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arXiv:2610.28705

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

Aryan Kumar

quant-phSubmitted Oct 20, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

Separately amplifying gate, latency, and mid-circuit measurement errors can simplify zero-noise extrapolation, but a weak interaction between amplification knobs does not establish an accurate zero-noise limit. We distinguish observable separability, response-model adequacy, and calibration of that limit, and test them in a pre-specified $2^3$ factorial experiment on three IBM dynamic circuits. For the primary echoed spectator observable, the latency-measurement coefficient is $b_{LM}=+0.0007$ with a nominal 95% bootstrap interval $[-0.0033,+0.0049]$. A separately planned follow-up meets a $\pm0.005$ equivalence criterion and recovers two injected interactions; both repeat runs nevertheless trigger the drift rule. Multi-knob extrapolations have smaller median absolute errors than the single-knob comparisons, yet all five original strategies exclude the known ideal value for three of six observables. Duration controls support a model-dependent contribution to the spectator deficit from unequal decay rates in native padding and added delay. We give an explicit construction showing why measurements at two measurement-knob levels cannot identify the extrapolated limit without response assumptions. A post-hoc precision analysis and an additive full-factorial estimator expose a further bias-variance trade-off hidden by point-error comparisons. The experiment provides a diagnostic of the implemented response surface, not a certification of physical channel independence or mitigation accuracy.

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Reliable benchmarking of Quantum Error Mitigation (QEM) requires distinguishing genuine improvements from artefacts of the post-processing arithmetic. In this paper, we expose a failure mode in Richardson Zero-Noise Extrapolation (ZNE), a widely used technique routinely (and often implicitly) relied upon in benchmarks and experiments. When noise amplification operates beyond usable signals - a regime that is quickly reached on current hardware for non-trivial circuits - we show that the extrapolation no longer reflects the underlying physics, but collapses into a fixed rescaling of a single noisy measurement, producing a bogus apparent improvement that is independent of noise amplification. This poses a rarely considered threat to the validity of many empirical evaluations in quantum computing. Measurements on real hardware (IQM Euro-Q-Exa) confirm this collapse with ordinary folding alone: as circuit depth erodes the signal, the reported estimate decouples from the truth and overshoots the ideal by up to 21%. We further introduce a matched-cost "garbage-folding" negative control that carries no usable signal yet reports a larger apparent improvement than genuine folding - showing that the magnitude of an improvement is not evidence of its correctness - alongside a zero-cost check flagging the artefact from data a benchmark already holds. We distil both into a short reporting checklist for ZNE benchmarks.

