UNIT III: Foundations of Quantum Computing
Short answer questions
1. What is a qubit?
Ans.) A qubit is a quantum bit that exists in a superposition of the basis states ∣0⟩ and ∣1⟩
2. Which visualization tool represents the state of a single qubit?
Ans.) The Bloch sphere.
3. Name the phenomenon that enables quantum computers to evaluate multiple
inputs at the same time.
Ans.) Quantum parallelism.
4. Which postulate collapses the quantum state to an eigenstate?
Ans.) Measurement postulate.
5. What type of matrix represents a unitary transformation?
Ans.) A matrix 𝑈satisfying 𝑈 † 𝑈 = 𝐼.
6. State the action of the Hadamard gate on ∣0⟩ and ∣1⟩.
Ans.) It will generate the superposition state
H ∣0⟩ = (|0⟩ + |1⟩)/√2 =|+⟩
H ∣1⟩ = (|0⟩ - |1⟩)/√2 =|-⟩
7. Which logic gate creates entanglement in quantum computing?
Ans.) The CNOT gate.
8. What is the key difference between a classical bit and a qubit?
Ans.) A classical bit takes either 0 or 1, while a qubit can be in a superposition of both.
9. Do measurements preserve quantum superpositions?
Ans.) No, measurement collapses the state to one of the eigenstates.
10. What is quantum entanglement?
Ans.) Quantum entanglement is a unique phenomenon in which two or more quantum
particles become linked so that the state of one particle is directly connected to the state of
the other(s), no matter how far apart they are in space.
11. Write the Bell state |Φ⁺⟩.
Ans.)
12. What gate combination typically creates a Bell state from |00⟩?
Ans.) Hadamard gate on the first qubit followed by a CNOT gate.
, Long Answer questions
1. Explain quantum states and measurement. Discuss how measurement affects a qubit.
A quantum state describes the complete physical information of a quantum system. For a
qubit, the general state is
∣ψ⟩ = α∣0⟩ + β∣1⟩
where 𝛼 and 𝛽 are complex probability amplitudes satisfying
∣ 𝛼 ∣2 +∣ 𝛽 ∣2 = 1.
Quantum measurement is the process by which a quantum system, such as a qubit,
interacts with a measuring device and produces a definite outcome. Before measurement,
a quantum system can exist in a superposition of multiple possible states.
Measurement in the computational basis projects the state onto ∣ 0⟩ or ∣ 1⟩. The
probabilities of outcomes are:
• 0 with probability ∣ 𝛼 ∣2
• 1 with probability ∣ 𝛽 ∣2
Every measurement corresponds to an observable, represented by a Hermitian operator.
Its eigenvalues = possible measurement outcomes.
Its eigenstates = states the system collapses into.
After measurement, the state collapses irreversibly to the observed eigenstate. Thus,
measurement destroys coherence and eliminates the superposition. For multi-qubit states,
measurement can also remove quantum correlations (entanglement). Measurement
distinguishes quantum mechanics from classical systems by its probabilistic nature and
collapse phenomenon.
2. What is quantum entanglement? Describe EPR pairs and discuss their role in
quantum information.
Ans.) Quantum entanglement is a unique phenomenon in which two or more quantum
particles become linked so that the state of one particle is directly connected to the state
of the other(s), no matter how far apart they are in space.
When particles are entangled, a measurement
performed on one particle instantaneously influences the state
of the other. This phenomenon Albert Einstein famously
described as “spooky action at a distance.
A standard entangled state is the EPR pair (Bell state):
∣ 00⟩+∣ 11⟩
∣ Φ+ ⟩ =
√2
Short answer questions
1. What is a qubit?
Ans.) A qubit is a quantum bit that exists in a superposition of the basis states ∣0⟩ and ∣1⟩
2. Which visualization tool represents the state of a single qubit?
Ans.) The Bloch sphere.
3. Name the phenomenon that enables quantum computers to evaluate multiple
inputs at the same time.
Ans.) Quantum parallelism.
4. Which postulate collapses the quantum state to an eigenstate?
Ans.) Measurement postulate.
5. What type of matrix represents a unitary transformation?
Ans.) A matrix 𝑈satisfying 𝑈 † 𝑈 = 𝐼.
6. State the action of the Hadamard gate on ∣0⟩ and ∣1⟩.
Ans.) It will generate the superposition state
H ∣0⟩ = (|0⟩ + |1⟩)/√2 =|+⟩
H ∣1⟩ = (|0⟩ - |1⟩)/√2 =|-⟩
7. Which logic gate creates entanglement in quantum computing?
Ans.) The CNOT gate.
8. What is the key difference between a classical bit and a qubit?
Ans.) A classical bit takes either 0 or 1, while a qubit can be in a superposition of both.
9. Do measurements preserve quantum superpositions?
Ans.) No, measurement collapses the state to one of the eigenstates.
10. What is quantum entanglement?
Ans.) Quantum entanglement is a unique phenomenon in which two or more quantum
particles become linked so that the state of one particle is directly connected to the state of
the other(s), no matter how far apart they are in space.
11. Write the Bell state |Φ⁺⟩.
Ans.)
12. What gate combination typically creates a Bell state from |00⟩?
Ans.) Hadamard gate on the first qubit followed by a CNOT gate.
, Long Answer questions
1. Explain quantum states and measurement. Discuss how measurement affects a qubit.
A quantum state describes the complete physical information of a quantum system. For a
qubit, the general state is
∣ψ⟩ = α∣0⟩ + β∣1⟩
where 𝛼 and 𝛽 are complex probability amplitudes satisfying
∣ 𝛼 ∣2 +∣ 𝛽 ∣2 = 1.
Quantum measurement is the process by which a quantum system, such as a qubit,
interacts with a measuring device and produces a definite outcome. Before measurement,
a quantum system can exist in a superposition of multiple possible states.
Measurement in the computational basis projects the state onto ∣ 0⟩ or ∣ 1⟩. The
probabilities of outcomes are:
• 0 with probability ∣ 𝛼 ∣2
• 1 with probability ∣ 𝛽 ∣2
Every measurement corresponds to an observable, represented by a Hermitian operator.
Its eigenvalues = possible measurement outcomes.
Its eigenstates = states the system collapses into.
After measurement, the state collapses irreversibly to the observed eigenstate. Thus,
measurement destroys coherence and eliminates the superposition. For multi-qubit states,
measurement can also remove quantum correlations (entanglement). Measurement
distinguishes quantum mechanics from classical systems by its probabilistic nature and
collapse phenomenon.
2. What is quantum entanglement? Describe EPR pairs and discuss their role in
quantum information.
Ans.) Quantum entanglement is a unique phenomenon in which two or more quantum
particles become linked so that the state of one particle is directly connected to the state
of the other(s), no matter how far apart they are in space.
When particles are entangled, a measurement
performed on one particle instantaneously influences the state
of the other. This phenomenon Albert Einstein famously
described as “spooky action at a distance.
A standard entangled state is the EPR pair (Bell state):
∣ 00⟩+∣ 11⟩
∣ Φ+ ⟩ =
√2
