GroverFigureOfMerit: An Agnostic Figure of Merit for Quantum Backend Characterization in the NISQ Era

Alejandro Fern\'andez; Marcos Guillermo Lammers; Tiago Restucha

REVIEW 3 major objections 7 minor 18 references

GroverFigureOfMerit: An Agnostic Figure of Merit for Quantum Backend Characterization in the NISQ Era

T0 review · 3 major / 7 minor · reviewed 2026-07-10 · glm-5.2

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Pith's one-line read Single Grover-based score cuts through quantum hardware noise

desk verdict A Grover-based benchmarking metric with working code and clean formulation, but no error bars, no parameter sensitivity analysis, and no real QPU validation — the scores may not actually discriminate backends yet. read the letter →

arxiv 2607.08636 v1 pith:FU2QQQBV submitted 2026-07-09 quant-ph

Tiago Restucha , Marcos Guillermo Lammers , Alejandro Fernández This is my paper · ORCID

classification quant-ph

keywords acrossbackendquantummetricsnoiseproviderstranspilationagnostic

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes running Grover's search algorithm on a quantum backend as a universal stress test, then compressing the outcome into a single number that captures how well the hardware actually performs under real algorithmic load. The core insight is that low-level hardware metrics like gate error rates and coherence times are poor predictors of whether a quantum algorithm will succeed, because they ignore transpilation overhead, topology constraints, and dynamic noise. By running the same Grover circuit on any backend through abstraction layers, and scoring the result with a formula that rewards hitting target states while penalizing uneven amplification and leakage to non-target states, the authors produce a provider-agnostic score. Experiments on noise models derived from real IBM and IonQ processors show the metric correctly ranks backends and captures the collapse of performance as circuit depth grows.

What carries the argument

The key machinery is the combination of (1) Grover's algorithm as a fixed algorithmic workload whose multi-controlled oracle and diffusion operator create deep, entangled circuits that stress-test routing and coherence, and (2) the scoring function that combines target-state success probability with two penalty terms—one for non-uniform amplification across targets (σ_T), one for probability leakage to non-marked states (P_N)—yielding a single number between 0 and 1. The implementation uses polymorphic backend adapters that transpile and execute the same logical circuit on different providers without code changes.

What would settle it

If, when executed on physical QPUs, the metric produces rankings inconsistent with those obtained from the noise models—or if the score variance across repeated runs on the same backend exceeds the score differences between backends—the metric's discriminating power would be undermined.

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Extended reading notes

Core claim

The central object is a scoring formula: Score = max(0, P_T − λ·σ_T − µ·P_N), where P_T is the cumulative probability of measuring target states, σ_T is the standard deviation of probabilities across targets (penalizing non-uniform amplification), and P_N is the probability leaked to non-target states. This formula, applied to the output distribution of a Grover circuit executed on any quantum backend, translates heterogeneous hardware characteristics—noise, topology, transpilation overhead—into a single comparable number. The experiments demonstrate that this number tracks known algorithmic limits (Grover's deterministic overshoot, the M=N/2 collapse) on ideal simulators while revealing the

Load-bearing premise

The experimental validation rests entirely on noise models derived from real processor characterizations rather than execution on physical quantum hardware, meaning the metric's ability to distinguish backend performance under actual dynamic conditions—crosstalk, thermal fluctuations, and run-to-run variability—remains unconfirmed.