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Separately amplifying gate, latency, and mid-circuit measurement errors can simplify zero-noise extrapolation, but a weak interaction between amplification knobs does not establish an accurate zero-noise limit. We distinguish observable separability, response-model adequacy, and calibration of that limit, and test them in a pre-specified 232^323 factorial experiment on three IBM dynamic circuits. For the primary echoed spectator observable, the latency-measurement coefficient is bLM=+0.0007b_{LM}=+0.0007bLM​=+0.0007 with a nominal 95% bootstrap interval [−0.0033,+0.0049][-0.0033,+0.0049][−0.0033,+0.0049]. A separately planned follow-up meets a ±0.005\pm0.005±0.005 equivalence criterion and recovers two injected interactions; both repeat runs nevertheless trigger the drift rule. Multi-knob extrapolations have smaller median absolute errors than the single-knob comparisons, yet all five original strategies exclude the known ideal value for three of six observables. Duration controls support a model-dependent contribution to the spectator deficit from unequal decay rates in native padding and added delay. We give an explicit construction showing why measurements at two measurement-knob levels cannot identify the extrapolated limit without response assumptions. A post-hoc precision analysis and an additive full-factorial estimator expose a further bias-variance trade-off hidden by point-error comparisons. The experiment provides a diagnostic of the implemented response surface, not a certification of physical channel independence or mitigation accuracy.
Dynamic quantum circuits integrate mid-circuit measurements and feed-forward operations to enable real-time classical processing and conditional quantum logic. These capabilities are central to key quantum protocols such as quantum error correction, and have recently demonstrated significant potential for reducing quantum resources, including circuit depth and gate count, across a range of applications. However, executing dynamic circuits on real quantum hardware introduces a critical trade-off: while resource requirements decrease, circuit fidelity degrades due to high error rates of mid-circuit measurements, as well as the decoherence errors accumulated during the extended idle periods introduced by both mid-circuit measurements and feed-forward operations. In this paper, we systematically investigate the impact of standard error mitigation techniques on dynamic circuit applications pertaining to Hamiltonian simulation and ground state estimation of physically relevant systems like the Heisenberg model. We explore dynamical decoupling (DD) as a strategy to suppress decoherence and crosstalk errors during idle windows introduced by mid-circuit measurements and feed-forward delays, and also examine error mitigation via zero-noise extrapolation (ZNE). Through experiments conducted on IBM quantum hardware, we benchmark effective combinations of these strategies that maximize the practical benefits of dynamic quantum circuits in these applications. We demonstrate that a combination of DD and ZNE is effective in mitigating the errors introduced during mid-circuit measurements and feed-forward operations, as well as the errors arising from faulty measurements. This approach yields a energy gap improvement of at least 60% in ground state estimation and reduces observed error of time-evolved states by up to 99% for the Ising model and up to 20% for the Heisenberg model.
Zero-noise extrapolation (ZNE) is an established quantum error mitigation technique that infers noise-free expectation values by scaling circuit noise and extrapolating to the zero-noise limit. Standard ZNE formulations implicitly assume memoryless Markovian gate noise, where error rates scale linearly with circuit duration. In physical solid-state quantum processors dominated by colored 1/fα1/f^\alpha1/fα environmental dephasing, this fundamental assumption fails: non-Markovian noise induces superlinear coherence decay χ(t)∝t1+α\chi(t) \propto t^{1+\alpha}χ(t)∝t1+α, causing standard polynomial extrapolation to diverge. In this work, we demonstrate how inserting dynamical decoupling (DD) sequences into idle circuit intervals spectrally filters low-frequency noise components, linearizing the effective decay profile into a Markovian regime χeff(t)∝t\chi_{\text{eff}}(t) \propto tχeff​(t)∝t. We benchmark this combined DD–ZNE framework by reconstructing the order parameter across the quantum phase transition of a 1D Transverse-Field Ising Model (TFIM). Our results demonstrate that while standalone ZNE fails under non-Markovian dephasing, the combined DD–ZNE approach eliminates extrapolation divergence and reliably recovers ideal quantum observables. Key Highlights: Theoretical Breakdown: Analytical derivation showing why polynomial ZNE diverges under 1/fα1/f^\alpha1/fα dephasing environments due to superlinear temporal scaling. Filter Function Engineering: Demonstration of spectator-qubit XY-4 pulse insertion as a high-pass spectral filter that suppresses low-frequency noise and restores linear decay dynamics. Phase Transition Benchmark: Validation on a 4-qubit 1D TFIM order parameter (⟨σz(0)σz(1)⟩\langle \sigma_z^{(0)} \sigma_z^{(1)} \rangle⟨σz(0)​σz(1)​⟩) across g∈[0.1,2.0]g \in [0.1, 2.0]g∈[0.1,2.0], demonstrating a >69%>69\%>69% error reduction over unmitigated circuits and preventing non-physical polynomial divergence.
Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.
We present the first experimental demonstration that quantum noise is informative, not merely destructive — validating Theorem 8 of IIT 5.1 (FEP-MaxCal Bridge). Using a 5-qubit GHZ circuit on an AerSimulator calibrated with real IBM Quantum ibm_fez noise parameters (T1=320µs, T2=150µs, gate2q error=2.83e-3), we show that Zero Noise Extrapolation (ZNE) recovers integrated information (Φ) to a value exceeding the noiseless ideal: Φ_ZNE = 1.1736 > Φ_ideal = 0.9999 (permutation test, p < 0.000001, N=20 runs, 8192 shots each, 10000 bootstrap resamples). ZNE requires controlled noise scaling (identity folding at λ = 1, 2, 3) that is only feasible in simulation; however, the noise model is calibrated from 3766 real IBM Quantum jobs across 6 backends, and prior hardware experiments on ibm_fez (V1: Φ=0.757, V2: Φ=0.832 with ZNE on real hardware) validate the baseline metrics. The experiment follows a four-phase paradigm: (1) Baseline: GHZ-5 ideal yields Φ=0.9999; (2) Attack: NISQ noise degrades Φ to 0.8804 (-12%); (3) Recovery: Hahn Echo alone shows no significant improvement (p=0.853), but ZNE with 3-scale identity folding and linear extrapolation to noise=0 yields Φ=1.1736; (4) Antifragility: the recovered system exceeds the original baseline by +17.4%. Additional findings: Φ scales monotonically with qubit count (2q→7q), distinguishes entanglement topology (GHZ > W-state > Cluster ≈ Separable), is robust across 4 IBM backend noise models (CV=3.35%), and decays exponentially with circuit depth (τ=29.52 layers). Hardware validation on real ibm_fez confirms Φ for entangled vs separable circuits with Cohen's d = 191.3 (p < 0.0001).
Zero-noise extrapolation provides a practical means of suppressing gate errors in current noisy intermediate-scale quantum hardware. The accuracy of the zero-noise estimate depends sensitively on the fidelity of the assumed noise model to the actual error scaling. This work introduces a hybrid Gaussian-exponential extrapolation scheme tailored for quantum circuits with periodic structure, which are ubiquitous in quantum algorithms. Under Pauli diagonal errors, by constructing and analyzing an approximate Markov process for the transfer of Pauli operators, we prove a central limit theorem: the noise amplification factor weakly approaches a log-normal distribution, which motivates augmenting the standard exponential model with Gaussian variance corrections. The resulting model requires no prior noise characterization and applies directly to arbitrary periodic circuits. Performance is assessed on Trotterized Ising dynamics, random circuits, and Grover search algorithm using Qiskit noise simulators. For moderate to large circuit depths, the hybrid model yields measurable reductions in bias relative to previous extrapolation variants, indicating its utility for error mitigation on near-term quantum hardware.
Reliable benchmarking of Quantum Error Mitigation (QEM) requires distinguishing genuine improvements from artefacts of the post-processing arithmetic. In this paper, we expose a failure mode in Richardson Zero-Noise Extrapolation (ZNE), a widely used technique routinely (and often implicitly) relied upon in benchmarks and experiments. When noise amplification operates beyond usable signals - a regime that is quickly reached on current hardware for non-trivial circuits - we show that the extrapolation no longer reflects the underlying physics, but collapses into a fixed rescaling of a single noisy measurement, producing a bogus apparent improvement that is independent of noise amplification. This poses a rarely considered threat to the validity of many empirical evaluations in quantum computing. Measurements on real hardware (IQM Euro-Q-Exa) confirm this collapse with ordinary folding alone: as circuit depth erodes the signal, the reported estimate decouples from the truth and overshoots the ideal by up to 21%. We further introduce a matched-cost "garbage-folding" negative control that carries no usable signal yet reports a larger apparent improvement than genuine folding - showing that the magnitude of an improvement is not evidence of its correctness - alongside a zero-cost check flagging the artefact from data a benchmark already holds. We distil both into a short reporting checklist for ZNE benchmarks.