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.

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.

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)

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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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 misconceptionThe glowing nodes are a single shot.

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

Limits

Typical misconceptionA revival proves a perfect memory device.

Better mental modelIdeal 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

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.
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.
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.