Show that the n-qubit Hadamard gate acts as
H
⊗n
|xi
n
=
1
√
2
n
2
n
−1
X
y=1
(−1)
x·y
|yi. (7.28)
where x · y is the bitwise product of x and y:
x · y = x
0
y
0
⊕ x
1
y
1
⊕ . . . x
n−1
y
n−1
. (7.29)
7.2. Often it helps to simplify circuits when we can identify equivalences be-
tween some combinations of gates. Prove, for example, the following circuit
identities:
a) HXH = Z
(b) HY H = −Y
(c) HZH = X
7.3. Show the following relations concerning rotation matrices:
(a) R
n
(θ
1
)R
n
(θ
2
) = R
n
(θ
1
+ θ
2
)
(b) XR
n
(θ)X = R
n
(−θ)
7.4. The “SWAP” gate S interchanges two inputs, defined by
S|xyi = |yxi.
(a) Give the matrix representing this gate.
(b) Show that it can be implemented by 3 CNOT gates as
S
12
= C
12
C
21
C
12
.
(c) Show that the matrix is equivalent to
S
12
=
1
2
( + X
1
X
2
+ Y
1
Y
2
+ Z
1
Z
2
)
7.5. The controlled phase-flip gate takes |11i to −|11i while leaving the other
basis states unchanged. It is sometimes represented as follows, since its
action is symmetric in the inputs:
|xi
•
|xi
|yi
•
(−1)
xy
|yi
(a) Construct the matrix for this gate.
(b) Build a CNOT gate using controlled phase-flip gates an another single-
qubit gate.
(c) What is the difference in the outputs of the following two circuits?
• •
• •
and
• •
• •
(d) Evaluate the output of the circuit
|xi
H
•
|yi
H
• •
|zi
H
•
142 Introduction to Quantum Physics and Information Processing
7.6. Show that classical conditional operations are equivalent to quantum con-
trol, i.e., show that the following two circuits are equivalent:
• •
≡
U U
7.7. Verify the following circuit identities:
(a)
X
≡
X
• •
X
(b)
X X
≡
• •
(c)
Z Z
≡
Z
• •
(d)
≡
Z
• •
Z
7.8. Consider the four possible 1-bit functions
f
0
: 0 → 0
1 → 0
,
f
1
: 0 → 0
1 → 1
,
f
2
: 0 → 1
1 → 0
,
f
3
: 0 → 1
1 → 1
.
Construct the matrix representation of U
f
for each. Also give a simple circuit
to implement each using basic 1-qubit gates.
7.9. Consider 1-bit integer addition. Write down the truth tables for sum and
carry bits. Then construct a quantum half-adder by implementing the truth
tables, using only CNOT gates.
7.10. Examine the following circuit and analyze the final output. Here, the input
is an unknown entangled state
|ψi = α|01i+ β|10i
and |GHZi =
1
√
2
(|000i + |111i) .
•
|ψi
H
•
H
•
• •
|GHZi
H H
?
•
H
•
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