<s>
The	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
,	O
also	O
called	O
HHL	B-Algorithm
algorithm	I-Algorithm
,	O
designed	O
by	O
Aram	O
Harrow	O
,	O
Avinatan	O
Hassidim	O
,	O
and	O
Seth	O
Lloyd	O
,	O
is	O
a	O
quantum	B-Device
algorithm	I-Device
published	O
in	O
2008	O
for	O
solving	O
linear	O
systems	O
.	O
</s>
<s>
The	O
algorithm	O
is	O
one	O
of	O
the	O
main	O
fundamental	O
algorithms	O
expected	O
to	O
provide	O
a	O
speedup	O
over	O
their	O
classical	O
counterparts	O
,	O
along	O
with	O
Shor	B-Algorithm
's	I-Algorithm
factoring	I-Algorithm
algorithm	I-Algorithm
,	O
Grover	B-Algorithm
's	I-Algorithm
search	I-Algorithm
algorithm	I-Algorithm
,	O
and	O
the	O
quantum	B-Algorithm
fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Provided	O
the	O
linear	O
system	O
is	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	O
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>
An	O
implementation	O
of	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
was	O
first	O
demonstrated	O
in	O
2013	O
by	O
Cai	O
et	O
al.	O
,	O
Barz	O
et	O
al	O
.	O
</s>
<s>
Due	O
to	O
the	O
prevalence	O
of	O
linear	O
systems	O
in	O
virtually	O
all	O
areas	O
of	O
science	O
and	O
engineering	O
,	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
has	O
the	O
potential	O
for	O
widespread	O
applicability	O
.	O
</s>
<s>
The	O
problem	O
we	O
are	O
trying	O
to	O
solve	O
is	O
:	O
given	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
and	O
a	O
unit	O
vector	O
,	O
find	O
the	O
solution	O
vector	O
satisfying	O
.	O
</s>
<s>
Next	O
,	O
Hamiltonian	O
simulation	O
techniques	O
are	O
used	O
to	O
apply	O
the	O
unitary	B-Algorithm
operator	I-Algorithm
to	O
for	O
a	O
superposition	O
of	O
different	O
times	O
.	O
</s>
<s>
The	O
ability	O
to	O
decompose	O
into	O
the	O
eigenbasis	O
of	O
and	O
to	O
find	O
the	O
corresponding	O
eigenvalues	O
is	O
facilitated	O
by	O
the	O
use	O
of	O
quantum	B-Algorithm
phase	I-Algorithm
estimation	I-Algorithm
.	O
</s>
<s>
Firstly	O
,	O
the	O
algorithm	O
requires	O
that	O
the	O
matrix	O
be	O
Hermitian	B-Algorithm
so	O
that	O
it	O
can	O
be	O
converted	O
into	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
As	O
is	O
Hermitian	B-Algorithm
,	O
the	O
algorithm	O
can	O
now	O
be	O
used	O
to	O
solve	O
to	O
obtain	O
.	O
</s>
<s>
Hamiltonian	O
simulation	O
is	O
used	O
to	O
transform	O
the	O
Hermitian	B-Algorithm
matrix	I-Algorithm
into	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
,	O
which	O
can	O
then	O
be	O
applied	O
at	O
will	O
.	O
</s>
<s>
This	O
is	O
possible	O
if	O
A	O
is	O
s-sparse	O
and	O
efficiently	O
row	O
computable	O
,	O
meaning	O
it	O
has	O
at	O
most	O
s	O
nonzero	O
entries	O
per	O
row	O
and	O
given	O
a	O
row	O
index	O
these	O
entries	O
can	O
be	O
computed	O
in	O
timeO(s )	O
.	O
</s>
<s>
The	O
key	O
subroutine	O
to	O
the	O
algorithm	O
,	O
denoted	O
,	O
is	O
defined	O
as	O
follows	O
