CSE 599d Quantum Computing Problem Set 2
Author: Dave Bacon (
)
Due: February 3, 2006
Exercise 1:
Fourinone Grover
Let
f
α
:
{
0
,
1
}
2
→ {
0
,
1
}
be one of four functions from two bits to one bit deﬁned as
f
α
(
x
1
, x
2
) =
δ
α
1
,x
1
δ
α
2
,x
2
where
α
∈ {
0
,
1
}
2
and
x
∈ {
0
,
1
}
2
(The four diﬀerent functions are label by the two bits of
α
.)
(a) Prove that in order to exactly (no probability of failure) distinguish between these four functions, you need to
query this function three times in the worst case.
(b) Suppose that you have a unitary gate which enacts this function in the standard reversible manner:
U
α
=
X
x
1
,x
2
∈{
0
,
1
}

x
1
, x
2
ih
x
1
, x
2
 ⊗
X
y
∈{
0
,
1
}

y
⊕
f
α
(
x
1
, x
2
)
ih
y

(1)
Explain how to use this unitary to create the state

α
i
=
1
2
X
x
1
,x
2
∈{
0
,
1
}
(

1)
f
α
(
x
1
,x
2
)

x
1
, x
2
i
(2)
(c) Show that the

α
i
states deﬁned in the last problem are all orthonormal.
(d) Since the four states deﬁned above are orthogonal, there is a measurement which distinguishes between the four
states. Write down a two qubit unitary matrix which transforms the

α
i
states into the four computational basis
states

00
i
,

01
i
,

10
i
, and

11
i
. Express the elements of this unitary matrix in the computational basis.
(e) Construct a circuit which transforms