Quantum networks / Designed couplings

Perfect Quantum State Transfer

How can a chain with fixed couplings deliver an excitation to the far end with certainty? Make the couplings strongest in the middle and weaker toward the ends, following a specific square-root profile. The chain's energy levels then become evenly spaced, so every mode comes back into step at the same moment and the excitation lands on the mirror site with certainty. A chain with equal couplings delivers only about 13 percent at that moment, and uneven site energies spoil the arrival.

Put one excitation on a chain of fifteen sites. Shape the couplings and watch its probability spread out and gather again at the mirror-image site. Then knock the site energies out of alignment and see the arrival fail.

The model is an ideal XY chain carrying a single excitation, with fixed nearest-neighbor couplings. The target readout is the probability that the excitation arrives, not the fidelity for an arbitrary qubit state. Dissipation, readout errors and real hardware are not modeled. The excitation travels directly along the chain; this is 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 shows averages; a shot gives one outcome

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

How the excitation travels to its mirror site

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

Each row is one complete readout.

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

05 / THE MECHANISM

How the couplings and site energies are set

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

Show the raw records

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: How can a chain with fixed couplings deliver an excitation to the far end with certainty?
One-sentence intuition: Designed couplings tune the single-excitation spectrum so that every mode comes back into phase at once, and the chain flips the excitation into its mirror image.

Core mathematical model

Evolution of the full state

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

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

The target probability

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

This is the probability that the single excitation arrives at the mirror site. On its own, this readout does not test the phase of a qubit.

Common difficulties

The scene is a model

Typical misconceptionThe glowing nodes show a single measurement.

Better mental modelThe scene shows the expected occupation of each site. A single shot is one complete row of bits in the record.

What the model leaves out

Typical misconceptionA revival proves a perfect memory device.

Better mental modelThe model is an ideal XY chain carrying a single excitation, with fixed nearest-neighbor couplings. The target readout is the probability that the excitation arrives, not the fidelity for an arbitrary qubit state. Dissipation, readout errors and real hardware are not modeled. The excitation travels directly along the chain; this is not quantum teleportation.

Run the experiment

01
Compare
Choose the first two presets in turn, keeping the time the same.
What to observe: At the designed arrival time, the chain with designed couplings refocuses the excitation; the uniform chain does not.
02
Break the effect
Try the third preset and edit the initial sites.
What to observe: Errors in the site energies spoil the phase alignment. In the ideal chain, an excitation that starts on an interior site still arrives at its mirror site.
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Read every shot
Collect snapshots, pin the run and export the complete record.
What to observe: Each row comes from a separate preparation. The measured fraction of target outcomes fluctuates around the model probability.