of	O
a	O
linear	O
objective	O
function	O
(	O
a	O
user-specified	O
function	O
to	O
be	O
minimized	O
or	O
maximized	O
)	O
,	O
over	O
the	O
intersection	O
of	O
the	O
cone	O
of	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrices	I-Algorithm
with	O
an	O
affine	O
space	O
.	O
</s>
<s>
The	O
problem	O
may	O
have	O
additional	O
constraints	B-Application
(	O
given	O
as	O
inputs	O
)	O
,	O
also	O
usually	O
formulated	O
as	O
inner	O
products	O
.	O
</s>
<s>
Each	O
constraint	B-Application
forces	O
the	O
inner	O
product	O
of	O
the	O
matrices	O
(	O
given	O
as	O
an	O
input	O
)	O
with	O
the	O
optimization	O
variable	O
to	O
be	O
smaller	O
than	O
a	O
specified	O
value	O
(	O
given	O
as	O
an	O
input	O
)	O
.	O
</s>
<s>
The	O
quantum	B-Device
algorithm	I-Device
consists	O
of	O
several	O
iterations	O
.	O
</s>
<s>
In	O
each	O
iteration	O
,	O
a	O
different	O
threshold	O
is	O
chosen	O
,	O
and	O
the	O
algorithm	O
outputs	O
either	O
a	O
solution	O
such	O
that	O
(	O
and	O
the	O
other	O
constraints	B-Application
are	O
satisfied	O
,	O
too	O
)	O
or	O
an	O
indication	O
that	O
no	O
such	O
solution	O
exists	O
.	O
</s>
<s>
The	O
quantum	B-Device
algorithm	I-Device
provides	O
a	O
quadratic	O
improvement	O
over	O
the	O
best	O
classical	O
algorithm	O
in	O
the	O
general	O
case	O
,	O
and	O
an	O
exponential	O
improvement	O
when	O
the	O
input	O
matrices	O
are	O
of	O
low	O
rank	O
.	O
</s>
<s>
Approximate	B-Algorithm
optimization	I-Algorithm
is	O
a	O
way	O
of	O
finding	O
an	O
approximate	O
solution	O
to	O
an	O
optimization	O
problem	O
,	O
which	O
is	O
often	O
NP-hard	O
.	O
</s>
<s>
For	O
combinatorial	O
optimization	O
,	O
the	O
quantum	B-Algorithm
approximate	I-Algorithm
optimization	I-Algorithm
algorithm	I-Algorithm
(	O
QAOA	O
)	O
briefly	O
had	O
a	O
better	O
approximation	B-Algorithm
ratio	I-Algorithm
than	O
any	O
known	O
polynomial	O
time	O
classical	O
algorithm	O
(	O
for	O
a	O
certain	O
problem	O
)	O
,	O
until	O
a	O
more	O
effective	O
classical	O
algorithm	O
was	O
proposed	O
.	O
</s>
<s>
The	O
relative	O
speed-up	O
of	O
the	O
quantum	B-Device
algorithm	I-Device
is	O
an	O
open	O
research	O
question	O
.	O
</s>
<s>
The	O
heart	O
of	O
QAOA	O
relies	O
on	O
the	O
use	O
of	O
unitary	B-Algorithm
operators	I-Algorithm
dependent	O
on	O
angles	O
,	O
where	O
is	O
an	O
input	O
integer	O
.	O
</s>
<s>
In	O
2020	O
,	O
it	O
was	O
shown	O
that	O
QAOA	O
exhibits	O
a	O
strong	O
dependence	O
on	O
the	O
ratio	O
of	O
a	O
problem	O
's	O
constraint	B-Application
to	O
variables	O
(	O
problem	O
density	O
)	O
placing	O
a	O
limiting	O
restriction	O
on	O
the	O
algorithm	O
's	O
capacity	O
to	O
minimize	O
a	O
corresponding	O
objective	O
function	O
.	O
</s>
<s>
In	O
the	O
paper	O
How	O
many	O
qubits	O
are	O
needed	O
for	O
quantum	B-Device
computational	I-Device
supremacy	I-Device
submitted	O
to	O
arXiv	O
,	O
the	O
authors	O
conclude	O
that	O
a	O
QAOA	O
circuit	O
with	O
420	O
qubits	O
and	O
500	O
constraints	B-Application
would	O
require	O
at	O
least	O
one	O
century	O
to	O
be	O
simulated	O
using	O
a	O
classical	O
simulation	O
algorithm	O
running	O
on	O
state-of-the-art	O
supercomputers	B-Architecture
so	O
that	O
would	O
be	O
sufficient	O
for	O
quantum	B-Device
computational	I-Device
supremacy	I-Device
.	O
</s>
<s>
A	O
rigorous	O
comparison	O
of	O
QAOA	O
with	O
classical	O
algorithms	O
can	O
give	O
estimates	O
on	O
depth	O
and	O
number	O
of	O
qubits	O
required	O
for	O
quantum	B-Device
advantage	I-Device
.	O
</s>