Editorial extensions

If this is right

If validated on physical hardware, this metric could serve as a standardized 'pass/fail' benchmark that quantum cloud providers expose directly, letting developers compare backends without interpreting provider-specific calibration data.
The approach generalizes beyond Grover: any algorithm with a known ideal output distribution could serve as a stress test, suggesting a catalog of algorithm-specific FoMs for different workload profiles (variational, Fourier transform, etc.).
The metric's sensitivity to transpilation overhead quantifies a concrete cost of provider-specific compilers, potentially motivating investment in provider-agnostic intermediate representations for quantum circuits.
The uniformity penalty (σ_T) may detect coherent error types that random benchmarking misses, since coherent errors produce systematic amplitude imbalances rather than uniform depolarization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The choice of Grover is strategic rather than practical: the authors explicitly note they do not claim algorithmic advantage, only that Grover's dense multi-controlled structure makes it a sensitive probe. Other equally dense circuits (e.g., QFT-based benchmarks) might yield different backend rankings, since the penalty structure is tailored to amplification behavior.
The λ and µ penalty weights are set to 1 by default but are configurable; the paper does not explore how sensitive backend rankings are to these choices, leaving open whether the metric is robust or can be gamed by parameter tuning.
The noise models used (IBM Fake Providers, IonQ Aria 1) preserve static error profiles but omit crosstalk and temporal variability; if these dynamic effects are significant, the metric's rankings could shift on real hardware, particularly for superconducting backends where crosstalk is architecture-dependent.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper takes the GRADE scoring function (Manor et al. 2025), wraps it in the Qonscious framework's polymorphic adapters for cross-provider execution, and runs it on simulated noise models from IBM and IonQ. The metric is clean, the code is public, and the heterogeneity analysis across nine providers is a useful reference. But the experimental validation can't actually show the metric discriminates between backends, because there are no error bars, no repeated trials, and no sensitivity analysis on the scoring parameters. The claim that it 'distinguishes backend performance' is not yet supported by the evidence presented.

What's new and done well: the integration of a GRADE-adapted score into a provider-agnostic execution framework is a legitimate engineering contribution. The scoring formula (Eq. 8: Score = max(0, P_T − λ·σ_T − µ·P_N)) is straightforward and well-motivated. The nine-provider heterogeneity taxonomy in Section 2 is a solid reference for anyone working in cross-provider quantum software. The code is on GitHub, which earns real credit for reproducibility. The paper is honest about its limitations — it explicitly flags QPU validation as future work and acknowledges that static noise models miss crosstalk and thermal effects.

The soft spot that actually matters: the stress-test concern lands. The paper reports single scores per backend (e.g., IBM scoring 0.495–0.792 at N=8, M=2) with S=2000 shots but no repeated trials, no confidence intervals, and no significance testing. At 2000 shots, the statistical noise on a probability estimate is roughly ±0.01–0.02 per state, and the spread between IBM backends could easily fall within run-to-run variance. The paper never checks this. Similarly, λ=1 and µ=1 are described as 'typical values' that 'can be adjusted,' but no sensitivity analysis shows whether the backend ranking survives different parameter choices. If changing λ to 0.5 flips the IBM vs. IonQ ordering, the metric isn't really distinguishing backends — it's just producing numbers that move with an unexamined knob.

The simulated-noise-only concern is real but less load-bearing than the reader suggests. The authors are upfront about it, and the more modest claim — that the metric is sensitive to noise, topology, and transpilation overhead — is supported by the simulation results. The problem is the stronger claim about discrimination, which requires statistical rigor the paper doesn't provide.

Who gets value: researchers building cross-provider quantum software tools, and anyone thinking about algorithmic benchmarking in NISQ. The Qonscious framework integration and the heterogeneity analysis are the real deliverables here.

Recommendation: deserves a serious referee. The core idea is sound, the code is public, and the gaps (error bars, parameter sensitivity, QPU validation) are addressable in revision. I'd push the authors to add repeated trials with confidence intervals and at least a basic λ/µ sensitivity sweep before accepting the discrimination claim.

Referee Report

3 major / 7 minor

Summary. The paper proposes GroverFigureOfMerit, a circuit-based figure of merit for quantum backend characterization that uses Grover's algorithm as an algorithmic stress test. The metric (Eq. 8) combines the cumulative success probability on target states (P_T) with penalties for non-uniform amplification among targets (σ_T) and probability leakage to non-target states (P_N). The approach is implemented on the Qonscious framework using polymorphic adapters that abstract provider-specific APIs, enabling agnostic execution on IBM, IonQ, and simulator backends. Experiments vary the number of target states M in search spaces of N=8 and N=32, and scale N from 8 to 128 with M=1, comparing ideal simulators against noise models derived from real IBM and IonQ processors. The authors find that ideal simulators achieve near-perfect scores, IBM noise models degrade severely with scaling, and IonQ's Aria 1 model shows greater resilience. The paper also provides a taxonomy of nine quantum providers and identifies three barriers (API fragmentation, metric semantics, opacity/volatility) that motivate agnostic metrics.

