Download Exercise sheet 11

Exercise Sheet 11
1. Which of the following expressions are possible superpositions of a qubit?
(a) 0.7 |0i + 0.3 |1i
(b) 0.8 |0i + 0.6 |1i
√
(c)
(d)
3
2
|1i −
cos2 θ |0i
1
2
|0i
+ sin2 θ |1i
Give for each valid superposition the probability that a 0 will be measured.
2. Consider a 2-qubit in the following superposition:
1
√ (|00i + 2 |01i + 3 |10i + 4 |11i)
30
The first qubit is measured, with as outcome 1. What is the superposition of the second
qubit after this measurement?
3. The Hadamard transformation transposes |0i as well as |1i into a superposition
in which
1
1
the outcomes 0 and 1 are equally likely. Its matrix definition is: √12
1 −1
Suppose that we apply the Hadamard transformation to:
(a) |0i
(b) |1i
(c)
√1
2
(|0i + |1i)
What is in each of these three cases the probability of measuring a 1?
4. Consider the 2-qubit √12 (|00i + |11i). Suppose that we first apply a rotation of − π8 on
the second qubit, and then a rotation of π8 on the first qubit. Show that the resulting
superposition of the 2-qubit is:
1
π
π
π
π
√ ((cos2 − sin2 ) |00i − 2· sin · cos |01i
8
8
8
8
2
π
π
π
π
+ 2· sin · cos |10i + (cos2 − sin2 ) |11i)
8
8
8
8
5. Suppose that in the parity game x = 0 en y = 1. Show that then in the quantum
computing solution of this game, a ⊕ b = 0 with probability cos2 π8 .
1