8 Quiz
Answer the following questions in order to test your understanding of the key ideas that you have been learning about.
Question 1
Which of the following statements about eigenvalues, eigenstates, eigenvectors and eigenfunctions are true?
Answer
The first five statements are all true. The last one is false: there may be more than one eigenvalue and corresponding eigenfunction associated with each eigenvalue equation.
Question 2
What are the eigenvalues and eigenvectors of the following matrix?
Answer
Following the prescription described in the course: 
, 
, 
and 
. So we first need to solve the quadratic equation
which is simply
This can be written as
So it has solutions 
and 
. These are the two eigenvalues.
We now write the two eigenvector equations:
For eigenvalue these reduce to
Both equations imply that 
, so 
and 
and the first eigenvector is
For eigenvalue these reduce to
Both equations imply that 
, so 
and and the second eigenvector is
Question 3
If a general spin state is written as 
, which of the following statements is true?
Answer
The probability of the outcome of a measurement indicating spin-up is 
and for spin-down is 
.
Question 4
Match the following two-particle spin states with the correct descriptions.
symmetric not entangled state
symmetric entangled state
antisymmetric entangled state
Using the following two lists, match each numbered item with the correct letter.
-
symmetric not entangled state
-
symmetric entangled state
-
antisymmetric entangled state
- 1 =
- 2 =
- 3 =
Answer
The triplet states (i.e. 
and 
and 
) are symmetric and the singlet state (i.e. 
) is antisymmetric under particle exchange. Two-particle states which cannot be factorised (i.e. and ) are known as entangled states. The other states (i.e. and ) are not entangled.
Question 5
The quantum NOT gate is represented by which of the following matrices?
Answer
The quantum NOT gate is represented by the Pauli-X operator, 
Question 6
The Hadamard gate is represented by which of the following matrices?
Answer
The Hadamard gate is represented by 
Question 7
Match the following quantum gates with the correct result.
NOT gate
Flips the state of a qubit
CNOT gate
Entangles a pair of disentangled qubits
Hadamard gate
Transforms a qubit into a superposition state
Using the following two lists, match each numbered item with the correct letter.
-
NOT gate
-
CNOT gate
-
Hadamard gate
-
Flips the state of a qubit
-
Entangles a pair of disentangled qubits
-
Transforms a qubit into a superposition state
- 1 =
- 2 =
- 3 =
Answer
A NOT gate flips the state of a qubit. A CNOT gate can entangle a pair of disentangled qubitts. A Hadamard gate can transform a qubit into a superposition state.
Question 8
If an input qubit 
is passed to a Hadamard gate, and the output from the Hadamard gate is then passed as input to another Hadamard gate, what will be the output from the second Hadamard gate?
Answer
The circuit can be written as 
. The action of the first Hadamard gate produces a superposition state:
Then passing this through the second Hadamard gate we have
The action of the second Hadamard gate is therefore to restore the original input qubit.
OpenLearn - Introduction to quantum computing
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