Significance. The problem addressed—heterogeneous and incomparable characterization metrics across quantum hardware providers—is real and well-motivated. The design choice of Grover's algorithm as a stress test is reasonable given its dense multi-controlled gate structure, which exercises routing, decomposition, and coherence. The scoring function (Eq. 8) is a clean adaptation of the externally cited GRADE approach, and the open-source implementation via Qonscious with polymorphic adapters is a practical contribution. The heterogeneity analysis across nine providers (Section 2) is a useful synthesis. However, the significance of the experimental validation is limited by the absence of physical QPU runs and the lack of statistical rigor in the reported results.

major comments (3)

§5, Figure 3: The central claim that the metric 'enables distinguishing backend performance under a unified score' (Abstract, §6) is not supported by the experimental methodology. All reported scores appear to be from single executions per backend per parameter setting (S=2000 shots, but no repeated trials). There are no error bars, confidence intervals, or significance tests. For example, at N=8, M=2, IBM architectures score between 0.495 and 0.792 (§5), but the reader cannot determine whether this spread reflects genuine backend differences or run-to-run variance from shot noise. At 2000 shots, the statistical uncertainty on a probability estimate is on the order of 1/sqrt(2000) ≈ 0.022 per outcome, and the composite score in Eq. 8 propagates this in a non-trivial way. Without repeated trials and variance estimates, the discriminative power of the metric is unverified. This is load-ba:
§4.3, Eq. (8): The scoring parameters λ and μ are described as 'typical values' set to λ=1, μ=1 that 'can be adjusted according to the application context or the characteristics of the evaluated backend.' No sensitivity analysis is provided. If the ranking of backends changes under different reasonable choices of λ and μ (e.g., λ=0.5, μ=2), then the metric's ability to 'distinguish backend performance' depends on parameter choices that are themselves arbitrary. The paper should demonstrate that the backend rankings reported in §5 are robust to perturbations of λ and μ, or else restrict the claim to sensitivity rather than discrimination.
§5: The entire experimental validation uses simulated noise models (IBM Fake Providers, IonQ Aria 1 model) rather than physical QPUs. The authors acknowledge this limitation (Abstract, §5, §6, §7) and note that crosstalk, thermal fluctuations, and temporal variability are not captured. This is acceptable for a proof-of-concept, but the abstract and conclusion should be carefully scoped: the current evidence supports that the metric is *sensitive to modeled noise, topology, and transpilation overhead*, not that it *distinguishes real backend performance*. The phrase 'distinguishes backend performance' in the abstract overstates what has been demonstrated.

minor comments (7)

§4.3, Eq. (8): The condition 'If µ·P_N ≥ P_T, then Score = 0' is stated in prose but not reflected in the equation. Consider writing Score = max(0, P_T − λ·σ_T − µ·P_N) and noting the saturation explicitly, or incorporating it as a piecewise definition.
§5, Figure 3: The figure caption references panels a) and b) but the axis labels, legend entries, and exact backend names are difficult to parse in the provided text. Ensure that the figure is self-contained with clearly labeled axes, a readable legend, and explicit identification of each backend.
§4.2, Eq. (4): The optimal iteration count R* is given as approximately π/4 · sqrt(N/M), but the text in §5 references R=2 for N=8 and R=4 for N=32 without showing the derivation. Adding a brief table of (N, M, R*) values for the experimental configurations would improve reproducibility.
§2: The heterogeneity analysis lists nine providers but Figure 1 is referenced without clear enumeration. A table summarizing provider, architecture type, native gate set, and calibration interface type would strengthen this section and make the 'three barriers' concrete.
§4.3: The number of shots S=2000 is stated without justification. A brief note on why 2000 is sufficient (or a reference to a convergence analysis) would be helpful.
§5: The text states 'IBM architectures suffer noise penalties, varying between 0.495 and 0.792' but does not specify which IBM Fake Providers were used or how many. The specific backends should be listed for reproducibility.
§6: The sentence 'While it would be possible to mitigate this by manually encoding the algorithm using the optimal native gates for each specific provider, this approach breaks the agnostic paradigm' is a reasonable point but could be strengthened by citing specific transpilation overhead ratios (e.g., native vs. transpiled gate counts) from the experiments.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: metric is computed from measurement outcomes, scoring function is externally sourced (GRADE), and the distinguishing claim is empirical, not definitional.