and	O
incorporates	O
a	O
phase	B-Algorithm
estimation	I-Algorithm
subroutine	O
:	O
</s>
<s>
The	O
phase	B-Algorithm
estimation	I-Algorithm
procedure	O
in	O
steps	O
1-3	O
allows	O
for	O
the	O
estimation	O
of	O
eigenvalues	O
of	O
A	O
up	O
to	O
error	O
.	O
</s>
<s>
The	O
states	O
'	O
nothing	O
 '	O
,	O
'	O
well	O
 '	O
and	O
'	O
ill	O
 '	O
are	O
used	O
to	O
instruct	O
the	O
loop	O
body	O
on	O
how	O
to	O
proceed	O
;	O
'	O
nothing	O
 '	O
indicates	O
that	O
the	O
desired	O
matrix	O
inversion	O
has	O
not	O
yet	O
taken	O
place	O
,	O
'	O
well	O
 '	O
indicates	O
that	O
the	O
inversion	O
has	O
taken	O
place	O
and	O
the	O
loop	O
should	O
halt	O
,	O
and	O
'	O
ill	O
 '	O
indicates	O
that	O
part	O
of	O
is	O
in	O
the	O
ill-conditioned	B-Algorithm
subspace	O
of	O
A	O
and	O
the	O
algorithm	O
will	O
not	O
be	O
able	O
to	O
produce	O
the	O
desired	O
inversion	O
.	O
</s>
<s>
The	O
body	O
of	O
the	O
algorithm	O
follows	O
the	O
amplitude	B-Algorithm
amplification	I-Algorithm
procedure	O
:	O
starting	O
with	O
,	O
the	O
following	O
operation	O
is	O
repeatedly	O
applied	O
:	O
</s>
<s>
Rather	O
than	O
repeating	O
times	O
to	O
minimize	O
error	O
,	O
amplitude	B-Algorithm
amplification	I-Algorithm
is	O
used	O
to	O
achieve	O
the	O
same	O
error	O
resilience	O
using	O
only	O
repetitions	O
.	O
</s>
<s>
The	O
best	O
classical	O
algorithm	O
which	O
produces	O
the	O
actual	O
solution	O
vector	O
is	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
which	O
runs	O
in	O
time	O
.	O
</s>
<s>
If	O
A	O
is	O
s-sparse	O
and	O
positive	O
semi-definite	O
,	O
then	O
the	O
Conjugate	B-Algorithm
Gradient	I-Algorithm
method	I-Algorithm
can	O
be	O
used	O
to	O
find	O
the	O
solution	O
vector	O
,	O
which	O
can	O
be	O
found	O
in	O
time	O
by	O
minimizing	O
the	O
quadratic	O
function	O
.	O
</s>
<s>
When	O
only	O
a	O
summary	O
statistic	O
of	O
the	O
solution	O
vector	O
is	O
needed	O
,	O
as	O
is	O
the	O
case	O
for	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
,	O
a	O
classical	O
computer	O
can	O
find	O
an	O
estimate	O
of	O
in	O
.	O
</s>
<s>
The	O
runtime	O
of	O
the	O
quantum	B-Device
algorithm	I-Device
for	O
solving	O
systems	O
of	O
linear	O
equations	O
originally	O
proposed	O
by	O
Harrow	O
et	O
al	O
.	O
</s>
<s>
was	O
shown	O
to	O
be	O
,	O
where	O
is	O
the	O
error	O
parameter	O
and	O
is	O
the	O
condition	B-Algorithm
number	I-Algorithm
of	O
.	O
</s>
<s>
This	O
was	O
subsequently	O
improved	O
to	O
by	O
Andris	O
Ambainis	O
and	O
a	O
quantum	B-Device
algorithm	I-Device
with	O
runtime	O
polynomial	O
in	O
was	O
developed	O
by	O
Childs	O
et	O
al	O
.	O
</s>
<s>
Since	O
the	O
HHL	B-Algorithm
algorithm	I-Algorithm
maintains	O
its	O
logarithmic	O
scaling	O
in	O
only	O
for	O
sparse	B-Algorithm
or	O
low	O
rank	O
matrices	O
,	O
Wossnig	O
et	O
al	O
.	O
</s>
<s>
extended	O
the	O
HHL	B-Algorithm
algorithm	I-Algorithm
based	O
on	O
a	O
quantum	O
singular	O
