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Validation tests of Gaussian boson samplers with photon-number resolving detectors

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that grouped count probabilities for photon-number-resolving Gaussian boson samplers can be simulated efficiently with the positive-P phase-space method, and that applying the test to large recent data shows disagreement…

desk verdict A practically useful, honest extension of GCP validation to PNR detectors; the central discrepancy claim is robust, while the fitted thermal model is clearly labeled and fails in higher dimensions. read the letter →

arxiv 2411.11228 v2 pith:NQQEPP6P submitted 2024-11-18 quant-ph

Alexander S. Dellios , Margaret D. Reid , Peter D. Drummond This is my paper · ORCID
classification quant-ph
keywords Gaussianbosonsamplingpositive-Pphase-spacerepresentationgroupedcountprobabilitiesphoton-number-resolvingdetectionquantumadvantagevalidationthermalizedsqueezedstatesstatisticalhypothesistesting
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

This paper claims that the positive-P phase-space representation can compute grouped count probabilities for Gaussian boson sampling with photon-number-resolving detectors far faster than direct classical simulation, making partial validation of large experiments practical. Applied to recent 72-, 216-, and 288-mode data, the method shows that the experimental counts as a whole disagree with the ideal pure-squeezed-state ground truth. Adding two corrections, a thermalization fraction and a transmission-matrix measurement correction, brings the one-dimensional total-count distribution into agreement within sampling error for the larger data sets. The same corrections fail for two-dimensional grouped counts and per-mode photon-number moments, indicating further systematic errors that the small number of experimental samples cannot resolve. The authors propose using such tests as feedback to improve Gaussian boson sampling hardware.

What carries the argument

The engine is the positive-P phase-space distribution paired with grouped count probabilities, or GCPs. The positive-P representation expands any density operator as a positive distribution over pairs of independent coherent-state amplitudes, so normally ordered expectation values, including photon-counting projectors of all orders, become ordinary moments that can be estimated by random sampling. A GCP is the probability that the total photon number in each of $d$ subsets of output modes takes specified values; in one dimension it is the total count distribution, and as $d$ approaches the mode number the GCP converges to the full Hafnian distribution, where the Hafnian is the matrix function whose evaluation for a count pattern is #P-hard. The positive-P samples are generated from Gaussian inputs through a quadrature-variance formula, with thermalized squeezed states parameterized by $\epsilon$, and propagation through the lossy transmission matrix is done at the amplitude level.

What would settle it

Compute exact grouped count probabilities for the actual 16-mode transmission matrix using exact Hafnian evaluation, and compare them with positive-P GCPs; a mismatch beyond the quoted sampling error would show the method does not transfer to the structured network, and this check is feasible because the 16-mode data set is small enough for exact probabilities.

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Extended reading notes

Core claim

The central claim is that binning a photon-number-resolving Gaussian boson sampling output into grouped count probabilities turns an exponentially hard validation problem into an efficiently samplable one, provided the normally ordered positive-P distribution is used. The paper generalizes Mandel's binning of classical photon statistics to arbitrary quantum states: each stochastic sample of the positive-P distribution directly estimates a multidimensional grouped count probability, so the method captures arbitrarily high-order correlations without computing a single Hafnian. In the loss-dominated regime of current experiments the sampled moments converge to exact distributions with errors below $10^{-3}$. When the same simulation is run on the published transmission matrix of a 216- and 288-mode processor, the ideal squeezed-state ground truth is rejected by chi-square and Z-statistic tests, while a thermalized ground truth with two fitted parameters ($\epsilon$ and $t$) reproduces the one-dimensional total count distribution within sampling error. The paper therefore establishes that current data are inconsistent with the ideal target distribution, that low-dimensional tests can be passed by a decoherent effective theory, and that higher-dimensional and per-mode tests expose residual systematic errors.

Load-bearing premise

Everything rests on assuming the positive-P sampling errors measured on uniform-loss random networks transfer to the real processor's structured, mode-dependent-loss network—and, for the 288-mode data, that a privately corrected ideal ground truth is the right comparison.

