A single qubit is in the state (3/5)|0> + (4/5)|1>. What is the probability of measuring |1>?
A3/5
B4/5
C9/25
D16/25
The probability of measuring |1> is |beta|^2 = (4/5)^2 = 16/25. A common mistake is confusing the amplitude with the probability — the amplitude is 4/5, but probability is the squared magnitude of the amplitude.
Question 2 True / False
A system of 3 qubits requires a state vector with 8 complex amplitudes to fully describe it.
TTrue
FFalse
Answer: True
Three qubits live in a 2^3 = 8-dimensional Hilbert space. The state vector has 8 complex components, one for each computational basis state |000> through |111>. This exponential growth — 2^n amplitudes for n qubits — is what makes classical simulation of quantum systems intractable for large n.
Question 3 Short Answer
What distinguishes a qubit from a classical probabilistic bit that has, say, a 70% chance of being 0 and a 30% chance of being 1?
Think about your answer, then reveal below.
Model answer: A qubit's state is described by complex amplitudes, not just probabilities, and these amplitudes can interfere constructively or destructively. A probabilistic classical bit has real, non-negative probabilities that always add — there is no interference. This interference between amplitudes is what enables quantum algorithms to amplify correct answers and suppress wrong ones.
The key distinction is interference. A probabilistic bit is a mixture described by real numbers that combine incoherently. A qubit is a coherent superposition described by complex amplitudes whose phases matter. Two paths to the same outcome can cancel (destructive interference) or reinforce (constructive interference), which has no classical analog.
Question 4 Multiple Choice
On the Bloch sphere, where is the state |+> = (|0> + |1>)/sqrt(2) located relative to |0> and |1>?
AAt the north pole, same as |0>
BAt the south pole, same as |1>
COn the equator, pointing along the +x axis
DExactly halfway between the poles along the z axis
|0> is the north pole and |1> is the south pole of the Bloch sphere. Equal superpositions like |+> lie on the equator. The phase of the superposition determines the azimuthal angle: |+> = (|0> + |1>)/sqrt(2) points along the +x direction, while |-> = (|0> - |1>)/sqrt(2) points along -x. States with relative phase i or -i point along +y or -y respectively.