Stage 2 · Lesson 6 of 17
Quantum gates and circuits
1 · Big question
How do X, H, Z and CX change one- and two-qubit circuits?
- Describe the introductory actions of X, H and Z.
- Identify control and target in CX.
- Step through one- and two-qubit state probabilities.
- Explain why H is not merely a randomness button.
2 · Before we begin
Ideas to bring with you
- A circuit is read from left to right.
- X exchanges the reference states.
3 · New words
Meet the words before we use them
- Hadamard gate (H)
- A gate that can create or analyse equal-probability states.
- phase and Z gate
- Phase is a state feature that affects later combinations; Z is a gate that changes relative phase.
- controlled-X (CX)
- A two-qubit gate that applies X to a target when its control is in the relevant basis state.
4 · Simple explanation
Build one idea at a time
X exchanges |0⟩ and |1⟩. H can prepare an equal-probability state from either reference state, but it does not create a classical random value until measurement.
The Z gate changes relative phase. It can leave immediate 0/1 probabilities unchanged while changing a later gate’s result.
CX has a control qubit and a target qubit. In basis-state examples, the target flips when the control is 1; for other valid quantum states, the same operation can help create entanglement.
Watch it happen
One- and two-qubit gate laboratory
Choose X, H, Z, CX or measurement, then inspect state and probability checkpoints.
Text description of the animation
A keyboard circuit builder shows one or two labelled wires, gate definitions, calculated state checkpoints, exact probabilities and a sampled histogram.
- Open each gate definition before adding it.
- Build H then measurement and predict the probability bars.
- Insert Z after H, compare immediate bars, then add a final H and compare again.
Evidence to calculate or record: Z leaves the immediate H-state 0/1 probabilities equal but changes the final H analysis result.
Predict
Commit to an idea before the reveal
Starting in |0⟩, what probabilities do you predict after H, and what changes if Z is added before an immediate measurement?
Choose a prediction to enable the experiment.
Try it
Define, build, predict
Open each gate definition before adding it.
Make and lock a prediction first.
Built-in circuit lab
Build from left to right
Choose the number of qubits, add instructions with the buttons, then predict before you run. Every drag action has a keyboard button alternative.
- No instructions yet.
Circuit results
Simulator| Result | Exact chance | Count | Sample percent |
|---|---|---|---|
| 0 | 100% | — | — |
Run the simulator to create a text summary of the chart.
Bit-order legend
Pi Leo labels wires q0, q1 and q2 from top to bottom. In displayed result strings, the highest-numbered bit is written on the left, so a two-qubit result is shown as q1q0. This matches the convention used in the Pi Leo simulator and common Qiskit count strings.
Gate definitions
- X
- Exchanges the |0⟩ and |1⟩ reference states in these examples.
- H
- Changes the state direction and can create equal 0/1 measurement probabilities from |0⟩.
- Z
- Changes relative phase. Its effect may appear only after a later analysing gate.
- CX
- Applies X to a target when its control is in the |1⟩ reference state.
- Measurement
- Produces a classical bit result.
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.
H produces equal immediate probabilities from |0⟩. Z changes phase without changing those bars; a following H converts the phase change into a different final probability.
9 · Explain the result
Connect the evidence to the idea
Gates transform the whole quantum state. Measurement samples the resulting probabilities, so randomness belongs to the preparation-plus-measurement experiment, not to H alone.
10 · Model and limitation
Useful model, honest boundary
The calculated checkpoint view connects each gate to exact state probabilities.
State labels and arrows are mathematical representations; the display does not show the physical pulses used by a particular processor.
11 · Common mix-ups
Careful wording prevents big mistakes
H is a classical randomness button.
H prepares a quantum state; random classical output appears only when that state is measured.
Z does nothing when probability bars stay the same.
Z can change relative phase and therefore later interference.
CX always copies a qubit.
CX is a reversible two-qubit operation and cannot generally copy an unknown state.
12 · Real quantum-computing connection
Where this appears in circuit work
X, H, Z and CX are standard circuit-level instructions, though hardware may implement or translate them using different native operations.
13 · Show me moreOptional deeper explanation
Show me more
For |+⟩=(|0⟩+|1⟩)/√2, Z produces |−⟩=(|0⟩−|1⟩)/√2. A later H maps these phase-distinct states to different reference states.
Try this
Explain the deeper idea in your own words, including one limitation.
14 · Quick summary
Keep these ideas
- X exchanges reference states.
- H prepares or analyses equal-amplitude states.
- Z changes relative phase.
- CX has a control and target.
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-06-q-01, ql-06-q-03, ql-06-q-04, ql-06-q-05, ql-06-q-08, ql-06-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-06-q-02, ql-06-q-06, ql-06-q-07 and their related lesson explanations about state vectors; normalisation; single-system measurement probabilities.
- Quantum Computation and Quantum InformationCambridge University Press · 2010 · Sections 1.2–1.3 and Chapters 4, 6 and 8 · accessed 2026-08-02
Supports quiz questions ql-06-q-09 and their related lesson explanations about quantum states and circuits; quantum algorithms; teleportation, noise and error correction.
- This model is deliberately limited: State labels and arrows are mathematical representations; the display does not show the physical pulses used by a particular processor.
- Predictions, simulations and physical-hardware evidence are labelled separately.