<s>
A	O
quantum	B-Architecture
computer	I-Architecture
is	O
a	O
computer	O
that	O
exploits	O
quantum	O
mechanical	O
phenomena	O
.	O
</s>
<s>
At	O
small	O
scales	O
,	O
physical	O
matter	O
exhibits	O
properties	O
of	O
both	O
particles	O
and	O
waves	O
,	O
and	O
quantum	B-Architecture
computing	I-Architecture
leverages	O
this	O
behavior	O
using	O
specialized	O
hardware	O
.	O
</s>
<s>
Classical	O
physics	O
cannot	O
explain	O
the	O
operation	O
of	O
these	O
quantum	O
devices	O
,	O
and	O
a	O
scalable	O
quantum	B-Architecture
computer	I-Architecture
could	O
perform	O
some	O
calculations	O
exponentially	O
faster	O
than	O
any	O
modern	O
"	O
classical	O
"	O
computer	O
.	O
</s>
<s>
In	O
particular	O
,	O
a	O
large-scale	O
quantum	B-Architecture
computer	I-Architecture
could	O
break	O
widely	O
used	O
encryption	O
schemes	O
and	O
aid	O
physicists	O
in	O
performing	O
physical	B-Device
simulations	I-Device
;	O
however	O
,	O
the	O
current	O
state	O
of	O
the	O
art	O
is	O
still	O
largely	O
experimental	O
and	O
impractical	O
.	O
</s>
<s>
The	O
basic	O
unit	O
of	O
information	O
in	O
quantum	B-Architecture
computing	I-Architecture
is	O
the	O
qubit	O
,	O
similar	O
to	O
the	O
bit	O
in	O
traditional	O
digital	O
electronics	O
.	O
</s>
<s>
If	O
a	O
quantum	B-Architecture
computer	I-Architecture
manipulates	O
the	O
qubit	O
in	O
a	O
particular	O
way	O
,	O
wave	O
interference	O
effects	O
can	O
amplify	O
the	O
desired	O
measurement	O
results	O
.	O
</s>
<s>
The	O
design	O
of	O
quantum	B-Device
algorithms	I-Device
involves	O
creating	O
procedures	O
that	O
allow	O
a	O
quantum	B-Architecture
computer	I-Architecture
to	O
perform	O
calculations	O
efficiently	O
.	O
</s>
<s>
If	O
a	O
physical	O
qubit	O
is	O
not	O
sufficiently	O
isolated	O
from	O
its	O
environment	O
,	O
it	O
suffers	O
from	O
quantum	O
decoherence	O
,	O
introducing	O
noise	B-Algorithm
into	O
calculations	O
.	O
</s>
<s>
Two	O
of	O
the	O
most	O
promising	O
technologies	O
are	O
superconductors	O
(	O
which	O
isolate	O
an	O
electrical	O
current	O
by	O
eliminating	O
electrical	O
resistance	O
)	O
and	O
ion	O
traps	O
(	O
which	O
confine	O
a	O
single	O
atomic	B-Language
particle	I-Language
using	O
electromagnetic	O
fields	O
)	O
.	O
</s>
<s>
Any	O
computational	O
problem	O
that	O
can	O
be	O
solved	O
by	O
a	O
classical	O
computer	O
can	O
also	O
be	O
solved	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
Conversely	O
,	O
any	O
problem	O
that	O
can	O
be	O
solved	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
can	O
also	O
be	O
solved	O
by	O
a	O
classical	O
computer	O
,	O
at	O
least	O
in	O
principle	O
given	O
enough	O
time	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
quantum	B-Architecture
computers	I-Architecture
obey	O
the	O
Church	O
–	O
Turing	O
thesis	O
.	O
</s>
<s>
This	O
means	O
that	O
while	O
quantum	B-Architecture
computers	I-Architecture
provide	O
no	O
additional	O
advantages	O
over	O
classical	O
computers	O
in	O
terms	O
of	O
computability	O
,	O
quantum	B-Device
algorithms	I-Device
for	O
certain	O
problems	O
have	O
significantly	O
lower	O
time	O
complexities	O
than	O
corresponding	O
known	O
classical	O
algorithms	O
.	O
</s>
<s>
Notably	O
,	O
quantum	B-Architecture
computers	I-Architecture
are	O
believed	O
to	O
be	O
able	O
to	O
solve	O
certain	O
problems	O
quickly	O
that	O
no	O
classical	O
computer	O
could	O
solve	O
in	O
any	O
feasible	O
amount	O
of	O
time	O
—	O
a	O
feat	O
known	O
as	O
"	O
quantum	O
supremacy.	O
"	O
</s>
<s>
The	O
study	O
of	O
the	O
computational	O
complexity	O
of	O
problems	O
with	O
respect	O
to	O
quantum	B-Architecture
computers	I-Architecture
is	O
known	O
as	O
quantum	O
complexity	O
theory	O
.	O
</s>
<s>
For	O
many	O
years	O
,	O
the	O
fields	O
of	O
quantum	O
mechanics	O
and	O
computer	B-General_Concept
science	I-General_Concept
formed	O
distinct	O
academic	O
communities	O
.	O
</s>
<s>
As	O
physicists	O
applied	O
quantum	O
mechanical	O
models	O
to	O
computational	O
problems	O
and	O
swapped	O
digital	O
bits	O
for	O
qubits	O
,	O
the	O
fields	O
of	O
quantum	O
mechanics	O
and	O
computer	B-General_Concept
science	I-General_Concept
began	O
to	O
converge	O
.	O
</s>
<s>
In	O
1980	O
,	O
Paul	O
Benioff	O
introduced	O
the	O
quantum	O
Turing	B-Architecture
machine	I-Architecture
,	O
which	O
uses	O
quantum	O
theory	O
to	O
describe	O
a	O
simplified	O
computer	O
.	O
</s>
<s>
When	O
digital	O
computers	O
became	O
faster	O
,	O
physicists	O
faced	O
an	O
exponential	O
increase	O
in	O
overhead	O
when	O
simulating	B-Device
quantum	I-Device
dynamics	I-Device
,	O
prompting	O
Yuri	O
Manin	O
and	O
Richard	O
Feynman	O
to	O
independently	O
suggest	O
that	O
hardware	O
based	O
on	O
quantum	O
phenomena	O
might	O
be	O
more	O
efficient	O
for	O
computer	O
simulation	O
.	O
</s>
<s>
Quantum	B-Device
algorithms	I-Device
then	O
emerged	O
