Separability does not certify zero-noise extrapolation: Factorial diagnostics in dynamic quantum circuits

Technical Summary: Separability Does Not Certify Zero-Noise Extrapolation

Problem Statement
Zero-noise extrapolation (ZNE) estimates ideal quantum observables by deliberately amplifying noise and fitting a response model to extrapolate to the zero-noise limit. While single-parameter noise amplification (e.g., gate folding) is common, dynamic circuits introduce distinct noise sources: gate errors ($G$), feedforward latency decoherence ($L$), and mid-circuit measurement (MCM) errors ($M$). A critical assumption in multi-knob ZNE is observable separability: the hypothesis that these noise sources act independently such that their combined effect on an observable is multiplicative (additive on a logarithmic scale). If separability fails, unmodeled interactions between noise channels can bias the extrapolated zero-noise limit. The paper investigates whether separately amplifying $G$, $L$, and $M$ on real hardware yields a separable response surface and whether this separability guarantees accurate mitigation.

Methodology
The author conducted a pre-specified $2^3$ factorial experiment on three IBM Quantum dynamic circuits (a spectator probe, a double teleportation circuit, and a GHZ state preparation).

• Knobs: Three noise amplification "knobs" were implemented:
  1. Gate ($\lambda_G$): Standard gate folding (replacing $U$ with $UU^\dagger U$).
  2. Latency ($\lambda_L$): Stretching the feedforward window using idle time, protected by a staggered echo sequence to suppress coherent phase rotations.
  3. Measurement ($\lambda_M$): Repeating MCMs and conditioning on outcome parity, with duration-matching padding to prevent latency noise from contaminating the measurement knob.
• Design: Each knob was set to levels 1 (native) and 3, creating eight corner settings. Two center points $(2,2,1)$ and $(2,2,3)$ were included to test for curvature. The experiment ran on ibm marrakesh (main) and ibm kingston (trial runs), with follow-up runs to inject known interactions and test drift.
• Analysis: The response was modeled as a log-linear function with interaction terms ($b_{ij}$). The study distinguished between:
  • Separability: Whether interaction coefficients (e.g., $b_{LM}$) vanish.
  • Model Adequacy: Whether the response model holds outside the measured range.
  • Target Calibration: Whether the extrapolated zero coordinate corresponds to the ideal noise-free value ($E_{id}$).
• Controls: The study included "positive controls" (injecting virtual rotations to create known interactions) and "negative controls" (trial runs without echoes to demonstrate coherent artifacts).

Key Results

1. Separability of the Primary Observable: For the primary "spectator" observable, the latency–measurement interaction coefficient was $b_{LM} = +0.0007$ with a 95% bootstrap interval $[-0.0033, +0.0049]$. This interval is contained within the pre-specified equivalence margin of $\pm 0.005$, though the main run's interval does not formally exclude an interaction as large as the minimum detectable effect (+0.0044). A separately planned follow-up run met a strict equivalence criterion (90% interval inside $\pm 0.005$) and confirmed the main run's bound at the 5% level, while also recovering injected interactions of known size.
2. Artifact of Coherent Phases: Trial runs without the staggered echo showed a large apparent interaction ($b_{LM} \approx 0.02$) caused by coherent phase rotations accumulating during the latency window. The echo successfully suppressed this, demonstrating that apparent interactions can arise from coherent errors masquerading as noise channel interactions.
3. Accuracy vs. Separability: Despite the primary interaction being negligible, separability did not certify accuracy. For three of the six observables, all five tested extrapolation strategies (joint fit, one-at-a-time, uniform scaling, latency-only, gate-only) produced 95% confidence intervals that excluded the known ideal value ($E_{id}=1$).
4. Bias Sources:
   • Unamplified Noise: Noise sources not controlled by the knobs (e.g., state preparation, single-qubit gates, terminal readout) contribute to a gap between the extrapolated limit ($E^*$) and the ideal value ($E_{id}$).
   • Non-Faithful Scaling: Duration control experiments revealed that the "padding" used to match measurement times decayed approximately 1.5–1.8 times faster than the added latency stretch. This suggests the latency knob does not scale the native window noise faithfully, introducing a model-dependent bias.
   • Curvature: Center points indicated non-linear decay rates for some observables, which single-level extrapolations fail to capture.
5. Strategy Performance: Multi-knob strategies (joint, one-at-a-time, uniform) generally outperformed single-knob strategies (which left noise sources unamplified). However, among the multi-knob strategies, no single approach consistently outperformed the others across all observables, and all suffered from the biases mentioned above.

Key Contributions

• Diagnostic Framework: The paper provides a rigorous factorial diagnostic for testing the separability assumption in dynamic circuits, distinguishing between observable separability, response-model validity, and target calibration.
• Identifiability Limits: It constructs an explicit proof that measurements at only two levels per knob cannot uniquely identify the zero-noise limit without assuming a specific response class (e.g., linearity), as unobserved quadratic terms can mimic different zero-coordinates.
• Coherent Error Masquerade: It demonstrates that coherent phase errors can produce non-zero interaction coefficients in log-scale models, necessitating echo sequences or phase-refocusing before interpreting interactions as physical channel coupling.
• Empirical Evidence of Bias: The study offers concrete evidence that even when noise channels appear separable, ZNE can fail to recover the ideal value due to unamplified noise sources and non-ideal scaling coordinates (e.g., unequal decay rates in padding vs. stretch).

Significance and Claims
The paper modestly claims that while factorial diagnostics can verify the separability of implemented knobs on specific hardware, separability is not a sufficient condition for accurate zero-noise extrapolation. The experiment serves as a diagnostic of the implemented response surface rather than a certification of physical channel independence or mitigation accuracy.

The author concludes that:

1. Small interaction coefficients do not guarantee that the extrapolated limit is the ideal value.
2. Accurate ZNE requires not only separable noise amplification but also faithful scaling coordinates (where the added noise matches the native noise) and the amplification of all relevant noise sources.
3. Future experiments must address the identifiability of the zero coordinate (e.g., via additional physical amplification levels or independent calibration of latency origins) and account for the bias-variance trade-off inherent in reducing model complexity (e.g., using additive fits).

The work does not propose a new amplification method but rather establishes a reproducible case study showing that current ZNE practices on dynamic circuits may yield biased results even when standard separability assumptions appear to hold.