Q047 · Mark / reflect / stop

Grover Amplitude Amplification

Choose a small search space, mark its target entries and move the round selector. Follow the phase oracle and reflection as probability flows toward the targets and back out again. Sample the readout and find a stopping point from the recorded success rates.

Ideal marked probability before readout $\text{—}$

Circuit oracle calls per trial $\text{—}$

MAKE A PREDICTION

Will more oracle calls always make the target more likely?

Orange tiles are the marked targets. Drag the round selector, acquire a round scan and use the measured success rates to choose when to stop.

Try the contrasting cases, then test your prediction with a record.

Inspect an operation; acquisition always runs the complete circuit

Replay shows the model and adds no events.

CHANGE ONE THING

Compare the cases

Oracle–reflection rounds

k

3

Search qubits

n

4

Marked entries

M

1

Drag the round selector or use its slider. Changing rounds retains the scan; changing the oracle starts a new record.

Errors, resources and sampling

First marked index; targets wrap

j_0

0

Oracle phase offset

\epsilon_O

0^{\circ}

Reflection phase offset

\epsilon_D

0^{\circ}

Readout flip probability per bit

p_{\mathrm{read}}

0.000

Trials per setting

N_{\mathrm{shots}}

400

Random seed

s

1996

Full complex state-vector simulation for four to sixty-four entries. The marked set is a known teaching oracle, not a real database lookup. Oracle and reflection phase offsets are coherent errors; classical bit flips affect the final readout. Circuit queries and one extra target-verification query per result are counted separately. Oracle construction and hardware costs are not priced.

Evidence and query budget

SIMULATED MEASUREMENT RECORD

Find the stopping point

Acquire a record to begin.

Colored bars and points come from records; thin gray marks are the model with the configured errors. Whiskers are pointwise 95% Wilson intervals for output or success fractions. They are not phase confidence intervals, simultaneous bands or guarantees after optional stopping.

Recorded success, 95% Wilson range

\text{—}

Best observed round among sampled settings

\text{—}

All circuit / verification queries

\text{—}

The best observed round is a descriptive choice from this finite scan, not a certified optimum. Every output is checked against the known teaching oracle; that adds one verification query. No real-data oracle construction or hardware advantage is demonstrated.

Recomputed from the exported trials
Rounds	Trials	Successes	Circuit / verification queries

Latest 12 trials; CSV retains all trials
Trial	Setting	Reported bits	Integer	Circuit queries

Start with the essentials

Focus question: Which tested round has the greatest recorded success, and what queries did the scan consume?
One-sentence intuition: The oracle alone leaves probabilities unchanged. Interference in the reflection changes their magnitudes. The pair rotates the ideal state repeatedly, so stopping matters.

Core mathematical model

Uniform preparation

|s\rangle=\frac{1}{\sqrt N}\sum_{x=0}^{N-1}|x\rangle,\quad N=2^n

Targets are a cyclic consecutive set starting at the selected index. Their values are already known to the simulator.

Phase oracle

O_{\epsilon_O}|x\rangle=\begin{cases}e^{i(\pi+\epsilon_O)}|x\rangle&x\in\mathcal M\\|x\rangle&x\notin\mathcal M\end{cases}

The phase offset is entered in degrees and converted to radians. A coherent error is not a probabilistic bit flip.

Reflection with phase error

D_{\epsilon_D}=-I+\bigl(1-e^{i(\pi+\epsilon_D)}\bigr)|s\rangle\langle s|

The leading global minus sign makes the zero-error operator the usual inversion about the mean. The imperfect operator remains unitary.

The complete circuit

|\psi_k\rangle=(D_{\epsilon_D}O_{\epsilon_O})^k|s\rangle

All complex components are updated explicitly. The operation slider can show the state immediately after an oracle or reflection.

Independent ideal reference

P_{\mathcal M}(k)=\sin^2((2k+1)\alpha),\quad\sin^2\alpha=M/N

This rotation formula applies to ideal phases before readout errors. It is an independent benchmark, not the rule used to generate the state vector.

Recorded success and circuit cost

\widehat P_{\mathcal M}(k)=N_{\mathrm{success},k}/N_{\mathrm{shots},k},\quad C_O=kN_{\mathrm{shots},k}

Each reported output is additionally checked once against the target set. The separate verification counter includes that extra query.

Common difficulties

An oracle is a resource

Typical misconceptionThe simulator discovers a secret in a real database.

Better mental modelThe target set is supplied to the teaching oracle. Constructing a real oracle can be costly and is outside this model.

Amplitude versus probability

Typical misconceptionA negative arrow represents a negative chance.

Better mental modelArrow direction is complex phase and length is amplitude magnitude. Probability is the squared magnitude and remains nonnegative.

No stopwatch proof

Typical misconceptionFast browser animation proves quantum advantage.

Better mental modelThis is a small classical simulation. The query ledger describes a declared model, not measured quantum hardware or a complete classical comparison.

Run the experiment

01
Watch phase before probability
Choose Before amplification, set one round and inspect its oracle step.
What to observe: The marked amplitude changes sign but its probability does not yet increase.
02
Find and pass the peak
Compare Stop near the peak with Go too far.
What to observe: The same target set can become much less likely after extra rounds.
03
Use records to choose a round
Acquire the full round scan.
What to observe: Success fractions fluctuate. The highest observed setting is a descriptive result from the finite record, not a guaranteed optimum.
04
Introduce a mismatch
Try an imperfect oracle and readout flips separately.
What to observe: Coherent phase errors change the evolved amplitudes. Readout flips distort the reported bit strings after evolution.