A two-qubit Bell-state check on IBM hardware: pre-registered, run once, reported as-is

# A two-qubit Bell-state check on IBM hardware: pre-registered, run once, reported as-is

**QMAI Research · 2026-10-30**

## What we tested
We prepared the two-qubit Bell state Φ⁺ = (|00⟩ + |11⟩)/√2 with a Hadamard and a CNOT. We measured it on one physical qubit pair of IBM's **ibm_marrakesh**, qubits **53 and 54**, through IBM Quantum's free Open plan. IBM's backend configuration, read through its SDK on 2026-10-30, reports the processor family as Heron, revision 2.

We ran three circuits in a single job, one per measurement basis (XX, YY, ZZ), with 1,024 shots each. That makes 3,072 shots in total. The job ran under IBM's free Open plan with no paid spend. IBM reported **3 seconds** of usage for it. Billing was not separately verified.

The pair was chosen by a rule fixed before any run: lowest reported two-qubit error, then lowest summed readout error.

## The rule we committed to before running
The estimator is the raw parity-based score

**F = (1 + E_XX − E_YY + E_ZZ) / 4**, where E = (N_even − N_odd) / N for each basis.

We used no readout mitigation, twirling, dynamical decoupling, post-selection or discarded shots. The confidence bound is a simultaneous 95% Hoeffding bound over the three expectations: ε = √(2·ln(6/0.05)/1024) ≈ 0.0967 per expectation, giving a radius of 3ε/4 ≈ 0.0725 on F.

The pre-registered verdicts were:
- **PASS** only if the lower bound is strictly above 0.5;
- **INCONSISTENT** if the upper bound is below 0;
- **MISS** otherwise;
- invalid or ambiguous data counts as **INVALID**, not as a score.

## Result
| Basis | 00 | 01 | 10 | 11 | E |
|---|---|---|---|---|---|
| XX | 494 | 5 | 60 | 465 | +0.8730 |
| YY | 58 | 444 | 508 | 14 | −0.8594 |
| ZZ | 512 | 7 | 63 | 442 | +0.8633 |

(Bit strings read c1c0, where c0 is the measured logical qubit 0 on physical qubit 53 and c1 is qubit 1 on physical 54.)

**F_raw = 0.89893.** The simultaneous 95% interval is **[0.8264, 0.9714]**. The lower bound is above 0.5, so the verdict is **PASS** under the rule above.

Our analysis and an independent reviewer's recount from the raw counts agree.

## What this does and does not show
**It shows** that our pipeline works end to end on real IBM hardware. That covers:
- circuit design;
- a compiled-circuit equivalence check (the transpiled circuits act only on the chosen pair, with the measured unitary matching the intended one up to global phase);
- a single logged submission;
- a pre-registered verdict.

On this pair, at this time, the raw score cleared the bar.

**It does not show:**
- **Certified entanglement, a device benchmark, or any quantum advantage.** This is one job on one qubit pair at one point in time.
- **A mitigated fidelity.** The score is raw. Readout and state-preparation errors are not corrected and can bias it in either direction. The counts show a visible asymmetry (for example "10" appears 60–63 times where "01" appears 5–7 times in the XX and ZZ bases). That is consistent with uneven readout error between the two qubits, which we did not characterize separately.
- **Systematic error coverage.** The interval covers sampling error under the stated independence assumptions, not systematic measurement bias.

## An earlier attempt that did not run
An earlier submission today stopped on one of our runner's refusal paths. No job id was returned or recorded. We did not keep the exact refusal message, so we do not state its cause. When we recompiled, the device's reported calibration timestamp had changed. With a separate owner approval, we then compiled fresh and submitted immediately, and that run is the one reported here. We report it because it is part of how the result was produced.

## Reproducibility
- **Job id:** dbib37mne4hc73be7r5g.
- **Sealed packet:** sha256 285c1c67…
- **Result file:** sha256 cca599dc…
- **SDK:** qiskit 2.6.1, qiskit-ibm-runtime 0.51.0.
- **Transpile:** seed 20261030, optimization level 1.



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