for	O
solving	O
oracle	O
problems	O
,	O
such	O
as	O
Deutsch	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
in	O
1985	O
,	O
the	O
BernsteinVazirani	O
algorithm	O
in	O
1993	O
,	O
and	O
Simon	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
in	O
1994	O
.	O
</s>
<s>
These	O
algorithms	O
did	O
not	O
solve	O
practical	O
problems	O
,	O
but	O
demonstrated	O
mathematically	O
that	O
one	O
could	O
gain	O
more	O
information	O
by	O
querying	O
a	O
black	B-Device
box	I-Device
with	O
a	O
quantum	O
state	O
in	O
superposition	O
,	O
sometimes	O
referred	O
to	O
as	O
quantum	B-Architecture
parallelism	I-Architecture
.	O
</s>
<s>
Peter	O
Shor	O
built	O
on	O
these	O
results	O
with	O
his	B-Algorithm
1994	I-Algorithm
algorithms	I-Algorithm
for	O
breaking	O
the	O
widely	O
used	O
RSA	B-Architecture
and	O
DiffieHellman	O
encryption	O
protocols	O
,	O
which	O
drew	O
significant	O
attention	O
to	O
the	O
field	O
of	O
quantum	B-Architecture
computing	I-Architecture
.	O
</s>
<s>
In	O
1996	O
,	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
established	O
a	O
quantum	O
speedup	O
for	O
the	O
widely	O
applicable	O
unstructured	B-Application
search	O
problem	O
.	O
</s>
<s>
The	O
same	O
year	O
,	O
Seth	O
Lloyd	O
proved	O
that	O
quantum	B-Architecture
computers	I-Architecture
could	O
simulate	O
quantum	O
systems	O
without	O
the	O
exponential	O
overhead	O
present	O
in	O
classical	O
simulations	O
,	O
validating	O
Feynman	O
's	O
1982	O
conjecture	O
.	O
</s>
<s>
Over	O
the	O
years	O
,	O
experimentalists	O
have	O
constructed	O
small-scale	O
quantum	B-Architecture
computers	I-Architecture
using	O
trapped	O
ions	O
and	O
superconductors	O
.	O
</s>
<s>
In	O
1998	O
,	O
a	O
two-qubit	O
quantum	B-Architecture
computer	I-Architecture
demonstrated	O
the	O
feasibility	O
of	O
the	O
technology	O
,	O
and	O
subsequent	O
experiments	O
have	O
increased	O
the	O
number	O
of	O
qubits	O
and	O
reduced	O
error	O
rates	O
.	O
</s>
<s>
In	O
2019	O
,	O
Google	B-Application
AI	I-Application
and	O
NASA	O
announced	O
that	O
they	O
had	O
achieved	O
quantum	B-Device
supremacy	I-Device
with	O
a	O
54-qubit	O
machine	O
,	O
performing	O
a	O
computation	O
that	O
is	O
impossible	O
for	O
any	O
classical	O
computer	O
.	O
</s>
<s>
The	O
threshold	O
theorem	O
shows	O
how	O
increasing	O
the	O
number	O
of	O
qubits	O
can	O
mitigate	O
errors	O
,	O
yet	O
fully	O
fault-tolerant	O
quantum	B-Architecture
computing	I-Architecture
remains	O
"	O
a	O
rather	O
distant	O
dream	O
"	O
.	O
</s>
<s>
According	O
to	O
some	O
researchers	O
,	O
noisy	O
intermediate-scale	O
quantum	O
(	O
NISQ	O
)	O
machines	O
may	O
have	O
specialized	O
uses	O
in	O
the	O
near	O
future	O
,	O
but	O
noise	B-Algorithm
in	O
quantum	B-Application
gates	I-Application
limits	O
their	O
reliability	O
.	O
</s>
<s>
In	O
recent	O
years	O
,	O
investment	O
in	O
quantum	B-Architecture
computing	I-Architecture
research	O
has	O
increased	O
in	O
the	O
public	O
and	O
private	O
sectors	O
.	O
</s>
<s>
While	O
programmers	B-Application
may	O
depend	O
on	O
probability	O
theory	O
when	O
designing	O
a	O
randomized	B-General_Concept
algorithm	I-General_Concept
,	O
quantum	O
mechanical	O
notions	O
like	O
superposition	O
and	O
interference	O
are	O
largely	O
irrelevant	O
for	O
program	O
analysis	O
.	O
</s>
<s>
Quantum	B-Device
programs	I-Device
,	O
in	O
contrast	O
,	O
rely	O
on	O
precise	O
control	O
of	O
coherent	O
quantum	O
systems	O
.	O
</s>
<s>
Physicists	O
describe	O
these	O
systems	O
mathematically	O
using	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Complex	O
numbers	O
model	O
probability	O
amplitudes	O
,	O
vectors	O
model	O
quantum	O
states	O
,	O
and	O
matrices	B-Architecture
model	O
the	O
operations	O
that	O
can	O
be	O
performed	O
on	O
these	O
states	O
.	O
</s>
<s>
Programming	O
a	O
quantum	B-Architecture
computer	I-Architecture
is	O
then	O
a	O
matter	O
of	O
composing	B-Application
operations	O
in	O
such	O
a	O
way	O
that	O
the	O
resulting	O
program	O
computes	O
a	O
useful	O
result	O
in	O
theory	O
and	O
is	O
implementable	O
in	O
practice	O
.	O
</s>
<s>
The	O
prevailing	O
model	O
of	O
quantum	B-Architecture
computation	I-Architecture
describes	O
the	O
computation	O
in	O
terms	O
of	O
a	O
network	O
of	O
quantum	O
logic	O
gates	O
.	O
</s>
<s>
This	O
model	O
is	O
a	O
complex	O
linear-algebraic	B-Language
generalization	O
of	O
boolean	O
circuits	O
.	O
</s>
<s>
However	O
,	O
a	O
quantum	B-Architecture
computation	I-Architecture
can	O
be	O
influenced	O
by	O
both	O
values	O
at	O
once	O
,	O
inexplicable	O
by	O
either	O
state	O
individually	O
.	O
</s>
<s>
Physicists	O
typically	O
use	O
Dirac	O
notation	O
for	O
quantum	O
mechanical	O
linear	B-Language
algebra	I-Language
,	O
writing	O
for	O
a	O
vector	O
labeled	O
ψ	O
.	O
</s>
<s>
Mathematically	O
,	O
the	O
application	O
of	O
such	O
a	O
