QC Study Notes

Dynamic GHZ via Qubit Reuse

3 min read

Dynamic GHZ via Qubit Reuse

qc/entanglement qc/dynamic

A GHZ state can be built with far less connectivity and depth than the naive fan-out approach (see GHZ States) if you’re willing to use mid-circuit measurement and feedforward — see Dynamic Circuits. This construction follows the paper Efficient Long-Range Entanglement Using Dynamic Circuits (PRX Quantum, 2024).

The core idea, step by step

  1. Hadamards on data qubits — turn each “even” qubit () into : independent random coins.
  2. First CNOT layer — CX(2,1), CX(4,3), … — creates a Bell pair between each even qubit and the odd qubit just before it, forming a “bridge.”
  3. Second CNOT layer — CX(0,1), CX(2,3), … — folds each even qubit into the bridge and onto the next edge qubit. After this, the bridge qubit holds the parity of its two neighboring coins.
  4. Measure the bridge, then correct — measuring the bridge qubit reveals whether its two neighboring coins agreed or disagreed. Reset the bridge, and only if they disagreed, flip the next pair to force agreement. This is the feedforward step: a random local outcome gets locally corrected so the overall state stays coherent, without ever needing a direct physical connection between far-apart qubits.
  5. Repeat down the chain — each measured connector qubit only needs to correct the next pair, not every qubit downstream — corrections propagate link by link rather than needing global knowledge.
  6. Boundary qubit — the very last qubit in the chain has no downstream pair to correct, so it’s just measured and reset with no feedforward.
  7. Final CNOT layer — same pattern as step 3 — brings the last pair of qubits into full agreement, producing the final GHZ signature: every run reads out as all-0s or all-1s, roughly 50/50.
def _apply_dynamic_section(qc, qr, cr, num_qubits) -> None:
    last = num_qubits - 1
    for i in range(1, last - 1, 2):
        _measure_reset_and_correct(qc, qr, cr, meas_qubit=i, x_targets=[i + 1, i + 2])
    _measure_and_reset_only(qc, qr, cr, qubit=last)

Why this matters practically: the naive GHZ fan-out (CX(0,1), CX(0,2), ..., CX(0,n-1)) needs qubit 0 directly connected to every other qubit — physically unrealistic on real hardware topologies like heavy-hex. The dynamic version trades that impossible connectivity requirement for classical feedforward, at the cost of needing hardware that supports mid-circuit measurement (Eagle/Heron-class devices generally do; check for the if_else operation in a backend’s supported operations).

Self-Check

  • Could you walk someone through the 7-step construction, in your own words, without looking?
  • Why does the “measure the bridge, then correct” step only need to fix the next pair, not every qubit downstream?
  • What connectivity requirement does this construction trade away, and what does it trade it for?

Graph View

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