full rationale

The paper's derivation chain is self-contained. The GroverFigureOfMerit score (Eq. 8: Score = max(0, P_T − λ·σ_T − µ·P_N)) is computed directly from measurement frequencies on a circuit whose ideal behavior is independently known from Grover's algorithm (Grover, 1996, external). The scoring function is adapted from GRADE (Manor et al., 2025, external citation), not derived from the authors' own prior work. The Qonscious framework (Lammers et al., 2025) is self-cited but serves only as an implementation platform providing polymorphic adapters — it is not invoked as evidence for the metric's validity or as a mathematical justification for the scoring formula. No parameters are fitted to data and then presented as predictions; λ and μ are set to fixed typical values (1, 1) without fitting. The central claim that the metric 'distinguishes backend performance' is an empirical observation from experiments showing different scores across noise models, not a result forced by construction. The weaknesses of the paper (lack of statistical significance testing, no parameter sensitivity analysis, simulated rather than physical hardware) are correctness and empirical-rigor concerns, not circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new physical entities, particles, forces, or mathematical objects. The metric is a deterministic function of measurement outcomes. The Qonscious framework is a software platform, not a theoretical entity. The free parameters (λ, µ) are hand-set weights, not fitted constants. The core axiom is that Grover's algorithm and the linear scoring form are appropriate for backend characterization.

free parameters (3)

lambda_factor (λ) = 1
Penalty weight for non-uniform amplification among target states. Set to 1 by the authors as a 'typical value' to equally penalize non-uniformity and dispersion (§4.3). Not fitted to data; chosen by hand.
mu_factor (µ) = 1
Penalty weight for probability leakage to non-target states. Set to 1 by the authors as a 'typical value' (§4.3). Not fitted to data; chosen by hand.
shots (S) = 2000
Number of measurement repetitions used to estimate the probability distribution. Fixed at 2000 (§4.3). Affects statistical precision of the score but is not a fitted model parameter.

assumptions (3)

domain assumption Grover's algorithm is an appropriate stress test for evaluating quantum backend performance across architectures.
Invoked in §1.1 and §4.1 to justify the choice of Grover as the benchmark circuit. The authors argue its dense multi-controlled gate structure and depth scaling make it suitable, but this is a modeling choice, not a proven fact.
domain assumption Noise models derived from real processor characterizations (IBM Fake Providers, IonQ Aria 1) preserve enough topological and error-profile fidelity to validate the metric's sensitivity to hardware differences.
Invoked in §5: 'they are models derived from real hardware characterizations that preserve the topological properties and error profiles.' The authors acknowledge in §7 that crosstalk and thermal fluctuations are not captured.
ad hoc to paper A single scalar score combining success probability, uniformity, and leakage captures sufficient information to distinguish backend performance for practical selection.
The entire metric design (Eq. 8) assumes that collapsing three distinct quantities (P_T, σ_T, P_N) into one number via linear combination with hand-chosen weights is meaningful. No justification is given for the linear form or the equal weighting beyond 'typical values.'

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Pith. "Pith review of GroverFigureOfMerit: An Agnostic Figure of Merit for Quantum Backend Characterization in the NISQ Era." pith.science (2026). https://pith.science/paper/FU2QQQBV

@misc{pith2026260708636,
  author       = {Pith},
  title        = {Pith review of: GroverFigureOfMerit: An Agnostic Figure of Merit for Quantum Backend Characterization in the NISQ Era},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FU2QQQBV}},
  note         = {Machine review of arXiv:2607.08636}
}

read the original abstract

The Noisy Intermediate-Scale Quantum (NISQ) era poses a challenge for developers: hardware providers expose capabilities through heterogeneous interfaces with proprietary metrics varying widely across providers, hindering informed backend selection. Static characterization metrics - coherence times T1/T2, gate error rates - exhibit limitations: they fail to capture dynamic variability across successive executions, overlook the impact of transpilation, and lack architectural comparability across physically distinct technologies. We propose a Figure of Merit (FoM) based on Grover's algorithm as an algorithmic stress test evaluating quantum backend performance holistically. The metric combines success probability on target states with penalties for non-uniform amplification and leakage to non-marked states, yielding a unified score across hardware architectures. Implemented via the Qonscious framework - a conditional execution platform using polymorphic adapters, it executes agnostically on IBM, IonQ backends, and simulators. Main contributions: (1) proposal and validation of GroverFigureOfMerit, incorporating uniformity and leakage penalties (adapted from GRADE) with emphasis on noise, transpilation, and topological constraints; (2) systematic analysis of heterogeneity across nine quantum providers motivating agnostic metrics; and (3) experimental demonstration via ideal simulators and real-processor noise models, confirming sensitivity to noise, topology, and transpilation overhead. Results confirm the metric distinguishes backend performance under a unified score, capturing intrinsic algorithmic limits. Validation on physical QPUs is identified as a natural next step.