value	O
estimation	O
technique	O
and	O
provided	O
a	O
linear	O
system	O
algorithm	O
for	O
dense	O
matrices	O
which	O
runs	O
in	O
time	O
compared	O
to	O
the	O
of	O
the	O
standard	O
HHL	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
An	O
important	O
factor	O
in	O
the	O
performance	O
of	O
the	O
matrix	O
inversion	O
algorithm	O
is	O
the	O
condition	B-Algorithm
number	I-Algorithm
,	O
which	O
represents	O
the	O
ratio	O
of	O
'	O
s	O
largest	O
and	O
smallest	O
eigenvalues	O
.	O
</s>
<s>
As	O
the	O
condition	B-Algorithm
number	I-Algorithm
increases	O
,	O
the	O
ease	O
with	O
which	O
the	O
solution	O
vector	O
can	O
be	O
found	O
using	O
gradient	O
descent	O
methods	O
such	O
as	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
decreases	O
,	O
as	O
becomes	O
closer	O
to	O
a	O
matrix	O
which	O
cannot	O
be	O
inverted	O
and	O
the	O
solution	O
vector	O
becomes	O
less	O
stable	O
.	O
</s>
<s>
Assuming	O
that	O
is	O
s-sparse	O
,	O
this	O
can	O
be	O
done	O
with	O
an	O
error	O
bounded	O
by	O
a	O
constant	O
,	O
which	O
will	O
translate	O
to	O
the	O
additive	O
error	O
achieved	O
in	O
the	O
output	O
state	O
.	O
</s>
<s>
The	O
phase	B-Algorithm
estimation	I-Algorithm
step	O
errs	O
by	O
in	O
estimating	O
,	O
which	O
translates	O
into	O
a	O
relative	O
error	O
of	O
in	O
.	O
</s>
<s>
While	O
there	O
does	O
not	O
yet	O
exist	O
a	O
quantum	O
computer	O
that	O
can	O
truly	O
offer	O
a	O
speedup	O
over	O
a	O
classical	O
computer	O
,	O
implementation	O
of	O
a	O
"	O
proof	O
of	O
concept	O
"	O
remains	O
an	O
important	O
milestone	O
in	O
the	O
development	O
of	O
a	O
new	O
quantum	B-Device
algorithm	I-Device
.	O
</s>
<s>
Demonstrating	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
remained	O
a	O
challenge	O
for	O
years	O
after	O
its	O
proposal	O
until	O
2013	O
when	O
it	O
was	O
demonstrated	O
by	O
Cai	O
et	O
al.	O
,	O
Barz	O
et	O
al	O
.	O
</s>
<s>
On	O
February	O
5	O
,	O
2013	O
,	O
Stefanie	O
Barz	O
and	O
co-workers	O
demonstrated	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
on	O
a	O
photonic	O
quantum	O
computing	O
architecture	O
.	O
</s>
<s>
reported	O
a	O
proof-of-concept	O
experimental	O
demonstration	O
of	O
the	O
quantum	B-Device
algorithm	I-Device
using	O
a	O
4-qubit	O
nuclear	O
magnetic	O
resonance	O
quantum	O
information	O
processor	O
.	O
</s>
<s>
For	O
certain	O
problems	O
,	O
quantum	B-Device
algorithms	I-Device
supply	O
exponential	O
speedups	O
over	O
their	O
classical	O
counterparts	O
,	O
the	O
most	O
famous	O
example	O
being	O
Shor	B-Algorithm
's	I-Algorithm
factoring	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
First	O
,	O
they	O
demonstrated	O
how	O
a	O
preconditioner	O
could	O
be	O
included	O
within	O
the	O
quantum	B-Device
algorithm	I-Device
.	O
</s>
<s>
This	O
expands	O
the	O
class	O
of	O
problems	O
that	O
can	O
achieve	O
the	O
promised	O
exponential	O
speedup	O
,	O
since	O
the	O
