Stage 2 · Lesson 7 of 17
Why instruction order matters
1 · Big question
Can the same gates perform different computations when their order changes?
- Explain why circuit order is part of a computation.
- Predict H→Z→H on |0⟩.
- Predict Z→H→H on |0⟩.
- Use state checkpoints to justify different outcomes.
2 · Before we begin
Ideas to bring with you
- H can prepare and analyse phase.
- Z changes relative phase.
3 · New words
Meet the words before we use them
- instruction order
- The left-to-right sequence in which circuit operations act.
- checkpoint
- A calculated state or probability view after a circuit step.
- analyse
- Use a later operation and measurement to reveal a state feature.
4 · Simple explanation
Build one idea at a time
Circuit instructions act in sequence. A later gate receives the state produced by every gate before it.
Starting from |0⟩, H→Z→H produces |1⟩. The first H creates |+⟩, Z changes it to |−⟩, and the last H maps |−⟩ to |1⟩.
Starting from |0⟩, Z→H→H produces |0⟩. Z leaves |0⟩ unchanged and the two H gates cancel in this case.
Watch it happen
Same gates, different order
Rearrange H, Z and H, predict first, then reveal calculated state checkpoints.
Text description of the animation
Two circuits start at |0⟩. H-Z-H ends at |1⟩; Z-H-H ends at |0⟩. A table names the state after each gate.
- Predict both required circuits.
- Step through Circuit A and record each state label.
- Step through Circuit B, then compare and justify the two required orders.
Evidence to calculate or record: Circuit A is certain 1 and Circuit B certain 0 in the ideal model, supported by their state checkpoints.
Predict
Commit to an idea before the reveal
Circuit A and Circuit B contain the same gate names. Must they have the same ideal measurement result?
Choose a prediction to enable the experiment.
Try it
Compare and justify
Predict both required circuits.
Make and lock a prediction first.
8 · Observe
What did the result actually show?
Look at the displayed values before reading the explanation. Record a pattern, an exception or something that changed.
The two final probability panels differ even though the multiset of gate names is the same.
9 · Explain the result
Connect the evidence to the idea
Quantum operations combine in order. Swapping H and Z can change the state supplied to later operations, so their order cannot be ignored.
10 · Model and limitation
Useful model, honest boundary
Checkpoints expose the reason for the final result rather than showing only an answer.
A recipe analogy helps with sequence, but quantum gates transform amplitudes and phase rather than food ingredients. Where this analogy stops: recipes do not model quantum interference.
11 · Common mix-ups
Careful wording prevents big mistakes
The same gate names always mean the same computation.
Order can change the state entering each later gate.
Z has no effect because it leaves |0⟩ unchanged.
Its effect depends on the state at the moment it acts.
Only the final gate matters.
Every earlier gate helps determine the state supplied to the final gate.
12 · Real quantum-computing connection
Where this appears in circuit work
Circuit optimisation and debugging both preserve or test the ordered mathematical operation, not merely the list of gate names.
13 · Show me moreOptional deeper explanation
Show me more
In matrix notation, the rightmost operation acts first on a state vector. This course keeps the learner-facing circuit order left to right and uses checkpoints to avoid notation confusion.
Try this
Explain the deeper idea in your own words, including one limitation.
14 · Quick summary
Keep these ideas
- Gate order is part of the computation.
- HZH|0⟩ gives |1⟩.
- ZHH|0⟩ gives |0⟩.
- State checkpoints explain the difference.
Ten-question quiz
Check the ideas—not decorative details
Feedback appears after submission. Retry whenever you like; 8/10 or above means “Topic understood”.
Sources and accuracy notes3 checked references · reviewed 2026-08-15
These records identify the claim each source supports. External documentation can change; dated platform claims were checked on the shown access date.
- CircuitsIBM Quantum Learning · Quantum circuits — Circuits · accessed 2026-08-15
Supports quiz questions ql-07-q-01, ql-07-q-04, ql-07-q-07, ql-07-q-08, ql-07-q-10 and their related lesson explanations about classical and quantum circuit models; gates and wires; standard-basis measurement.
- Quantum informationIBM Quantum Learning · Single systems · accessed 2026-08-15
Supports quiz questions ql-07-q-02, ql-07-q-03, ql-07-q-06 and their related lesson explanations about state vectors; normalisation; single-system measurement probabilities.
- Quantum Computer Science: An IntroductionCambridge University Press · 2007 · Chapters 1–4 · accessed 2026-08-15
Supports quiz questions ql-07-q-05, ql-07-q-09 and their related lesson explanations about qubits, gates and circuits; entanglement and teleportation; introductory quantum algorithms.
- This model is deliberately limited: A recipe analogy helps with sequence, but quantum gates transform amplitudes and phase rather than food ingredients. Where this analogy stops: recipes do not model quantum interference.
- Predictions, simulations and physical-hardware evidence are labelled separately.