arxiv: 2604.24475 · v1 · submitted 2026-04-27 · 🪐 quant-ph · cs.ET

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Improving Zero-Noise Extrapolation via Physically Bounded Models

Andriy Miranskyy , Adam Sorrenti , Jasmine Thind , Claude Gravel

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Pith reviewed 2026-05-08 04:12 UTC · model grok-4.3

classification 🪐 quant-ph cs.ET

keywords zero-noise extrapolationquantum error mitigationphysically bounded modelsextrapolation fittingnear-term quantum devicesunphysical predictions

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The pith

Physically bounding the zero-noise estimate during fitting reduces unphysical predictions in zero-noise extrapolation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces physically bounded variants of polynomial, exponential, and polynomial-exponential extrapolation models for zero-noise extrapolation by explicitly parameterizing the zero-noise value and constraining it to the valid physical range during optimization. This prevents the common problem of fitted models producing expectation values outside the possible range for quantum observables. On a benchmark of 180,000 synthetic circuits and millions of ZNE experiments under realistic IBM-derived noise, the bounded versions cut unphysical outputs and stabilize the exponential-family models, while polynomial models change little. Preliminary real-hardware tests on GHZ and W-state circuits show the same qualitative pattern of avoiding pathological extrapolations.

Core claim

By reparameterizing the extrapolation function so that the zero-noise estimate appears as an explicit parameter that is then constrained to the physical interval during optimization, bounded models for zero-noise extrapolation substantially reduce unphysical predictions and improve stability for exponential and polynomial-exponential families relative to their unconstrained counterparts.

What carries the argument

Physically bounded extrapolation models, formed by isolating the zero-noise term as a free parameter that is clamped to the valid observable range [0,1] (or equivalent) inside the fitting procedure.

If this is right

Bounded extrapolation cuts unphysical predictions across the 180,000-circuit synthetic benchmark.
Exponential and polynomial-exponential models gain measurable stability; polynomial models show little change.
On real hardware the bounded variants avoid pathological extrapolations and give a more usable accuracy-coverage trade-off.
The method slots into existing ZNE pipelines with only a change to the optimizer constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reparameterization trick could be applied to other extrapolation-based mitigation techniques that currently ignore physical bounds.
Device-specific calibration data could be used to set tighter, non-uniform bounds rather than the universal [0,1] interval.
The observed gap between simulation and hardware suggests that bounded models may help surface when noise models are incomplete.

Load-bearing premise

Constraining the zero-noise estimate to the physical interval during optimization does not systematically bias the extrapolated value away from the true noise-free result under realistic device noise.

What would settle it

On small circuits whose exact noise-free expectation value is known classically, measure whether the mean squared error of bounded-model predictions is not materially larger than that of unbounded models while the fraction of unphysical outputs drops sharply.

Figures

Figures reproduced from arXiv: 2604.24475 by Adam Sorrenti, Andriy Miranskyy, Claude Gravel, Jasmine Thind.

**Figure 1.** Figure 1: Histogram of ideal expectation values in the synthetic benchmark (bin view at source ↗

**Figure 2.** Figure 2: Distribution of transpiled circuit depths in the synthetic benchmark. view at source ↗

**Figure 3.** Figure 3: Empirical cumulative distribution functions (CDFs) of the improve view at source ↗

**Figure 4.** Figure 4: Expectation values as a function of the noise scale factor view at source ↗

read the original abstract

Zero-noise extrapolation (ZNE) mitigates errors in near-term quantum devices by extrapolating measurements obtained at amplified noise levels to estimate noise-free expectation values. In practice, commonly used extrapolation models are fitted without enforcing physical constraints, which can yield predictions outside the valid range of quantum observables. In this work, we introduce physically bounded variants of polynomial, exponential, and polynomial--exponential extrapolation models by explicitly parameterizing the zero-noise estimate and constraining it during optimization. We evaluate the approach using a large synthetic benchmark comprising 180,000 circuits and approximately 3.6 million ZNE experiments generated under realistic device noise models derived from IBM quantum backends. We also perform preliminary validation on real quantum hardware using GHZ and W-state circuits. Across the synthetic benchmark, bounded extrapolation substantially reduces unphysical predictions and improves the stability of exponential- and polynomial--exponential-family models, whereas polynomial models show little difference between bounded and unbounded variants. Hardware experiments show similar qualitative behaviour: bounded models generally avoid pathological extrapolations and often provide a more reliable balance between accuracy and usable coverage. At the same time, the results highlight practical limitations of current devices, including stronger-than-expected noise effects and variability not fully captured by simulation models. These results suggest that enforcing physical constraints during extrapolation improves the reliability of ZNE and that this approach can be incorporated into existing workflows with minimal modification.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Bounded ZNE models cut unphysical extrapolations on a large synthetic benchmark, but hardware results stay preliminary and bias checks are light.