logic	O
gate	O
to	O
a	O
quantum	O
state	O
vector	O
is	O
modelled	O
with	O
matrix	B-Architecture
multiplication	O
.	O
</s>
<s>
The	O
CNOT	O
gate	O
can	O
then	O
be	O
represented	O
using	O
the	O
following	O
matrix	B-Architecture
:	O
</s>
<s>
In	O
summary	O
,	O
a	O
quantum	B-Architecture
computation	I-Architecture
can	O
be	O
described	O
as	O
a	O
network	O
of	O
quantum	O
logic	O
gates	O
and	O
measurements	O
.	O
</s>
<s>
However	O
,	O
any	O
measurement	O
can	O
be	O
deferred	O
to	O
the	O
end	O
of	O
quantum	B-Architecture
computation	I-Architecture
,	O
though	O
this	O
deferment	O
may	O
come	O
at	O
a	O
computational	O
cost	O
,	O
so	O
most	O
quantum	B-Application
circuits	I-Application
depict	O
a	O
network	O
consisting	O
only	O
of	O
quantum	O
logic	O
gates	O
and	O
no	O
measurements	O
.	O
</s>
<s>
Quantum	B-Architecture
parallelism	I-Architecture
refers	O
to	O
the	O
ability	O
of	O
quantum	B-Architecture
computers	I-Architecture
to	O
evaluate	O
a	O
function	O
for	O
multiple	O
input	O
values	O
simultaneously	O
.	O
</s>
<s>
This	O
property	O
is	O
key	O
to	O
the	O
speedup	O
of	O
many	O
quantum	B-Device
algorithms	I-Device
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
models	O
of	O
computation	O
for	O
quantum	B-Architecture
computing	I-Architecture
,	O
distinguished	O
by	O
the	O
basic	O
elements	O
in	O
which	O
the	O
computation	O
is	O
decomposed	O
.	O
</s>
<s>
A	O
quantum	B-Application
gate	I-Application
array	I-Application
decomposes	O
computation	O
into	O
a	O
sequence	O
of	O
few-qubit	O
quantum	B-Application
gates	I-Application
.	O
</s>
<s>
A	O
quantum	B-Architecture
computation	I-Architecture
can	O
be	O
described	O
as	O
a	O
network	O
of	O
quantum	O
logic	O
gates	O
and	O
measurements	O
.	O
</s>
<s>
However	O
,	O
any	O
measurement	O
can	O
be	O
deferred	O
to	O
the	O
end	O
of	O
quantum	B-Architecture
computation	I-Architecture
,	O
though	O
this	O
deferment	O
may	O
come	O
at	O
a	O
computational	O
cost	O
,	O
so	O
most	O
quantum	B-Application
circuits	I-Application
depict	O
a	O
network	O
consisting	O
only	O
of	O
quantum	O
logic	O
gates	O
and	O
no	O
measurements	O
.	O
</s>
<s>
Any	O
quantum	B-Architecture
computation	I-Architecture
(	O
which	O
is	O
,	O
in	O
the	O
above	O
formalism	O
,	O
any	O
unitary	B-Algorithm
matrix	I-Algorithm
of	O
size	O
over	O
qubits	O
)	O
can	O
be	O
represented	O
as	O
a	O
network	O
of	O
quantum	O
logic	O
gates	O
from	O
a	O
fairly	O
small	O
family	O
of	O
gates	O
.	O
</s>
<s>
A	O
choice	O
of	O
gate	O
family	O
that	O
enables	O
this	O
construction	O
is	O
known	O
as	O
a	O
universal	O
gate	O
set	O
,	O
since	O
a	O
computer	O
that	O
can	O
run	O
such	O
circuits	O
is	O
a	O
universal	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
This	O
means	O
any	O
quantum	B-Architecture
computation	I-Architecture
can	O
be	O
performed	O
by	O
executing	O
a	O
sequence	O
of	O
single-qubit	O
gates	O
together	O
with	O
CNOT	O
gates	O
.	O
</s>
<s>
A	O
measurement-based	O
quantum	B-Architecture
computer	I-Architecture
decomposes	O
computation	O
into	O
a	O
sequence	O
of	O
Bell	O
state	O
measurements	O
and	O
single-qubit	O
quantum	B-Application
gates	I-Application
applied	O
to	O
a	O
highly	O
entangled	O
initial	O
state	O
(	O
a	O
cluster	O
state	O
)	O
,	O
using	O
a	O
technique	O
called	O
quantum	B-Application
gate	I-Application
teleportation	O
.	O
</s>
<s>
An	O
adiabatic	O
quantum	B-Architecture
computer	I-Architecture
,	O
based	O
on	O
quantum	B-Algorithm
annealing	I-Algorithm
,	O
decomposes	O
computation	O
into	O
a	O
slow	O
continuous	O
transformation	O
of	O
an	O
initial	O
Hamiltonian	O
into	O
a	O
final	O
Hamiltonian	O
,	O
whose	O
ground	O
states	O
contain	O
the	O
solution	O
.	O
</s>
<s>
A	O
topological	B-Device
quantum	I-Device
computer	I-Device
decomposes	O
computation	O
into	O
the	O
braiding	O
of	O
anyons	O
in	O
a	O
2D	O
lattice	O
.	O
</s>
<s>
The	O
quantum	O
Turing	B-Architecture
machine	I-Architecture
is	O
theoretically	O
important	O
but	O
the	O
physical	O
implementation	O
of	O
this	O
model	O
is	O
not	O
feasible	O
.	O
</s>
<s>
All	O
of	O
these	O
models	O
of	O
computation	O
—	O
quantum	B-Application
circuits	I-Application
,	O
one-way	O
quantum	B-Architecture
computation	I-Architecture
,	O
adiabatic	O
quantum	B-Architecture
computation	I-Architecture
,	O
and	O
topological	B-Device
quantum	I-Device
computation	I-Device
—	O
have	O
been	O
shown	O
to	O
be	O
equivalent	O
to	O
the	O
quantum	O
Turing	B-Architecture
machine	I-Architecture
;	O
given	O
a	O
perfect	O
implementation	O
of	O
one	O
such	O
quantum	B-Architecture
computer	I-Architecture
,	O
it	O
can	O
simulate	O
all	O
the	O
others	O
with	O
no	O
more	O
than	O
polynomial	O
overhead	O
.	O
</s>
<s>
This	O
equivalence	O
need	O
not	O
hold	O
for	O
practical	B-Architecture
quantum	I-Architecture
computers	I-Architecture
,	O
since	O
