---
title: "Grover Amplitude Amplification Interactive Lab | Sandbox Physics"
description: "Can one extra round make the answer harder to find? Past the optimal number of rounds, yes."
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language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/grover-amplitude-amplification/"
last_modified: "2026-10-27"
---

Q047 · Mark / reflect / stop

# Grover Amplitude Amplification

**Can one extra round make the answer harder to find?** Past the optimal number of rounds, yes. Each oracle call plus reflection turns the state a fixed step toward the targets. Once it gets there, another round turns it beyond them and the chance of success falls, so knowing when to stop matters.

Pick a small search space, mark the target entries and step through the rounds. Follow the phase oracle and the reflection as probability flows toward the targets and then back out again. Sample the readout and use the measured success rates to choose when to stop.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The Lab simulates the full complex state vector for four to sixty-four entries. The marked set is built into a teaching oracle; nothing is looked up in a real database. Phase offsets in the oracle and the reflection are coherent errors, and classical bit flips act on the final readout. Queries made by the circuit are counted separately from the one extra query that checks each result against the targets. The cost of building the oracle and of real hardware is not included.

Evidence and query counts

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

$\text{—}$

Round with the highest observed success among those sampled

$\text{—}$

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 |
| --- | --- | --- | --- | --- |

Physics tutorial

## Amplification has a stopping point

**Background** The state starts with equal amplitudes on every entry of a small search space. An oracle changes the phase of the marked entries. A second unitary then reflects the amplitudes about the uniform state.

**Why it matters** The algorithm moves probability around; it does not create any. Once the state has turned past the best direction, repeating the same operations can lower the chance of success.

Start with the essentials

- **Focus question**: Which of the tested rounds gave the highest measured success, and how many queries did the scan use?
- **One-sentence intuition**: The oracle by itself does not change any probability; interference in the reflection does. Together the two steps keep rotating the ideal state, so when you stop matters.

### Core mathematical model

#### Uniform preparation

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

The targets are consecutive entries starting at the selected index, wrapping around at the end. The simulator knows them in advance.

#### Phase oracle

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

You enter the phase offset in degrees, and it is converted to radians. This coherent error is different from a random bit flip.

#### Reflection with phase error

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

The overall minus sign in front makes the error-free operator the usual inversion about the mean. With an error, the operator is still unitary.

#### The complete circuit

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

Every complex amplitude is updated step by step. You can view the state right after any oracle or reflection step.

#### Ideal rotation formula

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

This rotation formula holds for ideal phases, before any readout error. It serves as an independent check; the state vector is not computed from it.

#### Measured 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 also checked once against the target set. That extra query is counted separately as a verification query.

### Common difficulties

#### The oracle is given the targets

**Typical misconception** The simulator discovers a secret in a real database.

**Better mental model** The target set is handed to the teaching oracle in advance. Building a real oracle can be expensive, and this model leaves that cost out.

#### A negative amplitude is not a negative chance

**Typical misconception** An arrow pointing the negative way means a negative chance.

**Better mental model** The arrow's direction is the complex phase and its length is the size of the amplitude. Probability is that size squared, so it is never negative.

#### Fast animation is not quantum advantage

**Typical misconception** The browser animation runs fast, so it proves quantum advantage.

**Better mental model** This is a small simulation on a classical computer. The query counts describe this model; they are not measurements on quantum hardware or a complete comparison with classical search.

### Run the experiment

1. 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.

2. 02

   #### Stop at the peak, then overshoot

   Compare Stop near the peak with Go too far.

   **What to observe:** After a few extra rounds, the same targets can become much less likely.

3. 03

   #### Choose a round from the data

   Acquire the full round scan.

   **What to observe:** The measured success fractions fluctuate. The round with the highest observed success describes this finite data set; it is not guaranteed to be the best round.

4. 04

   #### Add errors

   Try an imperfect oracle and readout flips separately.

   **What to observe:** Coherent phase errors change the amplitudes during the evolution. Readout flips corrupt the reported bit strings after the evolution is over.

Physical conclusion

Amplitude amplification works through interference and costs a finite number of queries. Extra rounds past a good stopping point can undo the gain.

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)