Editorial extensions

If this is right

  • One-dimensional GCP validation runs in minutes on a desktop for 288 modes, a speed-up the paper estimates at more than $10^{18}$ over Hafnian-based direct simulation.
  • Every analyzed data set is at least 66 standard deviations from the ideal pure-squeezed ground truth in the one-dimensional total-count test.
  • A thermalized state with $\epsilon \approx 0.05$ and a transmission correction $t \approx 0.98$--$0.99$ brings the total-count distribution within sampling error for the 72-, 216-LS, 216-HS, and 288-mode data sets.
  • The same corrections fail to reconcile two-dimensional grouped counts and per-mode photon-number moments, and the 16-mode data set with the most samples remains far from both ideal and corrected theory.
  • The authors propose using these validation tests as feedback to adjust experimental parameters, which they suggest may be a more practical route to quantum advantage than hardware improvements alone.

Reading between the lines

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

  • Because one-dimensional counts can be fitted by two parameters while two-dimensional counts cannot, the real error model is likely underdetermined by total-count data alone; mode-dependent losses or phase noise would be needed to explain the higher-order discrepancies. (Editorial inference.)
  • The periodic structure visible in per-mode photon-number moments points to the time-multiplexed network layout, and a classical sampler exploiting that structure might pass one-dimensional validation tests even where the full distribution is nonclassical. (Editorial inference.)
  • The same generalized binning machinery applies to any nonclassical input state, not only squeezed states, so the validation protocol could be exported to other photon-counting devices beyond Gaussian boson sampling. (Editorial inference.)
  • A natural next step would be to scan a wider parameter space, including mode-dependent thermalization and per-mode transmission corrections, and test whether any higher-dimensional model can simultaneously fit the one- and two-dimensional GCPs before claiming further systematic errors. (Editorial inference.)
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper extends the positive-P phase-space representation method to compute grouped count probabilities (GCPs) for Gaussian boson sampling with photon-number-resolving (PNR) detectors. The method is validated against exact total-count distributions for Haar-random unitary networks with uniform amplitude loss, and its numerical scaling is measured for d=1,2,3 GCPs on up to 480 modes. The method is then applied to the Borealis experimental data (16, 72, 216-LS, 216-HS, and 288 modes). The authors report that the experimental data are far from the ideal pure-squeezed ground truth (Z_EI between 66 and 167 for the quantum-advantage-relevant sets), and that a two-parameter modification of the ground truth—a thermalization fraction ϵ and a transmission-matrix correction t—brings the one-dimensional total-count GCPs within sampling error for the 72, 216-LS, 216-HS, and 288-mode data sets. However, the same model fails to describe two-dimensional GCPs and per-mode photon-number moments, which the authors attribute to additional systematic errors.

Significance. If the claims hold, the paper provides a scalable and computationally efficient validation toolkit for GBS experiments with PNR detectors, with claimed speedups of order 10^18 over direct Hafnian-based simulation. The central discrepancy result—that the experimental data deviate strongly from the ideal pure-squeezed ground truth—is parameter-free, consistent across multiple data sets, and robust to the sampling-error concerns. The paper also provides an open-source implementation (xqsim) and makes concrete falsifiable predictions, namely that a two-parameter thermalized model can capture the d=1 total-count distributions but not the d=2 or per-mode statistics. These strengths make the paper a useful contribution to the validation methodology for photonic quantum advantage experiments.