the	O
overhead	O
of	O
simulation	O
may	O
be	O
too	O
large	O
to	O
be	O
practical	O
.	O
</s>
<s>
Quantum	O
cryptography	O
could	O
potentially	O
fulfill	O
some	O
of	O
the	O
functions	O
of	O
public	B-Application
key	I-Application
cryptography	I-Application
.	O
</s>
<s>
Progress	O
in	O
finding	O
quantum	B-Device
algorithms	I-Device
typically	O
focuses	O
on	O
this	O
quantum	B-Application
circuit	I-Application
model	O
,	O
though	O
exceptions	O
like	O
the	O
quantum	O
adiabatic	O
algorithm	O
exist	O
.	O
</s>
<s>
Quantum	B-Device
algorithms	I-Device
can	O
be	O
roughly	O
categorized	O
by	O
the	O
type	O
of	O
speedup	O
achieved	O
over	O
corresponding	O
classical	O
algorithms	O
.	O
</s>
<s>
Quantum	B-Device
algorithms	I-Device
that	O
offer	O
more	O
than	O
a	O
polynomial	O
speedup	O
over	O
the	O
best-known	O
classical	O
algorithm	O
include	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
factoring	O
and	O
the	O
related	O
quantum	B-Device
algorithms	I-Device
for	O
computing	O
discrete	O
logarithms	O
,	O
solving	O
Pell	O
's	O
equation	O
,	O
and	O
more	O
generally	O
solving	O
the	O
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
for	O
abelian	O
finite	O
groups	O
.	O
</s>
<s>
These	O
algorithms	O
depend	O
on	O
the	O
primitive	O
of	O
the	O
quantum	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Certain	O
oracle	O
problems	O
like	O
Simon	B-Algorithm
's	I-Algorithm
problem	I-Algorithm
and	O
the	O
Bernstein	O
–	O
Vazirani	O
problem	O
do	O
give	O
provable	O
speedups	O
,	O
though	O
this	O
is	O
in	O
the	O
quantum	O
query	O
model	O
,	O
which	O
is	O
a	O
restricted	O
model	O
where	O
lower	O
bounds	O
are	O
much	O
easier	O
to	O
prove	O
and	O
does	O
n't	O
necessarily	O
translate	O
to	O
speedups	O
for	O
practical	O
problems	O
.	O
</s>
<s>
Other	O
problems	O
,	O
including	O
the	O
simulation	O
of	O
quantum	O
physical	O
processes	O
from	O
chemistry	O
and	O
solid-state	O
physics	O
,	O
the	O
approximation	O
of	O
certain	O
Jones	O
polynomials	O
,	O
and	O
the	O
quantum	B-Algorithm
algorithm	I-Algorithm
for	I-Algorithm
linear	I-Algorithm
systems	I-Algorithm
of	I-Algorithm
equations	I-Algorithm
have	O
quantum	B-Device
algorithms	I-Device
appearing	O
to	O
give	O
super-polynomial	O
speedups	O
and	O
are	O
BQP-complete	O
.	O
</s>
<s>
Because	O
these	O
problems	O
are	O
BQP-complete	O
,	O
an	O
equally	O
fast	O
classical	O
algorithm	O
for	O
them	O
would	O
imply	O
that	O
no	O
quantum	B-Device
algorithm	I-Device
gives	O
a	O
super-polynomial	O
speedup	O
,	O
which	O
is	O
believed	O
to	O
be	O
unlikely	O
.	O
</s>
<s>
Some	O
quantum	B-Device
algorithms	I-Device
,	O
like	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
amplitude	B-Algorithm
amplification	I-Algorithm
,	O
give	O
polynomial	O
speedups	O
over	O
corresponding	O
classical	O
algorithms	O
.	O
</s>
<s>
Many	O
examples	O
of	O
provable	O
quantum	O
speedups	O
for	O
query	O
problems	O
are	O
related	O
to	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
including	O
Brassard	B-Algorithm
,	I-Algorithm
Høyer	I-Algorithm
,	I-Algorithm
and	I-Algorithm
Tapp	I-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
for	O
finding	O
collisions	O
in	O
two-to-one	O
functions	O
,	O
which	O
uses	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
and	O
Farhi	O
,	O
Goldstone	O
,	O
and	O
Gutmann	O
's	O
algorithm	O
for	O
evaluating	O
NAND	O
trees	O
,	O
which	O
is	O
a	O
variant	O
of	O
the	O
search	O
problem	O
.	O
</s>
<s>
A	O
notable	O
application	O
of	O
quantum	B-Architecture
computation	I-Architecture
is	O
for	O
attacks	O
on	O
cryptographic	O
systems	O
that	O
are	O
currently	O
in	O
use	O
.	O
</s>
<s>
Integer	O
factorization	O
,	O
which	O
underpins	O
the	O
security	O
of	O
public	B-Application
key	I-Application
cryptographic	I-Application
systems	O
,	O
is	O
believed	O
to	O
be	O
computationally	O
infeasible	O
with	O
an	O
ordinary	O
computer	O
for	O
large	O
integers	O
if	O
they	O
are	O
the	O
product	O
of	O
few	O
prime	O
numbers	O
(	O
e.g.	O
,	O
products	O
of	O
two	O
300-digit	O
primes	O
)	O
.	O
</s>
<s>
By	O
comparison	O
,	O
a	O
quantum	B-Architecture
computer	I-Architecture
could	O
solve	O
this	O
problem	O
exponentially	O
faster	O
using	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
find	O
its	O
factors	O
.	O
</s>
<s>
This	O
ability	O
would	O
allow	O
a	O
quantum	B-Architecture
computer	I-Architecture
to	O
break	O
many	O
of	O
the	O
cryptographic	O
systems	O
in	O
use	O
today	O
,	O
in	O
the	O
sense	O
that	O
there	O
would	O
be	O
a	O
polynomial	O
time	O
(	O
in	O
the	O
number	O
of	O
digits	O
of	O
the	O
integer	O
)	O
algorithm	O
for	O
solving	O
the	O
problem	O
.	O
</s>
<s>
In	O
particular	O
,	O
most	O
of	O
the	O
popular	O
public	B-Application
key	I-Application
ciphers	I-Application
are	O
based	O
on	O
the	O
difficulty	O
of	O
factoring	O
integers	O
or	O
