Case Study: Measuring Teleportation Fidelity on Hardware

Executive Summary

Teleportation is the standard "we have a working quantum computer" demonstration, and it is also one of the easiest results to overstate. Teleporting $|0\rangle$ and recovering $|0\rangle$ proves nothing — a classical wire does that.

This case study builds the demonstration properly: teleport a set of states that spans the Bloch sphere, compute the average fidelity, and compare it against the classical bound of 2/3 that any measure-and-resend strategy can achieve. Then attribute the gap between the measured 0.83 and the ideal 1.0 to specific hardware errors.

Skills applied

• Implementing the three-qubit teleportation circuit with corrections (§9.4).
• Choosing input states that make the demonstration meaningful (§9.11).
• Computing average fidelity over a state ensemble.
• Comparing against the classical fidelity bound (§9.12).

Background

The classical bound

Suppose Alice has an unknown qubit and no entanglement. Her best strategy is to measure it and send Bob the classical outcome; Bob prepares that state. Averaged over all pure input states drawn uniformly from the Bloch sphere, this yields a fidelity of exactly

$$F_{\text{classical}} = \frac{2}{3} \approx 0.667$$

Any protocol beating 2/3 on average over a uniformly sampled ensemble cannot be explained classically. That is the threshold your experiment must clear, and it is why the choice of input states is not a detail.

Why $|0\rangle$ alone proves nothing

A classical strategy measuring in the $Z$ basis teleports $|0\rangle$ and $|1\rangle$ with fidelity 1.0. It fails only on superpositions, dropping to 0.5 on equatorial states. A demonstration restricted to computational-basis inputs therefore has no power to distinguish quantum from classical.

Phase 1: The circuit

from qiskit import QuantumCircuit

def teleport(prep):
    """prep: circuit fragment preparing the state to teleport on qubit 0."""
    qc = QuantumCircuit(3, 3)
    qc.compose(prep, qubits=[0], inplace=True)   # state to send
    qc.barrier()
    qc.h(1); qc.cx(1, 2)                          # shared Bell pair
    qc.barrier()
    qc.cx(0, 1); qc.h(0)                          # Alice's Bell measurement
    qc.measure(0, 0); qc.measure(1, 1)
    qc.barrier()
    with qc.if_test((qc.clbits[1], 1)):           # corrections
        qc.x(2)
    with qc.if_test((qc.clbits[0], 1)):
        qc.z(2)
    return qc

On backends without dynamic circuits, use the deferred version from Chapter 4 — replace the conditionals with CX(1→2) and CZ(0→2) and measure everything at the end. The results are equivalent for this experiment.

Phase 2: Choose a spanning ensemble

Six states — the ± eigenstates of $X$, $Y$, and $Z$ — form a tomographically complete set and are the standard minimal ensemble:

For each, Bob rotates the target state back to $|0\rangle$ before measuring, so fidelity is the probability of reading 0. This trick avoids full tomography and costs one circuit per input state.

Phase 3: Results

8,192 shots per state:

Average over the six: $\bar F = 0.826$.

Two features stand out.

The average clears the bound. $0.826 \gg 0.667$. With shot noise of $\approx 0.004$ per state, the margin is enormous. The demonstration is sound.

Equatorial states do worse than polar ones. $|0\rangle$ and $|1\rangle$ average 0.90; the four equatorial states average 0.79. This is diagnostic, not incidental: polar states are immune to dephasing, equatorial states are maximally exposed to it. The 11-point gap is the dephasing signature, and its presence confirms the experiment is sensitive to exactly the thing a classical strategy would fail on.

Phase 4: Attribute the missing 0.17

Four contributions, estimated from device properties:

The prediction under-explains the loss by about 7 points. Rather than shrug, chase it: the usual culprit is that the idle time — qubit 2 sitting untouched from Bell-pair creation until the corrections — is longer than the nominal circuit duration suggests, because measurement takes 1.2 μs and the classical feed-forward latency adds more. On dynamic-circuit hardware, the conditional block's latency is often the single largest contributor and is not visible in a gate count.

Finding. Teleportation fidelity on current hardware is dominated by readout error and by idle-time dephasing during the classical round trip — not by gate error. Optimizing the gates would buy almost nothing.

Phase 5: Reporting it honestly

Defensible: "Average teleportation fidelity of $0.826 \pm 0.004$ over the six-state ensemble, exceeding the classical bound of 2/3 by more than 30σ. Polar states averaged 0.90 and equatorial states 0.79, consistent with dephasing-limited performance."

Not defensible: "We achieved 91% teleportation fidelity" (cherry-picking $|0\rangle$). "We demonstrated teleportation" without an ensemble or a bound comparison. Any claim of instantaneous transfer.

Discussion Questions

1. Why is 2/3 the classical bound, and why must the ensemble be uniform over the Bloch sphere for the comparison to be meaningful?
2. Equatorial states fared 11 points worse than polar. Explain the physics and say what you would measure to confirm it.
3. The error budget under-predicted the loss. List three candidate explanations and an experiment to discriminate between them.
4. Would this experiment be more or less convincing if run with the deferred-measurement circuit? What changes physically?

Your Turn: Extensions

• Implement both dynamic and deferred versions and compare average fidelity; explain any difference.
• Extend to full process tomography of the teleportation channel and extract the process fidelity.
• Vary an inserted idle delay on qubit 2 and plot fidelity against delay; fit $T_2$.
• Compute the classical bound yourself by simulating a measure-and-resend strategy over 1,000 random Bloch-sphere states.

Key Takeaways

• Teleporting $|0\rangle$ demonstrates nothing; the demonstration requires an ensemble spanning the Bloch sphere.
• The classical fidelity bound for an unknown qubit is 2/3 — clear it, with error bars, or the claim is unsupported.
• Polar-versus-equatorial fidelity asymmetry is a direct dephasing signature and a useful diagnostic.
• On real hardware, teleportation is limited by readout error and idle-time dephasing during classical feed-forward, not by gate error.
• Report the average over the ensemble, never the best single state.