major comments (3)
  1. [§IV.B, §V.D, Appendix B] The positive-P sampler is validated only against Haar-random unitary networks with uniform loss (tU) in Section IV.B. The Borealis simulations in Section V use the actual structured time-domain-multiplexed T-matrix with mode-dependent losses described in Section V.D, and the error bars entering the χ² tests are the sub-ensemble estimates σ_T,i from Appendix B (Eq. 29). If σ_T,i is underestimated for this non-uniform structured network, the reported |Z_ET| ≈ 1 agreement in Table II would be inflated. Please provide a direct convergence test on a Borealis-like structured matrix—for example, by computing exact GCPs for a small sub-network with the experimentally measured T-matrix and squeezing vector and comparing them to positive-P results—or explicitly discuss how sensitive the ET agreement is to the accuracy of the sampling-error estimates.
  2. [§V.B, Table I, Ref. [53]] The ideal ground truth for the 288-mode data set is corrected based on private communication (Ref. [53]), and the corrected distribution is reported to be closer to the experimental data than the distribution originally published in the Borealis paper. This correction is not publicly verifiable, yet it directly affects the Z_EI = 167 result in Table I and the associated discrepancy claim for the largest quantum-advantage data set. Please make the corrected ground-truth distribution publicly available (e.g., as a data file in the repository) or provide a detailed reproducible derivation so that this result can be independently checked.
  3. [§V.B, Table II, §V.C] The two-parameter (ϵ, t) modified ground truth is fitted separately to each data set using a Nelder-Mead simplex algorithm, but the paper does not specify the cost function minimized, the initial values and convergence criteria, or how the quoted ±0.0005 uncertainties on ϵ and t were obtained. Since the ET agreement in Table II is a central claim, please provide these details, along with confidence intervals or a χ² surface plot, so the reader can judge whether the fit is well constrained and whether the agreement is a meaningful test or an over-parameterized post-hoc fit.
minor comments (5)
  1. [§V.B] The statement that the 288-mode distribution is '98σN further' from the expected mean than the 216-mode HS distribution is arithmetically inconsistent with the reported Z_EI values of 70 and 167; the difference is 97σN, so '97σN' would be correct.
  2. [Abstract and Conclusion] The abstract states a speedup of '~10^18 times faster' without specifying the baseline; the Conclusion clarifies that this is relative to direct Hafnian computation on Fugaku. Please state the baseline in the abstract for clarity.
  3. [§III.B, Eq. (15), Eq. (18), Eq. (21), Appendix B] Several LaTeX/OCR artifacts appear in the text, including '/dispiint' in Eq. (15) and Eq. (21), and '/radicaltp' in Appendix B. These should be corrected to the intended integral and square-root symbols.
  4. [Table II] The caption states that ϵ and t each have error bars of ±0.0005 for all data sets, but the origin of these error bars is not described. Please clarify whether these are fit uncertainties, estimated from the spread over the ten simulation runs, or derived from the χ² surface.
  5. [§IV.A, Eq. (24)] The notation f_c = 2F1(a,b;c;z) is used without defining the hypergeometric parameters a, b, c explicitly; Eq. (24) would be easier to follow if the arguments of the hypergeometric function were written out.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ideal-ground-truth discrepancy is parameter-free, the modified-model agreement is an explicitly fitted d=1 result, and the d=2 and moment tests are out-of-sample and fail.

full rationale

The paper's central discrepancy claim is not circular: the ideal ground truth is computed from the experimental lossy transmission matrices and squeezing vectors ('The ideal ground truth distributions for each data set are formulated from their corresponding lossy transmission matrices T and simulated pure squeezed states using squeezing vectors r'), with no parameters fitted to the Borealis output. The positive-P sampling method is validated against independent exactly-known distributions for uniform-loss Haar-random networks in Section IV.B, where the simulated moments converge to Eq. (24) with errors below 10^-3; this is an external check, not a self-referential one. The improved agreement of the thermalized model is explicitly obtained by fitting: Table II states that 't and ϵ fitting parameters are found using a Nelder-Mead simplex algorithm which minimizes the distance between experimental and theoretical distributions,' and the quoted χ2_ET/k and ZET values are in-sample qualities of that fit, not predictions. The paper does not rename this fit as a prediction; it calls it a 'small modification of the GBS parameters [that] greatly improves agreement for some tests.' Moreover, the paper reports out-of-sample failures using the same fitted parameters: the d=2 grouped-count comparisons remain far from the thermalized ground truth (e.g., ZET = 9 ± 1 for 216-mode HS), and the per-mode photon-number moments also fail (ZET ≈ 45 for 216-mode HS). These failures are genuinely predictive and reduce any concern that the fitted d=1 agreement is being presented as forced evidence. The reliance on private communication [53] for the 288-mode ideal distribution is a reproducibility weakness, but it is not circularity, because the 288-mode discrepancy is consistent with the parameter-free discrepancies seen in all other data sets. Overall, no load-bearing step reduces by construction to its inputs.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the positive-P representation's exactness, the Hafnian formula for GBS, and the specific two-parameter thermalization/transmission model. The two parameters are fitted to the d=1 data, making the d=1 agreement in-sample. No new physical entities are introduced.