the	O
discrete	O
logarithm	O
problem	O
,	O
both	O
of	O
which	O
can	O
be	O
solved	O
by	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
RSA	B-Architecture
,	O
Diffie	B-Protocol
–	I-Protocol
Hellman	I-Protocol
,	O
and	O
elliptic	B-Protocol
curve	I-Protocol
Diffie	I-Protocol
–	I-Protocol
Hellman	I-Protocol
algorithms	O
could	O
be	O
broken	O
.	O
</s>
<s>
Identifying	O
cryptographic	O
systems	O
that	O
may	O
be	O
secure	O
against	O
quantum	B-Device
algorithms	I-Device
is	O
an	O
actively	O
researched	O
topic	O
under	O
the	O
field	O
of	O
post-quantum	O
cryptography	O
.	O
</s>
<s>
Some	O
public-key	B-Application
algorithms	O
are	O
based	O
on	O
problems	O
other	O
than	O
the	O
integer	O
factorization	O
and	O
discrete	O
logarithm	O
problems	O
to	O
which	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
applies	O
,	O
like	O
the	O
McEliece	O
cryptosystem	O
based	O
on	O
a	O
problem	O
in	O
coding	B-Error_Name
theory	I-Error_Name
.	O
</s>
<s>
Lattice-based	O
cryptosystems	O
are	O
also	O
not	O
known	O
to	O
be	O
broken	O
by	O
quantum	B-Architecture
computers	I-Architecture
,	O
and	O
finding	O
a	O
polynomial	O
time	O
algorithm	O
for	O
solving	O
the	O
dihedral	B-Algorithm
hidden	B-Algorithm
subgroup	I-Algorithm
problem	I-Algorithm
,	O
which	O
would	O
break	O
many	O
lattice	O
based	O
cryptosystems	O
,	O
is	O
a	O
well-studied	O
open	O
problem	O
.	O
</s>
<s>
It	O
has	O
been	O
proven	O
that	O
applying	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
break	O
a	O
symmetric	O
(	O
secret	O
key	O
)	O
algorithm	O
by	O
brute	O
force	O
requires	O
time	O
equal	O
to	O
roughly	O
2n/2	O
invocations	O
of	O
the	O
underlying	O
cryptographic	O
algorithm	O
,	O
compared	O
with	O
roughly	O
2n	O
in	O
the	O
classical	O
case	O
,	O
meaning	O
that	O
symmetric	B-Algorithm
key	I-Algorithm
lengths	O
are	O
effectively	O
halved	O
:	O
AES-256	O
would	O
have	O
the	O
same	O
security	O
against	O
an	O
attack	O
using	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
that	O
AES-128	O
has	O
against	O
classical	O
brute-force	O
search	O
(	O
see	O
Key	O
size	O
)	O
.	O
</s>
<s>
The	O
most	O
well-known	O
example	O
of	O
a	O
problem	O
that	O
allows	O
for	O
a	O
polynomial	O
quantum	O
speedup	O
is	O
unstructured	B-Application
search	O
,	O
which	O
involves	O
finding	O
a	O
marked	O
item	O
out	O
of	O
a	O
list	O
of	O
items	O
in	O
a	O
database	O
.	O
</s>
<s>
This	O
can	O
be	O
solved	O
by	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
using	O
queries	O
to	O
the	O
database	O
,	O
quadratically	O
fewer	O
than	O
the	O
queries	O
required	O
for	O
classical	O
algorithms	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
advantage	O
is	O
not	O
only	O
provable	O
but	O
also	O
optimal	O
:	O
it	O
has	O
been	O
shown	O
that	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
gives	O
the	O
maximal	O
possible	O
probability	O
of	O
finding	O
the	O
desired	O
element	O
for	O
any	O
number	O
of	O
oracle	O
lookups	O
.	O
</s>
<s>
Problems	O
that	O
can	O
be	O
efficiently	O
addressed	O
with	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
have	O
the	O
following	O
properties	O
:	O
</s>
<s>
For	O
problems	O
with	O
all	O
these	O
properties	O
,	O
the	O
running	O
time	O
of	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
on	O
a	O
quantum	B-Architecture
computer	I-Architecture
scales	O
as	O
the	O
square	O
root	O
of	O
the	O
number	O
of	O
inputs	O
(	O
or	O
elements	O
in	O
the	O
database	O
)	O
,	O
as	O
opposed	O
to	O
the	O
linear	O
scaling	O
of	O
classical	O
algorithms	O
.	O
</s>
<s>
A	O
general	O
class	O
of	O
problems	O
to	O
which	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
applied	O
is	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
,	O
where	O
the	O
database	O
through	O
which	O
the	O
algorithm	O
iterates	O
is	O
that	O
of	O
all	O
possible	O
answers	O
.	O
</s>
<s>
Breaking	O
symmetric	B-Algorithm
ciphers	I-Algorithm
with	O
this	O
algorithm	O
is	O
of	O
interest	O
to	O
government	O
agencies	O
.	O
</s>
<s>
Since	O
chemistry	O
and	O
nanotechnology	O
rely	O
on	O
understanding	O
quantum	O
systems	O
,	O
and	O
such	O
systems	O
are	O
impossible	O
to	O
simulate	O
in	O
an	O
efficient	O
manner	O
classically	O
,	O
quantum	B-Device
simulation	I-Device
may	O
be	O
an	O
important	O
application	O
of	O
quantum	B-Architecture
computing	I-Architecture
.	O
</s>
<s>
Quantum	B-Device
simulation	I-Device
could	O
also	O
be	O
used	O
to	O
simulate	O
the	O
behavior	O
of	O
atoms	O
and	O
particles	O
at	O
unusual	O
conditions	O
such	O
as	O
the	O
reactions	O
inside	O
a	O
collider	O
.	O
</s>
<s>
Quantum	B-Device
simulations	I-Device
might	O
be	O
used	O
to	O
understand	O
this	O
process	O
and	O
increase	O
the	O
energy	O
efficiency	O
of	O
production	O
.	O
</s>
<s>
Quantum	B-Algorithm
annealing	I-Algorithm
relies	O
on	O
the	O
adiabatic	O
theorem	O