Figures

Figures reproduced from arXiv: 2607.08636 by the authors.

**Figure 2.** Qonscious conditional execution flow. The FoM result (score) can feed the decision function to optimize quantum resource usage [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Comparative benchmarking of the Grover Score with a variable number of targets. a) For N = 8, a performance peak is observed at M = 2 coinciding with the optimal theoretical rotation, and generalized degradation to ∼ 0.5 at M = 4 due to Grover’s algorithmic limit (M = N/2). b) For N = 32, ideal simulators maintain high fidelity, while Fake Providers suffer severe decoherence, relegating their scores to the probabili… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Score comparison across different backends First, ideal simulator lines confirm Grover’s theoretical limit, converging on practically identical values. As observed in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

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Quantum Resource Management in the NISQ Era: Challenges, Vision, and a Runtime Framework

Lammers, Marcos Guillermo and Holik, Federico Hernán and Fernández, Alejandro , title =. 2025 , journal =. doi:10.48550/arXiv.2508.19276 , url =

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Manor, Shay and Kumar, Millan and Behera, Priyank and Khalid, Azain and Zeng, Oliver , title =. 2025 , journal =. doi:10.48550/arXiv.2504.19387 , url =

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Reviewed July 10, 2026 · model on record in the stance chip above.

[1] [1]

Quantum Resource Management in the NISQ Era: Challenges, Vision, and a Runtime Framework

Lammers, Marcos Guillermo and Holik, Federico Hernán and Fernández, Alejandro , title =. 2025 , journal =. doi:10.48550/arXiv.2508.19276 , url =

work page Pith review arXiv doi:10.48550/arxiv.2508.19276 2025

[2] [2]

GRADE: Grover-based Benchmarking Toolkit for Assessing Quantum Hardware

Manor, Shay and Kumar, Millan and Behera, Priyank and Khalid, Azain and Zeng, Oliver , title =. 2025 , journal =. doi:10.48550/arXiv.2504.19387 , url =

work page Pith review arXiv doi:10.48550/arxiv.2504.19387 2025

[3] [3]

and Blume-Kohout, Robin , title =

Proctor, Timothy and Young, Kevin and Baczewski, Andrew D. and Blume-Kohout, Robin , title =. Nature Reviews Physics , volume =. 2025 , doi =

work page 2025

[4] [4]

, title =

Grover, Lov K. , title =. Proceedings of the 28th Annual ACM Symposium on Theory of Computing (STOC '96) , year =. doi:10.1145/237814.237866 , eprint =

work page doi:10.1145/237814.237866

[5] [5]

and Mayer, Karl and Proctor, Timothy , journal=

Lubinski, Thomas and Johri, Sonika and Varosy, Paul and Coleman, Jeremiah and Zhao, Luning and Necaise, Jason and Baldwin, Charles H. and Mayer, Karl and Proctor, Timothy , journal=. Application-Oriented Performance Benchmarks for Quantum Computing , year=

work page

[6] [6]

2026 , eprint=

Machine Learning on Heterogeneous, Edge, and Quantum Hardware for Particle Physics (ML-HEQUPP) , author=. 2026 , eprint=

work page 2026

[7] [7]

National Science Review , volume=

Advancements in superconducting quantum computing , author=. National Science Review , volume=. 2025 , doi=

work page 2025

[8] [8]

IEEE Conference Publication , year=

Quantum Hardware Roofline: Evaluating the Impact of Gate Expressivity on Quantum Processor Design , author=. IEEE Conference Publication , year=. doi:10.1109/10821315 , url=

work page doi:10.1109/10821315