read the letter

The paper's main result is that reparameterizing common ZNE fits to keep the zero-noise estimate inside the physical interval reduces bad predictions and improves stability for exponential and poly-exponential models. Polynomials barely change. They back this with 180k synthetic circuits and 3.6M experiments that have known ground truth from IBM-style noise models, which gives the comparison real weight. The bounded versions simply avoid the cases where unconstrained fits spit out values outside [0,1] or whatever the observable range is. That is a concrete, usable tweak and the scale of the test set is the part that stands out. The hardware runs on GHZ and W states show the same qualitative pattern but are described as preliminary, and the authors note that real devices have more variability than the simulations capture. The weakest part is the lack of direct checks on whether the constraint systematically shifts the estimate away from the true noise-free value when the noise model is imperfect. The synthetic numbers hold up on their own terms, but anyone adopting this would still want to see how often the bound is active and what it does to accuracy on real hardware. This is aimed at people who already run ZNE on near-term devices and want a low-effort way to make it more reliable. It is not a deep theoretical advance, but the benchmark is large enough and the change is simple enough that it deserves referee time. A review could tighten the hardware section and add a short bias diagnostic without much extra work.

Referee Report

0 major / 4 minor

Summary. The paper introduces physically bounded variants of polynomial, exponential, and polynomial-exponential extrapolation models for zero-noise extrapolation (ZNE) by explicitly parameterizing the zero-noise estimate and constraining it to the physical interval during optimization. It evaluates the approach on a large synthetic benchmark of 180,000 circuits and 3.6 million ZNE experiments generated from realistic IBM-derived noise models, plus preliminary hardware tests on GHZ and W-state circuits. Key results show that bounded models substantially reduce unphysical predictions and improve stability for exponential and poly-exponential families (with little difference for polynomials), and exhibit similar qualitative behavior on hardware despite device limitations.

Significance. If the results hold, this provides a practical, low-overhead improvement to ZNE, a widely used error-mitigation technique for near-term quantum devices. The scale of the synthetic benchmark (with known ground truth) supplies robust statistical support for reduced unphysical extrapolations and enhanced stability in specific model families. The method integrates easily into existing workflows. Hardware results, though preliminary, add relevance while underscoring gaps between simulation and real devices.

minor comments (4)

Abstract and results: the synthetic benchmark size is given as 'approximately 3.6 million' experiments; report the exact count and breakdown by model family and noise level in the methods or results section for reproducibility.
Results section: define the stability metric explicitly (e.g., variance of extrapolated values across amplification factors or bootstrap resampling) and report quantitative effect sizes or statistical tests for the claimed improvements in stability and reduction of unphysical predictions.
Hardware experiments: the validation is described as preliminary; include the exact number of circuits, shots per circuit, and quantitative metrics (e.g., fraction of unphysical predictions) to allow direct comparison with the synthetic benchmark.
Methods: provide the explicit parameterization of the zero-noise estimate and the form of the constrained optimization objective (e.g., via Lagrange multipliers or reparameterization) to enable straightforward implementation by readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The assessment correctly identifies the core contribution—physically bounded extrapolation models that reduce unphysical predictions—and the value of the large-scale synthetic benchmark. We will incorporate minor editorial improvements and clarifications in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's contribution consists of defining bounded variants of standard extrapolation models (by explicitly parameterizing the zero-noise estimate and adding constraints to the optimizer) followed by an empirical comparison against unbounded variants. This comparison is performed on a synthetic benchmark with known ground-truth noise-free values (180k circuits, 3.6M experiments), so reported improvements in stability and reduction of unphysical outputs are measured directly rather than derived from the fitted parameters themselves. No load-bearing derivation, self-citation chain, or ansatz reduces any claimed result to an input quantity by construction. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that quantum expectation values lie in [-1,1] and that this interval can be enforced without invalidating the extrapolation under device noise.

axioms (1)

domain assumption Quantum observables have expectation values strictly bounded in [-1, 1]
Used to constrain the zero-noise parameter during optimization; standard in quantum mechanics.

pith-pipeline@v0.9.0 · 5552 in / 1147 out tokens · 34729 ms · 2026-05-08T04:12:00.573645+00:00 · methodology

discussion (0)

Reference graph

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