Years 7–10 · Week 7 of 12
Quantum Gates, Phase and Interference
Learning goals
By the end, you can…
- Predict the action of X, Z and H on selected basis and superposition states.
- Trace a short circuit one gate at a time.
- Compare constructive and destructive amplitude interference.
- Explain why phase can matter even when immediate probabilities are unchanged.
What you already know
Connect to a familiar idea
The |+⟩ and |−⟩ states give the same immediate 0/1 probabilities but have different relative phases. Gates can convert that phase difference into a difference that computational-basis measurement can reveal.
- |0⟩ and |1⟩ form the computational basis.
- A qubit state includes amplitudes and relative phase.
- Measurement probabilities come from squared amplitude magnitudes.
Opening story
Start with something familiar
Two circuits begin in |0⟩. One applies H twice and ends back at |0⟩. The other places Z between the H gates and ends at |1⟩. The Z gate does not change the middle probability bars, yet its phase change controls the final interference.
Plain-English explanation
Build the idea carefully
Three gates and their roles
The X gate exchanges |0⟩ and |1⟩ and is sometimes compared with a classical NOT, although it also acts on superpositions. The Z gate leaves |0⟩ unchanged and changes the sign of the |1⟩ amplitude, altering relative phase without necessarily changing immediate 0/1 probabilities.
Gate sequences reveal phase
The H gate maps basis states to equal-magnitude superpositions and can recombine amplitudes so that they reinforce for one output and cancel for another. Gate order matters: H–H returns |0⟩ to |0⟩, while H–Z–H maps |0⟩ to |1⟩ in the ideal model. Interference combines probability amplitudes, not already-calculated probabilities, which is why relative phase can alter later results.
Try the model
Gate-by-gate interference player
Build or choose a one-qubit circuit, step through each gate and compare the state, phase indicators and predicted measurement probabilities.
Qubit 0 is the least-significant state-vector bit. Displayed basis labels read q(n−1)…q0.
Ready. Adjust a control, then run the model.
What this model shows: An ideal state-vector simulation; signed bars are mathematical amplitudes, not extra observable particles.
Text alternative for this interactive
Circuit step table Select H–H or H–Z–H and read the state, relative phase and probability row after each gate.
Expected observation: H–H returns |0⟩ to |0⟩, whereas H–Z–H sends |0⟩ to |1⟩ because Z changes relative phase before amplitudes recombine.
Guided activity
Find the hidden phase change
- Predict the output of H–H from |0⟩.
- Step through H–H and record each state.
- Repeat for H–Z–H, noting the unchanged middle probabilities after Z.
- Explain which final observation reveals the phase difference.
Evidence to collect: Two completed step tables showing identical middle probabilities but different relative phase and final deterministic outcomes.
Glossary
Words to know
- quantum gate
- A reversible operation that transforms a quantum state.
- X gate
- A gate that exchanges the computational-basis states.
- Z gate
- A gate that changes the relative sign of the |1⟩ component.
- H gate
- The Hadamard gate, which creates and recombines equal-magnitude amplitudes.
- relative phase
- The phase difference between state components.
- constructive interference
- Amplitude combination that increases an outcome amplitude.
- destructive interference
- Amplitude combination that reduces or cancels an outcome amplitude.
Short recap
Keep these ideas
- X, Z and H transform both basis states and superpositions.
- Relative phase can be invisible to one immediate measurement yet affect later interference.
- Gate order and amplitude interference determine an ideal circuit's final probabilities.
Knowledge check
6 clear questions
Choose an answer for immediate feedback. You may retry, and your best submitted score is kept.
Go Further Optional extension for Years 9–10
For Years 9–10, follow the real-amplitude state vectors |0⟩ = [1, 0], |+⟩ = [1/√2, 1/√2] and |−⟩ = [1/√2, −1/√2] without calculating with complex numbers.
Try this
Use the supplied X, Z and H mapping cards to trace H–H and H–Z–H and verify the final state vectors.
Adult support Teacher and parent notes
Discuss
- Emphasise amplitudes before probabilities when explaining interference.
- Ask students to compare circuit order rather than memorising isolated gate slogans.
Answer guidance
A complete explanation mentions the Z-induced relative sign and the final H converting phase into a basis-state difference.
Offline activity
Use signed amplitude cards for |0⟩ and |1⟩ components and physically pair equal or opposite cards during the final H step.
Safety
No special hazards; the circuit is an ideal mathematical simulation.
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.
- Circuits IBM Quantum Learning · official learning module · checked 2026-08-02
- Quantum information IBM Quantum Learning · official learning module · checked 2026-08-02
- The qubit in quantum computing Microsoft Learn · official documentation · checked 2026-08-02
Content review: Reviewed on 2026-08-02.