---
title: "Perfect Quantum State Transfer Interactive Lab | Sandbox Physics"
description: "Scatter here. Reassemble there. Launch one excitation into a chain of fifteen sites. Shape the couplings and watch its probability gather at the mirror site. Break the energy alignment to see the arrival fail."
canonical_url: "https://sandboxphysics.com/labs/perfect-quantum-state-transfer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/perfect-quantum-state-transfer/"
last_modified: "2026-10-18"
---

QUANTUM NETWORKS / DESIGNED COUPLINGS

# Perfect Quantum State Transfer

Launch one excitation into a chain of fifteen sites. Shape the couplings and watch its probability gather at the mirror site. Break the energy alignment to see the arrival fail.

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

Ideal single-excitation XY chain with static nearest-neighbor couplings. The target readout is excitation probability, not arbitrary-qubit fidelity. No dissipation, readout error or hardware is modeled; this is direct propagation, not quantum teleportation.

15 SITES · ONE EXCITATION

AT THE MIRROR SITE

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MODEL PROBABILITY

Height and glow: square root of site probability. Gold: launch. Cyan ring: mirror target. Links: actual relative couplings. Nodes are fixed sites, not moving particles.

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EDIT THE INITIAL STATE

Choose a launch site. Its mirror is outlined in cyan.

THE PICTURE IS AN EXPECTATION. NOW TAKE A SHOT.

## Read the whole pattern.

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Each shot starts fresh, evolves to the selected time and reads every site once. Ideal projective readout; all bit strings are kept. The fixed simulation seed makes repeated runs reproducible.

Recorded target fraction

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95% Wilson interval

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Model excitation count

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Model odd–even contrast

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01 / TIMING

### One arrival, precisely timed.

$Jt/\hbar$

02 / COUNTS

### Where did the excitation arrive?

Gold: measured frequency. Cyan: model probability. Linear vertical scale; no rejected shots.

Site number

03 / THE FULL EVOLUTION

### The route to a mirror image.

Model site excitation over time. Horizontal: site; vertical: time. Fixed square-root color scale, from zero to one. Gold line: selected time.

04 / RAW SHOTS

### One row. One complete readout.

Latest 20 independent preparations. Filled: excited. Empty: ground. Rows are not a time sequence. CSV contains every shot.

05 / LOOK UNDER THE HOOD

### Shape the links. Align the energies.

Cyan bars: actual neighboring coupling. Gold dots: fixed site energy error, with a separate scale.

Read the raw ledger

| Trial | Sites, first to last | Target |
| --- | --- | --- |

Model, units & sources

$$
\frac{H}{J}=\sum_{j=1}^{14}c_j(|j\rangle\langle j+1|+\mathrm{h.c.})+\frac WJ\sum_{j=1}^{15}\epsilon_j|j\rangle\langle j|
$$

$$
c_j=1-\lambda+\lambda\sqrt{\frac{j(15-j)}{56}},\qquad \tau=Jt/\hbar
$$

Fifteen open sites; one excitation. The designed chain has maximum coupling J. The error profile is a fixed sine sequence on sites, scaled by W. The reference removes coupling design but keeps the same error profile and launch site.

$$
\tau_* = \frac{\pi\sqrt{56}}2,\qquad P_{15}(\tau)=\sin^{28}\!\left(\frac{\tau}{\sqrt{56}}\right)
$$

The last expression applies to an ideal designed chain launched at site one. At the mirror time every basis site is reflected. An arbitrary qubit also needs the known transfer phase accounted for; the displayed arrival probability does not measure that phase. No dynamically switched gates, teleportation, disorder averaging or environment is simulated.

[Christandl et al. — Perfect state transfer in quantum spin networks (2004)](https://arxiv.org/html/quant-ph/0309131)

$$
I=\frac{2}{N}\sum_{j=1}^N(-1)^{j-1}\langle n_j\rangle
$$

The odd–even contrast uses site one with positive sign. Model time is dimensionless. Evolution uses exponential action without renormalization; drift is reported below. Sampling intervals quantify only ideal finite-shot uncertainty.

$E_0=J$

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Physics tutorial

## Perfect Quantum State Transfer

**Background** Launch one excitation into a chain of fifteen sites. Shape the couplings and watch its probability gather at the mirror site. Break the energy alignment to see the arrival fail.

**Why it matters** A return needs an explanation. Change one cause and keep the observation time fixed.

Start with the essentials

- **Focus question**: Scatter here. Reassemble there.
- **One-sentence intuition**: Engineered couplings synchronize the single-excitation spectrum so the chain performs a mirror inversion.

### Core mathematical model

#### Evolve the full state

$$
|\psi(t)\rangle=e^{-iHt/\hbar}|\psi(0)\rangle
$$

Probabilities, correlations and sampled records all come from this same state.

#### A measurable target

$$
P_{\mathrm{target}}=|\langle N+1-j_0|\psi(t)\rangle|^2
$$

One excitation arriving at the mirror site. This readout alone does not test qubit phase.

### Common difficulties

#### A model picture

**Typical misconception** The glowing nodes are a single shot.

**Better mental model** The scene shows site expectations. A shot is one full binary row in the ledger.

#### Limits

**Typical misconception** A revival proves a perfect memory device.

**Better mental model** Ideal single-excitation XY chain with static nearest-neighbor couplings. The target readout is excitation probability, not arbitrary-qubit fidelity. No dissipation, readout error or hardware is modeled; this is direct propagation, not quantum teleportation.

### Run the experiment

1. 01

   #### Compare

   Choose the first two presets, keeping the same time.

   **What to observe:** At the designed arrival time, the engineered chain refocuses and the uniform chain does not.

2. 02

   #### Break it

   Try the third preset and edit the initial sites.

   **What to observe:** Energy errors spoil phase alignment. Interior sites still mirror in the ideal chain.

3. 03

   #### Read every shot

   Collect snapshots, pin the run and export the complete ledger.

   **What to observe:** Every row is a separate preparation. The recorded target fraction fluctuates around its model probability.

Physical conclusion

Ideal single-excitation XY chain with static nearest-neighbor couplings. The target readout is excitation probability, not arbitrary-qubit fidelity. No dissipation, readout error or hardware is modeled; this is direct propagation, not quantum teleportation.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired 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)