to	O
undertake	O
calculations	O
.	O
</s>
<s>
Since	O
quantum	B-Architecture
computers	I-Architecture
can	O
produce	O
outputs	O
that	O
classical	O
computers	O
cannot	O
produce	O
efficiently	O
,	O
and	O
since	O
quantum	B-Architecture
computation	I-Architecture
is	O
fundamentally	O
linear	O
algebraic	O
,	O
some	O
express	O
hope	O
in	O
developing	O
quantum	B-Device
algorithms	I-Device
that	O
can	O
speed	O
up	O
machine	O
learning	O
tasks	O
.	O
</s>
<s>
For	O
example	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
,	O
or	O
"	O
HHL	B-Algorithm
Algorithm	I-Algorithm
"	O
,	O
named	O
after	O
its	O
discoverers	O
Harrow	O
,	O
Hassidim	O
,	O
and	O
Lloyd	O
,	O
is	O
believed	O
to	O
provide	O
speedup	O
over	O
classical	O
counterparts	O
.	O
</s>
<s>
Some	O
research	O
groups	O
have	O
recently	O
explored	O
the	O
use	O
of	O
quantum	B-Algorithm
annealing	I-Algorithm
hardware	O
for	O
training	O
Boltzmann	B-Algorithm
machines	I-Algorithm
and	O
deep	O
neural	O
networks	O
.	O
</s>
<s>
However	O
,	O
the	O
immense	O
size	O
and	O
complexity	O
of	O
the	O
structural	O
space	O
of	O
all	O
possible	O
drug-like	O
molecules	O
pose	O
significant	O
obstacles	O
,	O
which	O
could	O
be	O
overcome	O
in	O
the	O
future	O
by	O
quantum	B-Architecture
computers	I-Architecture
.	O
</s>
<s>
Quantum	B-Architecture
computers	I-Architecture
are	O
naturally	O
good	O
for	O
solving	O
complex	O
quantum	O
many-body	O
problems	O
and	O
thus	O
may	O
be	O
instrumental	O
in	O
applications	O
involving	O
quantum	O
chemistry	O
.	O
</s>
<s>
There	O
are	O
a	O
number	O
of	O
technical	O
challenges	O
in	O
building	O
a	O
large-scale	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
Physicist	O
David	O
DiVincenzo	O
has	O
listed	O
these	O
requirements	O
for	O
a	O
practical	B-Architecture
quantum	I-Architecture
computer	I-Architecture
:	O
</s>
<s>
Sourcing	O
parts	O
for	O
quantum	B-Architecture
computers	I-Architecture
is	O
also	O
very	O
difficult	O
.	O
</s>
<s>
Superconducting	O
quantum	B-Architecture
computers	I-Architecture
,	O
like	O
those	O
constructed	O
by	O
Google	B-Application
and	O
IBM	O
,	O
need	O
helium-3	O
,	O
a	O
nuclear	O
research	O
byproduct	O
,	O
and	O
special	O
superconducting	O
cables	O
made	O
only	O
by	O
the	O
Japanese	O
company	O
Coax	O
Co	O
.	O
</s>
<s>
One	O
of	O
the	O
greatest	O
challenges	O
involved	O
with	O
constructing	O
quantum	B-Architecture
computers	I-Architecture
is	O
controlling	O
or	O
removing	O
quantum	O
decoherence	O
.	O
</s>
<s>
Examples	O
include	O
the	O
quantum	B-Application
gates	I-Application
,	O
and	O
the	O
lattice	O
vibrations	O
and	O
background	O
thermonuclear	O
spin	O
of	O
the	O
physical	O
system	O
used	O
to	O
implement	O
the	O
qubits	O
.	O
</s>
<s>
Decoherence	O
times	O
for	O
candidate	O
systems	O
in	O
particular	O
,	O
the	O
transverse	O
relaxation	O
time	O
T2	O
(	O
for	O
NMR	O
and	O
MRI	B-Algorithm
technology	O
,	O
also	O
called	O
the	O
dephasing	O
time	O
)	O
,	O
typically	O
range	O
between	O
nanoseconds	O
and	O
seconds	O
at	O
low	O
temperature	O
.	O
</s>
<s>
Currently	O
,	O
some	O
quantum	B-Architecture
computers	I-Architecture
require	O
their	O
qubits	O
to	O
be	O
cooled	O
to	O
20	O
millikelvin	O
(	O
usually	O
using	O
a	O
dilution	O
refrigerator	O
)	O
in	O
order	O
to	O
prevent	O
significant	O
decoherence	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
time-consuming	O
tasks	O
may	O
render	O
some	O
quantum	B-Device
algorithms	I-Device
inoperable	O
,	O
as	O
attempting	O
to	O
maintain	O
the	O
state	O
of	O
qubits	O
for	O
a	O
long	O
enough	O
duration	O
will	O
eventually	O
corrupt	O
the	O
superpositions	O
.	O
</s>
<s>
These	O
issues	O
are	O
more	O
difficult	O
for	O
optical	O
approaches	O
as	O
the	O
timescales	O
are	O
orders	O
of	O
magnitude	O
shorter	O
and	O
an	O
often-cited	O
approach	O
to	O
overcoming	O
them	O
is	O
optical	O
pulse	B-Algorithm
shaping	I-Algorithm
.	O
</s>
<s>
As	O
described	O
in	O
the	O
threshold	O
theorem	O
,	O
if	O
the	O
error	O
rate	O
is	O
small	O
enough	O
,	O
it	O
is	O
thought	O
to	O
be	O
possible	O
to	O
use	O
quantum	B-Device
error	I-Device
correction	I-Device
to	O
suppress	O
errors	O
and	O
decoherence	O
.	O
</s>
<s>
An	O
often-cited	O
figure	O
for	O
the	O
required	O
error	O
rate	O
in	O
each	O
gate	O
for	O
fault-tolerant	O
computation	O
is	O
10−3	O
,	O
assuming	O
the	O
noise	B-Algorithm
is	O
depolarizing	O
.	O
</s>
<s>
The	O
number	O
required	O
to	O
factor	O
integers	O
using	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
still	O
polynomial	O
,	O
and	O
thought	O
to	O
be	O
between	O
L	O
and	O
L2	O
,	O
where	O
L	O
is	O
the	O
number	O
of	O
digits	O
in	O
the	O
number	O
to	O
be	O
factored	O
;	O
error	O
correction	O
algorithms	O
would	O
inflate	O
this	O
figure	O
by	O
an	O
additional	O
factor	O
of	O
L	O
.	O
For	O
a	O
1000-bit	O
number	O
,	O
this	O
