<s>
In	O
quantum	B-Architecture
computing	I-Architecture
,	O
the	O
quantum	B-Algorithm
phase	I-Algorithm
estimation	I-Algorithm
algorithm	I-Algorithm
(	O
also	O
referred	O
to	O
as	O
quantum	O
eigenvalue	O
estimation	O
algorithm	O
)	O
,	O
is	O
a	O
quantum	B-Device
algorithm	I-Device
to	O
estimate	O
the	O
phase	O
(	O
or	O
eigenvalue	O
)	O
of	O
an	O
eigenvector	O
of	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
.	O
</s>
<s>
More	O
precisely	O
,	O
given	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
and	O
a	O
quantum	O
state	O
such	O
that	O
,	O
the	O
algorithm	O
estimates	O
the	O
value	O
of	O
with	O
high	O
probability	O
within	O
additive	O
error	O
,	O
using	O
qubits	O
(	O
without	O
counting	O
the	O
ones	O
used	O
to	O
encode	O
the	O
eigenvector	O
state	O
)	O
and	O
controlled-U	O
operations	O
.	O
</s>
<s>
Phase	B-Algorithm
estimation	I-Algorithm
is	O
frequently	O
used	O
as	O
a	O
subroutine	O
in	O
other	O
quantum	B-Device
algorithms	I-Device
,	O
such	O
as	O
Shor	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
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
.	O
</s>
<s>
Let	O
U	O
be	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
that	O
operates	O
on	O
m	O
qubits	O
with	O
an	O
eigenvector	O
such	O
that	O
.	O
</s>
<s>
We	O
can	O
write	O
the	O
eigenvalue	O
in	O
the	O
form	O
because	O
U	O
is	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
over	O
a	O
complex	O
vector	O
space	O
,	O
so	O
its	O
eigenvalues	O
must	O
be	O
complex	O
numbers	O
with	O
absolute	O
value	O
1	O
.	O
</s>
<s>
Let	O
be	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
with	O
eigenvector	O
such	O
that	O
.	O
</s>
<s>
It	O
can	O
be	O
reused	O
again	O
in	O
another	O
run	O
of	O
phase	B-Algorithm
estimation	I-Algorithm
.	O
</s>
