Years 11–12 · Week 4 of 12

Phase, Interference and the Bloch Sphere

55 minutes 7 possible star points Mathematical Extension for Year 12

Learning goals

By the end, you can…

Distinguish global phase from relative phase.
Predict the H–H and H–Z–H circuits from |0⟩.
Interpret the Bloch sphere as a representation of one pure qubit state.
Relate basis changes to measurements in the X and Z bases.

What you already know

Connect to a familiar idea

The states |+⟩ = (|0⟩ + |1⟩)/√2 and |−⟩ = (|0⟩ − |1⟩)/√2 have the same immediate Z-basis probabilities. Their relative signs nevertheless lead to different interference.

One-qubit state vectors and normalisation
Computational-basis measurement
The basic actions of H and Z gates

Opening story

Start with something familiar

Two wave signals can have equal strength yet combine differently because their peaks and troughs are aligned differently. Quantum amplitudes also combine with phase, so probabilities seen after a circuit depend on more than the size of each amplitude.

Plain-English explanation

Build the idea carefully

Relative phase creates distinguishable states

Multiplying an entire isolated state by one phase factor gives only global phase and changes no measurement prediction. Changing the phase of one component relative to another produces a different state that can be distinguished after suitable operations. The states |+⟩ and |−⟩ each give 50% zero and 50% one in the Z basis. However, H|+⟩ = |0⟩ and H|−⟩ = |1⟩, so the H gate converts their relative-phase difference into different computational-basis outcomes.

Bloch-sphere and measurement-basis representations

Worked circuits: H followed by H sends |0⟩ back to |0⟩. Inserting Z gives HZH|0⟩ = |1⟩. This is interference: amplitudes are transformed and combined before their squared magnitudes are measured. The Bloch sphere represents single-qubit pure states up to global phase. The north and south poles represent |0⟩ and |1⟩; opposite equatorial points can represent |+⟩ and |−⟩. It is a state-space diagram, not a physical ball containing a qubit.

Phase and basis on the Bloch sphere The sphere represents one pure qubit state up to global phase; the adjacent table provides the same information without 3D interaction.

Try the model

Relative-phase and Bloch representation

Adjust amplitude balance and relative phase, inspect Z- and X-basis probabilities, then compare H–H with H–Z–H step by step.

Interactive teaching model

State setting Number of trials

Ready. Adjust a control, then run the model.

What this model shows: Use either the rotatable sphere or the keyboard-accessible 2D state table; both control the same ideal one-qubit simulation.

Text alternative for this interactive

Text alternative: plus has amplitudes (1/√2, 1/√2) and minus has (1/√2, −1/√2). Applying H adds and subtracts these components, producing (1, 0) and (0, 1).

Expected observation: |+⟩ and |−⟩ have equal Z-basis probabilities, but an H basis change maps them deterministically to |0⟩ and |1⟩ respectively.

Guided activity

Same probabilities, different phase

Prepare |+⟩ and |−⟩ and record their Z-basis probabilities.
Predict each result after applying H.
Run 100 shots for each transformed state.
Explain which evidence shows that the original states were distinct.

Evidence to collect: A comparison showing identical initial Z-basis probabilities but different post-H histograms, with the difference attributed to relative phase and interference.

Glossary

Words to know

phase: An angular property of an amplitude that affects how amplitudes combine.
relative phase: The phase difference between state-vector components.
global phase: A common phase multiplying the whole state, with no observable effect by itself.
interference: The addition of amplitudes before probabilities are calculated.
Bloch sphere: A geometric representation of one pure qubit state up to global phase.
Z basis: The measurement basis {|0⟩, |1⟩}.
X basis: The measurement basis {|+⟩, |−⟩}.

Short recap

Keep these ideas

Global phase and relative phase play different roles.
|+⟩ and |−⟩ have identical Z-basis probabilities but are distinct states.
H–H returns |0⟩, whereas H–Z–H sends |0⟩ to |1⟩.
The Bloch sphere represents state space and must not be interpreted as a physical orbit.

Mathematical Extension Optional extension for Year 12

Up to global phase, a pure qubit can be written |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩. The angles θ and φ locate a point on the Bloch sphere. Measuring along different axes corresponds to choosing different orthonormal bases.

Try this

For θ = π/2, compare φ = 0, π and π/2. Write each state, predict its Z-basis probabilities and identify its labelled equatorial point.

Adult support Teacher and parent notes

Discuss

Why cannot Z-basis probabilities alone distinguish plus from minus?
What mathematical step causes interference in the H gate?
Which parts of the Bloch sphere are representation choices rather than physical locations?

Answer guidance

Students should connect relative sign or phase with addition and subtraction of amplitudes. Do not accept claims that the sphere is the qubit's physical shape.

Offline activity

Use arrow cards for amplitudes. Add or subtract the cards component by component for H–H and H–Z–H, then square the final magnitudes.

Safety

Provide the 2D alternative for students who cannot or prefer not to use a rotatable 3D control. No rapid rotation is automatic.

Open the full Quantum Computing Foundations adult guide

Sources and further reading

Checked references for this lesson

These sources support the lesson’s main scientific claims. Links open on the source organisation’s site.

The qubit in quantum computing Microsoft Learn · official documentation · checked 2026-08-02
Quantum information IBM Quantum Learning · official learning module · checked 2026-08-02
Quantum Computation and Quantum Information Cambridge University Press · textbook publisher page · checked 2026-08-02

Content review: Reviewed on 2026-08-02.