free parameters (2)
  • thermalization fractions ǫ (per data set) = 216HS: 0.0510; 288: 0.0547; 72: 0.0507; 216LS: 0.0105; 16: 0.0648 (±0.0005)
    Fitted via Nelder-Mead to minimize d=1 GCP distance (Table II); introduce decoherence into the input squeezed states.
  • transmission corrections t (per data set) = 216HS: 0.9941; 288: 0.9848; 72: 0.9861; 216LS: 0.9882; 16: 0.9841 (±0.0005)
    Fitted together with ǫ; multiplies the experimental T-matrix to model measurement errors (Table II).
assumptions (5)
  • standard math The positive-P representation provides exact normally-ordered moments for any quantum state.
    Used to derive Eq. (18) for GCPs; standard result (Refs [18,42]).
  • standard math The GBS output probability for PNR detectors is given by the Hafnian formula (Eq. (7)).
    Used to compare with exact distributions and define the ground truth.
  • ad hoc to paper Experimental decoherence can be modeled by a single thermalization fraction ǫ reducing the coherence parameter, and transmission errors by a single scalar t multiplying the T-matrix.
    Introduced in Eqs. (3) and Section V.B; this two-parameter model is not derived from first principles and is shown to be insufficient for d=2 and moment tests.
  • domain assumption Experimental sampling errors of GCPs follow Poissonian statistics σ_e,i = sqrt(G_S,i).
    Stated in Appendix B; validated indirectly for moments using a classical thermal-state test, but not for GCPs.
  • standard math The Wilson-Hilferty transformation makes χ²/k approximately normal for k≥10.
    Used to define Z-scores in Section V.A.

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Cite this review

Pith. "Pith review of Validation tests of Gaussian boson samplers with photon-number resolving detectors." pith.science (2026). https://pith.science/paper/NQQEPP6P

@misc{pith2026241111228,
  author       = {Pith},
  title        = {Pith review of: Validation tests of Gaussian boson samplers with photon-number resolving detectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQQEPP6P}},
  note         = {Machine review of arXiv:2411.11228}
}
read the original abstract

An important challenge with the current generation of noisy, large-scale quantum computers is the question of validation. Does the hardware generate correct answers? If not, what are the errors? This issue is often combined with questions of computational advantage, but it is a fundamentally distinct issue. In current experiments, complete validation of the output statistics is generally not possible because it is exponentially hard to do so. Here, we apply phase-space simulation methods to partially verify recent experiments on Gaussian boson sampling (GBS) implementing photon-number resolving (PNR) detectors. The positive-P phase-space distribution is employed, as it uses probabilistic sampling to reduce complexity. It is \sim10^{18} times faster than direct classical simulation for experiments on 288 modes where quantum computational advantage is claimed. When combined with binning and marginalization to improve statistics, multiple validation tests are efficiently computable, of which some tests can be carried out on experimental data. We show that the data as a whole shows discrepancies with theoretical predictions for perfect squeezing. However, a small modification of the GBS parameters greatly improves agreement for some tests. We suggest that such validation tests could form the basis of feedback methods to improve GBS experiments.

Figures

Figures reproduced from arXiv: 2411.11228 by the authors.

Figure 1
Figure 1. Validation of the positive-P GCP distributions [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Run-times in seconds versus mode number of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Comparisons of one-dimensional GCPs G (M) {1,...,M} (m) versus grouped counts m for experimental data obtained from the 216-mode HS Borealis experiment. a) Data is compared to positive-P simulations with pure squeezed state inputs, where simulations are performed for an ensemble size of ES = 1.2 × 106 , and the corresponding normalised difference between distributions is plotted in b). In c) experimental patterns ar… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Description follows Fig.(3) except experimental [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Surface plots of two-dimensional GCP comparisons [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Description follows Fig.(5) however in this case e [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 8
Figure 8. Figure 8: Figure description follows Fig.(7), except data [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: The exact mean photon number (solid black line) [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

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Forward citations

Cited by 1 Pith paper

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    The authors introduce a blocked loop Hafnian and finite-difference sieve that compute coarse-grained photon-number distributions of Gaussian states in exponential, not combinatorial, time.

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