Authors
Remo Pulcini (corresponding) — Independent Research / Quantum Horizon (Italy)
Technical support: AI assistant (methodology & code implementation guidance)

Abstract
We study the differential robustness of global correlation and global coherence in GHZ states under depolarizing noise using folding-based zero-noise extrapolation (ZNE), bootstrap uncertainty quantification, and model selection via Akaike Information Criterion (AIC). Two global observables are compared: ⟨Z⊗n⟩ (correlation) and ⟨X⊗n⟩ (coherence). We introduce a practical metric, the Quantum Fragility Index (QFI), defined as the ratio between the noise breakdown thresholds of coherence and correlation. Numerical simulations (Qiskit Aer) up to n=4 qubits show that global coherence degrades earlier than correlation, with a clear separation in breakdown thresholds. We provide a breakdown map across noise strengths and folding depths, highlighting stability regions where ZNE remains well-behaved and regimes where it becomes unreliable.

1. Introduction
Near-term (NISQ) quantum devices are limited by noise. Error mitigation techniques such as ZNE can partially recover noise-free estimates without full error correction. However, different physical features may degrade at different rates: phase-sensitive coherence may vanish earlier than correlation signatures that can persist as classical mixtures. Motivated by this, we compare global correlation and global coherence observables for GHZ states and propose a compact ratio metric that captures relative fragility.

2. Background
2.1 GHZ states

∣GHZn⟩=∣0⟩⊗n+∣1⟩⊗n2|GHZ_n\rangle=\frac{|0\rangle^{\otimes n}+|1\rangle^{\otimes n}}{\sqrt{2}}∣GHZn⟩=2∣0⟩⊗n+∣1⟩⊗n

2.2 Observables
We measure:

• correlation: OZ=Z⊗nO_Z = Z^{\otimes n}OZ=Z⊗n

• coherence: OX=X⊗nO_X = X^{\otimes n}OX=X⊗n
  Operationally, OZO_ZOZ is estimated by measuring in the computational basis; OXO_XOX is estimated by applying Hadamards on all qubits and measuring in Z.

2.3 Noise amplification by folding
We amplify effective noise by increasing circuit depth using folding factors λ∈{1,3,5,… }\lambda \in \{1,3,5,\dots\}λ∈{1,3,5,…}, exploiting self-inverses (H and CX) in the GHZ construction.

2.4 Zero-Noise Extrapolation
For each noise strength p, we compute expectation values at multiple λ\lambdaλ and fit polynomials E(λ)E(\lambda)E(λ) to extrapolate E(0)E(0)E(0). We compare linear and quadratic models; model choice is guided by AIC. Uncertainty is estimated via bootstrap resampling, yielding confidence intervals (CI95).

3. Methods
3.1 Simulation framework
Experiments were conducted using Qiskit Aer with depolarizing noise applied to single-qubit H and two-qubit CX gates (with CX noise scaled higher than H). Shots and bootstrap counts were chosen to balance stability and runtime.

3.2 Breakdown definition
For an observable OOO, define a breakdown threshold pbreak(O)p_{break}(O)pbreak(O) as the smallest noise strength p where instability arises, e.g.:

• CI95 includes unphysical regions or becomes excessively wide (wide CI)

• extrapolated value becomes out-of-range

• or CI95 includes 0 while the low-noise regime remains clearly non-zero (practical collapse)

3.3 Quantum Fragility Index (QFI)

QFI(n)=pbreak(OX)pbreak(OZ)QFI(n)=\frac{p_{break}(O_X)}{p_{break}(O_Z)}QFI(n)=pbreak(OZ)pbreak(OX)

Interpretation: QFI < 1 implies coherence breaks earlier than correlation.

4. Results
4.1 Baseline tests
Single-qubit and small-n runs validate the workflow and show consistent ZNE behavior at low noise.

4.2 GHZ 3-qubit
Bootstrap-ZNE on ⟨ZZZ⟩ showed stable confidence intervals and consistent model selection; coherence-sensitive behavior begins to show higher fragility relative to correlation as noise increases.

4.3 GHZ 4-qubit: ZNE breakdown mapping
We mapped ZNE behavior across a noise grid and folding depths (extended folding set). Across the explored range, ⟨XXXX⟩ decays towards zero faster than ⟨ZZZZ⟩, consistent with the intuition that phase coherence is more sensitive to depolarizing processes than classical correlation.

Representative outputs showed (example ranges from our runs):

• at moderate noise, ⟨XXXX⟩ approaches 0 while ⟨ZZZZ⟩ remains >0

• at higher noise, both approach ~0, but coherence collapses earlier
  AIC frequently favored the linear model in large portions of the parameter grid; some regimes showed improved fit for quadratic models, consistent with non-linear dependence on noise amplification.

4.4 QFI estimate (n=4)
From the breakdown map, we observed approximate thresholds:

pbreak(XXXX)≈0.28,pbreak(ZZZZ)≈0.34p_{break}(XXXX)\approx 0.28,\quad p_{break}(ZZZZ)\approx 0.34pbreak(XXXX)≈0.28,pbreak(ZZZZ)≈0.34

leading to:

QFI(4)≈0.82QFI(4)\approx 0.82QFI(4)≈0.82

indicating that coherence breaks ~18% earlier than correlation in this setup.

5. Discussion
Our results highlight a practical and often overlooked point: a circuit may retain correlation signatures while losing the phase-coherent component that underlies quantum advantage. This has implications for benchmarking: reporting only correlation-type metrics may overestimate usable “quantumness”. The proposed QFI offers a compact diagnostic dimensionless ratio, and breakdown maps provide actionable guidance on when ZNE should be trusted.

6. Limitations & future work
This study uses a simplified depolarizing noise model; real devices exhibit coherent errors, crosstalk, and non-Markovian effects. Future work can apply the same pipeline to hardware backends or to more realistic noise models, and can test alternative mitigation schemes.

7. Conclusion
We introduce QFI and present a folding-based ZNE breakdown map for GHZ states up to 4 qubits. Global coherence measured by ⟨X⊗n⟩ degrades earlier than correlation ⟨Z⊗n⟩. The proposed metrics and mapping procedure can serve as a lightweight diagnostic tool for NISQ error mitigation studies.

Data & code availability
CSV files are generated automatically during execution (filenames printed at runtime). The benchmark script is self-contained and can be run on standard Python installations with Qiskit Aer.

Data and Code Availability

CSV files are generated automatically during execution (filenames printed at runtime).

The benchmark script is self-contained and executable on standard Python installations with Qiskit Aer.

Additional materials are available via:

www.quantumhorizon.it