Stage 1 · Lesson 2 of 17
Introductory qubit models
1 · Big question
How can a simple direction model help us predict qubit measurements?
- Read |0⟩ and |1⟩ as labels for two reference states.
- Use a state-direction model to predict 0 and 1 probabilities.
- Recognise the |+⟩ and |−⟩ presets as states with equal immediate 0/1 probabilities.
- Explain that relative phase can affect later instructions even when immediate probabilities match.
2 · Before we begin
Ideas to bring with you
- A qubit measurement can record 0 or 1.
- A model can be useful without being a literal picture.
3 · New words
Meet the words before we use them
- ket and reference-state notation
- A ket is a state label written with a vertical bar and angle bracket; the reference-state label |0⟩ is read ‘ket zero’.
- probability
- A number from 0 to 1 that describes the predicted chance of an outcome.
- phase
- A feature of a quantum state that can change how later instructions combine possible outcomes.
4 · Simple explanation
Build one idea at a time
A reference state is a standard label used for preparations and calculations. The labels |0⟩ and |1⟩ are read ‘ket zero’ and ‘ket one’ and name two such reference states.
A prediction arrow represents a one-qubit mathematical model. At the |0⟩ end it predicts 0 with probability 1; at the |1⟩ end it predicts 1 with probability 1.
The |+⟩ and |−⟩ presets both predict equal immediate chances of 0 and 1. They differ in relative phase. A later Hadamard (H) instruction—an operation that analyses this difference—can therefore give different results.
Watch it happen
Controllable state-direction model
Choose |0⟩, |1⟩, |+⟩ or |−⟩, rotate the model, and compare the calculated probability bars.
Text description of the animation
A mathematical arrow has presets at |0⟩, |1⟩, |+⟩ and |−⟩. Probability bars show 1/0, 0/1, one-half/one-half and one-half/one-half respectively, while a phase label distinguishes plus from minus.
- Choose each preset in turn.
- Record P(0), P(1) and the phase label.
- Apply a later H in the plus and minus cases and compare the predicted result.
Evidence to calculate or record: |+⟩ and |−⟩ share immediate 0/1 probabilities but a later H maps them to different reference-state predictions.
Predict
Commit to an idea before the reveal
Which presets will show equal 0 and 1 probability bars, and will that make the states identical?
Choose a prediction to enable the experiment.
Try it
Match state presets to predictions
Choose each preset in turn.
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 probability bars for |+⟩ and |−⟩ match before an analysing gate, while their phase labels and later H results differ.
9 · Explain the result
Connect the evidence to the idea
Immediate 0/1 probabilities do not contain every state feature. Relative phase becomes observable through a suitable later instruction and measurement.
10 · Model and limitation
Useful model, honest boundary
The arrow model organises one-qubit states and connects its direction to calculated measurement probabilities.
The arrow is mathematical. It is not a tiny physical arrow, compass needle or spinning object inside a processor.
11 · Common mix-ups
Careful wording prevents big mistakes
The state arrow is a real needle inside the qubit.
The arrow is a representation of calculated state information.
Equal 0/1 probabilities mean |+⟩ and |−⟩ are identical.
Their relative phases differ and can affect a later gate sequence.
An amplitude is the same thing as a probability.
Probabilities are calculated from amplitudes; the two ideas are not interchangeable.
12 · Real quantum-computing connection
Where this appears in circuit work
Circuit diagrams commonly use |0⟩ as the starting reference state and use H and Z to create or analyse states with different directions and phases.
13 · Show me moreOptional deeper explanation
Show me more
A more precise model uses two calculation components called probability amplitudes. Their squared magnitudes give the two measurement probabilities, while their relative phase helps determine later interference.
Try this
Explain the deeper idea in your own words, including one limitation.
14 · Quick summary
Keep these ideas
- |0⟩ is read ‘ket zero’ and |1⟩ ‘ket one’.
- The arrow is a mathematical model.
- |+⟩ and |−⟩ have equal immediate 0/1 probabilities.
- Phase can change what later gates produce.
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 notes4 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.
- The qubit in quantum computingMicrosoft Learn · Qubit state, visualisation and measurement · accessed 2026-08-02
Supports quiz questions ql-02-q-02, ql-02-q-06, ql-02-q-08 and their related lesson explanations about qubit state vectors; measurement; Bloch-sphere representation; single-qubit operations.
- Quantum informationIBM Quantum Learning · Single systems · accessed 2026-08-15
Supports quiz questions ql-02-q-01, ql-02-q-03, ql-02-q-04, ql-02-q-07, ql-02-q-10 and their related lesson explanations about state vectors; normalisation; single-system measurement probabilities.
- CircuitsIBM Quantum Learning · Quantum circuits — Circuits · accessed 2026-08-15
Supports quiz questions ql-02-q-09 and their related lesson explanations about classical and quantum circuit models; gates and wires; standard-basis measurement.
- Quantum Computing for EveryoneMIT Press · 2019 · Parts II–III · accessed 2026-08-15
Supports quiz questions ql-02-q-05 and their related lesson explanations about school-accessible circuit explanations; measurement, entanglement and algorithms; limits of everyday analogies.
- This model is deliberately limited: The arrow is mathematical. It is not a tiny physical arrow, compass needle or spinning object inside a processor.
- Predictions, simulations and physical-hardware evidence are labelled separately.