implies	O
a	O
need	O
for	O
about	O
104	O
bits	O
without	O
error	O
correction	O
.	O
</s>
<s>
However	O
,	O
other	O
careful	O
estimates	O
lower	O
the	O
qubit	O
count	O
to	O
3million	O
for	O
factorizing	O
2,048	O
-bit	O
integer	O
in	O
5	O
months	O
on	O
a	O
trapped-ion	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
Another	O
approach	O
to	O
the	O
stability-decoherence	O
problem	O
is	O
to	O
create	O
a	O
topological	B-Device
quantum	I-Device
computer	I-Device
with	O
anyons	O
,	O
quasi-particles	O
used	O
as	O
threads	O
,	O
and	O
relying	O
on	O
braid	B-Algorithm
theory	I-Algorithm
to	O
form	O
stable	O
logic	O
gates	O
.	O
</s>
<s>
Quantum	B-Device
supremacy	I-Device
is	O
a	O
term	O
coined	O
by	O
John	O
Preskill	O
referring	O
to	O
the	O
engineering	O
feat	O
of	O
demonstrating	O
that	O
a	O
programmable	O
quantum	O
device	O
can	O
solve	O
a	O
problem	O
beyond	O
the	O
capabilities	O
of	O
state-of-the-art	O
classical	O
computers	O
.	O
</s>
<s>
The	O
problem	O
need	O
not	O
be	O
useful	O
,	O
so	O
some	O
view	O
the	O
quantum	B-Device
supremacy	I-Device
test	O
only	O
as	O
a	O
potential	O
future	O
benchmark	O
.	O
</s>
<s>
In	O
October	O
2019	O
,	O
Google	B-Application
AI	I-Application
Quantum	O
,	O
with	O
the	O
help	O
of	O
NASA	O
,	O
became	O
the	O
first	O
to	O
claim	O
to	O
have	O
achieved	O
quantum	B-Device
supremacy	I-Device
by	O
performing	O
calculations	O
on	O
the	O
Sycamore	B-Device
quantum	I-Device
computer	I-Device
more	O
than	O
3,000,000	O
times	O
faster	O
than	O
they	O
could	O
be	O
done	O
on	O
Summit	B-Device
,	O
generally	O
considered	O
the	O
world	O
's	O
fastest	O
computer	O
.	O
</s>
<s>
This	O
claim	O
has	O
been	O
subsequently	O
challenged	O
:	O
IBM	O
has	O
stated	O
that	O
Summit	B-Device
can	O
perform	O
samples	O
much	O
faster	O
than	O
claimed	O
,	O
and	O
researchers	O
have	O
since	O
developed	O
better	O
algorithms	O
for	O
the	O
sampling	O
problem	O
used	O
to	O
claim	O
quantum	B-Device
supremacy	I-Device
,	O
giving	O
substantial	O
reductions	O
to	O
the	O
gap	O
between	O
Sycamore	B-Device
and	O
classical	O
supercomputers	B-Architecture
and	O
even	O
beating	O
it	O
.	O
</s>
<s>
In	O
December	O
2020	O
,	O
a	O
group	O
at	O
USTC	O
implemented	O
a	O
type	O
of	O
Boson	O
sampling	O
on	O
76	O
photons	B-Application
with	O
a	O
photonic	O
quantum	B-Architecture
computer	I-Architecture
,	O
Jiuzhang	O
,	O
to	O
demonstrate	O
quantum	B-Device
supremacy	I-Device
.	O
</s>
<s>
The	O
authors	O
claim	O
that	O
a	O
classical	O
contemporary	O
supercomputer	B-Architecture
would	O
require	O
a	O
computational	O
time	O
of	O
600	O
million	O
years	O
to	O
generate	O
the	O
number	O
of	O
samples	O
their	O
quantum	B-Architecture
processor	I-Architecture
can	O
generate	O
in	O
20	O
seconds	O
.	O
</s>
<s>
On	O
November	O
16	O
,	O
2021	O
,	O
at	O
the	O
quantum	B-Architecture
computing	I-Architecture
summit	B-Device
,	O
IBM	O
presented	O
a	O
127-qubit	O
microprocessor	O
named	O
IBM	B-Device
Eagle	I-Device
.	O
</s>
<s>
Some	O
researchers	O
have	O
expressed	O
skepticism	O
that	O
scalable	O
quantum	B-Architecture
computers	I-Architecture
could	O
ever	O
be	O
built	O
,	O
typically	O
because	O
of	O
the	O
issue	O
of	O
maintaining	O
coherence	O
at	O
large	O
scales	O
,	O
but	O
also	O
for	O
other	O
reasons	O
.	O
</s>
<s>
Bill	O
Unruh	O
doubted	O
the	O
practicality	O
of	O
quantum	B-Architecture
computers	I-Architecture
in	O
a	O
paper	O
published	O
in	O
1994	O
.	O
</s>
<s>
Skeptics	O
like	O
Gil	O
Kalai	O
doubt	O
that	O
quantum	B-Device
supremacy	I-Device
will	O
ever	O
be	O
achieved	O
.	O
</s>
<s>
Physicist	O
Mikhail	O
Dyakonov	O
has	O
expressed	O
skepticism	O
of	O
quantum	B-Architecture
computing	I-Architecture
as	O
follows	O
:	O
</s>
<s>
"	O
So	O
the	O
number	O
of	O
continuous	O
parameters	O
describing	O
the	O
state	O
of	O
such	O
a	O
useful	O
quantum	B-Architecture
computer	I-Architecture
at	O
any	O
given	O
moment	O
must	O
be	O
...	O
about	O
10300	O
...	O
Could	O
we	O
ever	O
learn	O
to	O
control	O
the	O
more	O
than	O
10300	O
continuously	O
variable	O
parameters	O
defining	O
the	O
quantum	O
state	O
of	O
such	O
a	O
system	O
?	O
</s>
<s>
For	O
physically	O
implementing	O
a	O
quantum	B-Architecture
computer	I-Architecture
,	O
many	O
different	O
candidates	O
are	O
being	O
pursued	O
,	O
among	O
them	O
(	O
distinguished	O
by	O
the	O
physical	O
system	O
used	O
to	O
realize	O
the	O
qubits	O
)	O
:	O
</s>
<s>
Linear	O
optical	O
quantum	B-Architecture
computer	I-Architecture
(	O
LOQC	O
)	O
(	O
qubits	O
realized	O
by	O
processing	O
states	O
of	O
different	O
modes	O
of	O
light	O
through	O
linear	O
elements	O
e.g.	O
</s>
<s>
The	O
large	O
number	O
of	O
candidates	O
demonstrates	O
that	O
quantum	B-Architecture
computing	I-Architecture
,	O
despite	O
rapid	O
progress	O
,	O
is	O
still	O
in	O
its	O
infancy	O
.	O
</s>
<s>
Any	O