scaling	O
of	O
HHL	O
and	O
the	O
best	O
classical	O
algorithms	O
are	O
both	O
polynomial	O
in	O
the	O
condition	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
Dominic	O
Berry	O
proposed	O
a	O
new	O
algorithm	O
for	O
solving	O
linear	O
time	O
dependent	O
differential	O
equations	O
as	O
an	O
extension	O
of	O
the	O
quantum	B-Device
algorithm	I-Device
for	O
solving	O
linear	O
systems	O
of	O
equations	O
.	O
</s>
<s>
Berry	O
provides	O
an	O
efficient	O
algorithm	O
for	O
solving	O
the	O
full-time	O
evolution	O
under	O
sparse	B-Algorithm
linear	O
differential	O
equations	O
on	O
a	O
quantum	O
computer	O
.	O
</s>
<s>
The	O
Finite	B-Application
Element	I-Application
Method	I-Application
uses	O
large	O
systems	O
of	O
linear	O
equations	O
to	O
find	O
approximate	O
solutions	O
to	O
various	O
physical	O
and	O
mathematical	O
models	O
.	O
</s>
<s>
Montanaro	O
and	O
Pallister	O
demonstrate	O
that	O
the	O
HHL	B-Algorithm
algorithm	I-Algorithm
,	O
when	O
applied	O
to	O
certain	O
FEM	O
problems	O
,	O
can	O
achieve	O
a	O
polynomial	O
quantum	O
speedup	O
.	O
</s>
<s>
Quantum	O
speedups	O
for	O
the	O
finite	B-Application
element	I-Application
method	I-Application
are	O
higher	O
for	O
problems	O
which	O
include	O
solutions	O
with	O
higher-order	O
derivatives	O
and	O
large	O
spatial	O
dimensions	O
.	O
</s>
<s>
provide	O
a	O
new	O
quantum	B-Device
algorithm	I-Device
to	O
determine	O
the	O
quality	O
of	O
a	O
least-squares	B-Algorithm
fit	I-Algorithm
in	O
which	O
a	O
continuous	O
function	O
is	O
used	O
to	O
approximate	O
a	O
set	O
of	O
discrete	O
points	O
by	O
extending	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
.	O
</s>
<s>
As	O
the	O
number	O
of	O
discrete	O
points	O
increases	O
,	O
the	O
time	O
required	O
to	O
produce	O
a	O
least-squares	B-Algorithm
fit	I-Algorithm
using	O
even	O
a	O
quantum	O
computer	O
running	O
a	O
quantum	O
state	O
tomography	O
algorithm	O
becomes	O
very	O
large	O
.	O
</s>
<s>
The	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
has	O
been	O
applied	O
to	O
a	O
support	O
vector	O
machine	O
,	O
which	O
is	O
an	O
optimized	O
linear	O
or	O
non-linear	O
binary	O
classifier	O
.	O
</s>
<s>
show	O
that	O
a	O
quantum	O
support	O
vector	O
machine	O
can	O
be	O
used	O
for	O
big	B-Application
data	I-Application
classification	O
and	O
achieve	O
an	O
exponential	O
speedup	O
over	O
classical	O
computers	O
.	O
</s>
<s>
developed	O
an	O
algorithm	O
for	O
performing	O
Bayesian	O
training	O
of	O
deep	O
neural	O
networks	O
in	O
quantum	O
computers	O
with	O
an	O
exponential	O
speedup	O
over	O
classical	O
training	O
due	O
to	O
the	O
use	O
of	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
,	O
providing	O
also	O
the	O
first	O
general-purpose	O
implementation	O
of	O
the	O
algorithm	O
to	O
be	O
run	O
in	O
cloud-based	B-General_Concept
quantum	I-General_Concept
computers	I-General_Concept
.	O
</s>
