Algorithm guide · foundations
Quantum Zeno effect
Frequent measurements inhibit a rotation. Compare a watched qubit with an unmeasured reference under the same drive.
Circuit size: 2 qubits
RY rotations · Projective measurement · Conditional versus total probability
Review quantum fundamentals →Notation used in this lesson
- |ψ⟩ and ⟨ψ|
- A state vector (ket) and its conjugate transpose (bra). ⟨a|b⟩ is their inner product.
- α, β and i
- α and β are complex amplitudes; i²=−1. A probability is the squared magnitude |α|².
- XOR, · and ⊗
- XOR adds bits modulo two. In s·x, multiply corresponding bits and XOR them. ⊗ combines registers or operations on different wires.
- U†, Σ and Tr
- U† is the adjoint (inverse for a unitary). Σ sums terms. Tr is the matrix trace, the sum of diagonal entries.
- Probability, shots and angles
- Ideal probabilities describe the model; measured shot frequencies fluctuate. Angles inside trigonometric expressions are in radians unless explicitly labeled turns.
1. The problem
Compare two qubits under the same total π rotation. Measure one repeatedly during its evolution and the other only at the end. Quantify how monitoring suppresses transitions.
2. How it works
A small rotation initially gives only a small probability of leaving |0⟩. Projecting after every slice interrupts the coherent buildup that would otherwise rotate all the way to |1⟩. Measurement is physical state change here, not a passive look at an unchanged state.
Slice the drive
Apply RY(π/N) to watched and reference qubits in each of N steps.
Measure watched q0
After every slice, record its actual projected result in a new classical bit. Do not reset it.
Measure reference q1 at the end
Its uninterrupted rotations add to RY(π).
Compare events
Distinguish staying zero at every check from merely ending zero after possible departures and returns.
3. The mathematics, one idea at a time
Each successful projection returns the watched state to |0⟩, so multiply the same conditional survival probability N times.
This includes histories that left zero and returned. It follows from the two-state transition recurrence after measurements.
The reference completes the full coherent π rotation.
Work this through by hand
4. Two checks: the first nontrivial suppression
The circuit has two physical qubits for any N. N=1 is the supported baseline with no intermediate suppression; N=2 is the smallest useful comparison.
Choose two measurement steps. Each slice is RY(π/2).
| Stage | Calculation or state | What it means |
|---|---|---|
| First watched check | P(0)=P(1)=1/2 | The result is projected and stored in c0. |
| Second watched check | Each transition has probability 1/2 | All four two-check histories have probability 1/4. |
| Compare | P(all-zero)=1/4; P(final 0)=1/2; P(reference 0)=0 | Final zero also includes the history 1→0. |
Expected outcomes for this example
These are ideal predictions for the stated configuration, not a result already measured by your browser. Rows describing alternative modes or combined events are labeled explicitly.
| Outcome or event | Ideal probability | Interpretation |
|---|---|---|
| 100,101,110,111 | 25% each ideally | c2 is reference=1; c1 is the last watched check and c0 the first. |
5. Load the worked example in Composer
The worked example can use smaller or simpler inputs than the gallery defaults. The button below loads the exact circuit used in this lesson.
- Measurement steps (N)
- 2
- Save any current circuit you want to keep, then open the worked example. Loading it does not execute a paid hardware job.
- Trace the preparation, main operations and measurements against the worked table. Check the gate editor for controls, targets and angles.
- Start with an ideal noiseless simulator compatible with the circuit. Mid-circuit measurement, reset and feed-forward require support from the chosen backend.
- Inspect the live preview. For a recorded experiment, open Execute, set the available backend options and shots, and Run. Sign in when prompted to retain it in My Jobs.
- Read the output as described below before changing one setting and repeating. The guide describes the original example; edits can change the answer.
6. Read the output correctly
The Classical bits summary displays overlapping events. All-zero history is a subset of final watched zero, so these bars do not sum to 100%.
At N=2, c1c0=00 and 01 both end with watched zero. The final qubit chart cannot tell which history occurred.
A bit label is a register value; its bar height is a probability or count. For S shots and observed count k, the sampled frequency is k/S. A typical sampling scale for probability p is √[p(1−p)/S]; this is not a strict error bound. Finite samples do not have to match the ideal percentages exactly.
7. Common mistakes & limits
- The circuit does not reset, discard failed histories or postselect survivors.
- Perfect freezing is a limiting idealization, not the finite-N result.
8. Check your understanding
Why is final watched zero more likely than an all-zero history?
Reveal the reasoning
A history may first produce 1 and later return to 0. With two checks that extra history contributes another 1/4.
9. Further study
The lesson’s register mappings, supported inputs and loadable example follow Quantket’s implemented circuit. A reference may use different wire names, bit order or resource conventions.
Gallery reference: defaults, definitions & implementation notes
Inputs & default settings
Increase measurement frequency at fixed total rotation, then compare history survival and final populations in Classical bits.
These are the gallery defaults. The worked example above may override them. Configure the card to change the inputs before loading its editable circuit.
- Measurement steps (N)
- Default: 8 · Whole number from 1 to 16.
Implementation notes
Both qubits start at |0> and receive N rotations RY(pi/N), for total angle pi. Watched q0 is genuinely measured in Z after every slice; reference q1 is measured only at the end. Each projection interrupts coherence without resetting or recovering the qubit. All attempts are retained. History survival means every watched result was 0: probability cos(pi/(2N))^(2N). Final watched P(0)=[1+cos(pi/N)^N]/2 also includes paths that left and returned. At N=2 these are 25% and 50%, respectively. The unmeasured reference completes the rotation to |1>. Finite N suppresses transitions; perfect freezing is the ideal large-N limit. Registers c0..c(N-1) retain the checks, earliest first by index; cN is the reference. Displayed cN..c0 strings place the reference first and the latest check next. Classical bits shows overlapping per-shot event frequencies, not exclusive bins. Qubits shows only the final state, not whether every check survived. This is an ideal projective-measurement model; circuit edits change its predictions.
Default example: predictions & circuit facts
These values describe the unedited default setup. They are predictions and circuit properties, not measured results from your saved execution.
- Step angle
- pi/8 rad
- Ideal all-zero history
- 73.31%
- Ideal watched final 0
- 76.54%
- Ideal reference final 0
- 0.00%
Read your saved result
Check the submitted circuit, measured register and execution settings in Run details. Exact results give computed probabilities; sampled results show frequencies from the recorded shots. Edited circuits and noise can change the expected outcome. The algorithm label records the circuit's origin; it does not certify an edited circuit or a measured result.