computational	O
problem	O
solvable	O
by	O
a	O
classical	O
computer	O
is	O
also	O
solvable	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
Intuitively	O
,	O
this	O
is	O
because	O
it	O
is	O
believed	O
that	O
all	O
physical	O
phenomena	O
,	O
including	O
the	O
operation	O
of	O
classical	O
computers	O
,	O
can	O
be	O
described	O
using	O
quantum	O
mechanics	O
,	O
which	O
underlies	O
the	O
operation	O
of	O
quantum	B-Architecture
computers	I-Architecture
.	O
</s>
<s>
Conversely	O
,	O
any	O
problem	O
solvable	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
is	O
also	O
solvable	O
by	O
a	O
classical	O
computer	O
.	O
</s>
<s>
More	O
formally	O
,	O
any	O
quantum	B-Architecture
computer	I-Architecture
can	O
be	O
simulated	O
by	O
a	O
Turing	B-Architecture
machine	I-Architecture
.	O
</s>
<s>
In	O
other	O
words	O
,	O
quantum	B-Architecture
computers	I-Architecture
provide	O
no	O
additional	O
power	O
over	O
classical	O
computers	O
in	O
terms	O
of	O
computability	O
.	O
</s>
<s>
This	O
means	O
that	O
quantum	B-Architecture
computers	I-Architecture
cannot	O
solve	O
undecidable	O
problems	O
like	O
the	O
halting	O
problem	O
,	O
and	O
the	O
existence	O
of	O
quantum	B-Architecture
computers	I-Architecture
does	O
not	O
disprove	O
the	O
Church	O
–	O
Turing	O
thesis	O
.	O
</s>
<s>
While	O
quantum	B-Architecture
computers	I-Architecture
cannot	O
solve	O
any	O
problems	O
that	O
classical	O
computers	O
cannot	O
already	O
solve	O
,	O
it	O
is	O
suspected	O
that	O
they	O
can	O
solve	O
certain	O
problems	O
faster	O
than	O
classical	O
computers	O
.	O
</s>
<s>
For	O
instance	O
,	O
it	O
is	O
known	O
that	O
quantum	B-Architecture
computers	I-Architecture
can	O
efficiently	O
factor	O
integers	O
,	O
while	O
this	O
is	O
not	O
believed	O
to	O
be	O
the	O
case	O
for	O
classical	O
computers	O
.	O
</s>
<s>
The	O
class	O
of	O
problems	O
that	O
can	O
be	O
efficiently	O
solved	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
with	O
bounded	O
error	O
is	O
called	O
BQP	O
,	O
for	O
"	O
bounded	O
error	O
,	O
quantum	O
,	O
polynomial	O
time	O
"	O
.	O
</s>
<s>
More	O
formally	O
,	O
BQP	O
is	O
the	O
class	O
of	O
problems	O
that	O
can	O
be	O
solved	O
by	O
a	O
polynomial-time	O
quantum	O
Turing	B-Architecture
machine	I-Architecture
with	O
an	O
error	O
probability	O
of	O
at	O
most	O
1/3	O
.	O
</s>
<s>
As	O
a	O
class	O
of	O
probabilistic	O
problems	O
,	O
BQP	O
is	O
the	O
quantum	O
counterpart	O
to	O
BPP	O
(	O
"	O
bounded	O
error	O
,	O
probabilistic	O
,	O
polynomial	O
time	O
"	O
)	O
,	O
the	O
class	O
of	O
problems	O
that	O
can	O
be	O
solved	O
by	O
polynomial-time	O
probabilistic	O
Turing	B-Architecture
machines	I-Architecture
with	O
bounded	O
error	O
.	O
</s>
<s>
It	O
is	O
known	O
that	O
and	O
is	O
widely	O
suspected	O
that	O
,	O
which	O
intuitively	O
would	O
mean	O
that	O
quantum	B-Architecture
computers	I-Architecture
are	O
more	O
powerful	O
than	O
classical	O
computers	O
in	O
terms	O
of	O
time	O
complexity	O
.	O
</s>
<s>
However	O
,	O
it	O
is	O
known	O
that	O
;	O
that	O
is	O
,	O
all	O
problems	O
that	O
can	O
be	O
efficiently	O
solved	O
by	O
a	O
deterministic	O
classical	O
computer	O
can	O
also	O
be	O
efficiently	O
solved	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
,	O
and	O
all	O
problems	O
that	O
can	O
be	O
efficiently	O
solved	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
can	O
also	O
be	O
solved	O
by	O
a	O
deterministic	O
classical	O
computer	O
with	O
polynomial	O
space	O
resources	O
.	O
</s>
<s>
It	O
is	O
further	O
suspected	O
that	O
BQP	O
is	O
a	O
strict	O
superset	O
of	O
P	O
,	O
meaning	O
there	O
are	O
problems	O
that	O
are	O
efficiently	O
solvable	O
by	O
quantum	B-Architecture
computers	I-Architecture
that	O
are	O
not	O
efficiently	O
solvable	O
by	O
deterministic	O
classical	O
computers	O
.	O
</s>
<s>
It	O
is	O
suspected	O
that	O
;	O
that	O
is	O
,	O
it	O
is	O
believed	O
that	O
there	O
are	O
efficiently	O
checkable	O
problems	O
that	O
are	O
not	O
efficiently	O
solvable	O
by	O
a	O
quantum	B-Architecture
computer	I-Architecture
.	O
</s>
<s>
For	O
instance	O
,	O
it	O
has	O
been	O
shown	O
that	O
a	O
non-local	O
hidden	O
variable	O
quantum	B-Architecture
computer	I-Architecture
based	O
on	O
Bohmian	O
Mechanics	O
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
a	O
slight	O
speedup	O
over	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
,	O
which	O
runs	O
in	O
steps	O
.	O
</s>
<s>
Note	O
,	O
however	O
,	O
that	O
neither	O
search	O
method	O
would	O
allow	O
quantum	B-Architecture
computers	I-Architecture
to	O
solve	O
NP-complete	O
problems	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
Theories	O
of	O
quantum	O
gravity	O
,	O
such	O
as	O
M-theory	B-General_Concept
and	O
loop	O
quantum	O
gravity	O
,	O
may	O
allow	O
even	O
faster	O
computers	O
to	O
be	O
built	O
.	O
</s>
