Boosting Photon-Number-Resolved Detection Rates of Transition-Edge Sensors by Machine Learning

Aonan Zhang; Ewan Mer; Gerard J. Machado; Hao Zhan; Ian A. Walmsley; Matthew J.H. Kendall; Raj B. Patel; Ruidi Zhu; Shang Yu; Zhenghao Li

REVIEW 3 major objections 4 minor 51 references

Boosting Photon-Number-Resolved Detection Rates of Transition-Edge Sensors by Machine Learning

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Two machine-learning classifiers let transition-edge sensors resolve photon numbers at 800 kHz, four times their intrinsic thermal recovery rate, while preserving accurate photon-number assignment up to at least five photons.

desk verdict A practical four-fold TES speed-up with honest benchmarking; the main caveat is the synthetic-training assumption and missing error bars on the headline TVD plot. read the letter →

arxiv 2411.15360 v1 pith:BLF5N6IR submitted 2024-11-22 quant-ph physics.ins-det

Zhenghao Li , Matthew J.H. Kendall , Gerard J. Machado , Ruidi Zhu , Ewan Mer , Hao Zhan , Aonan Zhang , Shang Yu

show 2 more authors

Ian A. Walmsley Raj B. Patel

This is my paper · ORCID

classification quant-phphysics.ins-det

keywords transition-edgesensorphoton-numberresolutionmachinelearningK-nearestneighboursHDBSCANprincipalcomponentanalysiscoherentstatessqueezedlight

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 machine-learning signal processing can overcome the main practical limitation of transition-edge sensors (TESs) for photon-number-resolving detection: their slow thermal recovery, which normally forces operation below a few hundred kilohertz. The authors propose a supervised classifier (K-nearest neighbours, KNN) trained on low-rate voltage traces that are computationally overlapped to emulate high-rate pile-up, and an unsupervised clustering method (HDBSCAN) that groups PCA-reduced traces. Benchmarking with coherent and squeezed light, they report that the supervised method maintains accurate photon-number assignment up to at least five photons at 800 kHz, a four-fold improvement, and that the unsupervised method resolves up to ten photons for thermal states at the same rate. A sympathetic reader would care because this is a hardware-agnostic route to faster PNR detection for quantum information and metrology applications.

What carries the argument

The load-bearing objects are (1) the voltage trace, a digitized time series of the SQUID-amplified TES response to one light pulse; (2) principal component analysis, which reduces each trace to a small set of factor scores and separates photon-number clusters in two dimensions; and (3) the two classifiers: KNN, which labels a new trace by majority vote of its five nearest labelled neighbours in Euclidean distance, and HDBSCAN, a hierarchical density-based clustering algorithm that identifies high-density regions and labels low-density points as noise. The critical training construction is the overlap of 100 kHz traces to emulate higher repetition rates, which transfers reliable low-rate labels to the high-rate regime.

What would settle it

Measure the raw voltage traces at 800 kHz from a source whose photon numbers are independently known (e.g., a heralded single-photon source that provides a ground-truth tag per pulse), then compare the KNN labels to the heralded truth. If the KNN labels systematically overcount or degrade for exactly those traces that show the flattening seen in the supplementary material, the training emulation missed a nonlinearity that undermines the reported accuracy.

Watch

Extended reading notes

Core claim

The central claim is that photon-number assignment for TES signals does not require waiting for full thermal recovery if the classifier accounts for the influence of preceding pulses. The supervised method builds a labelled training set by taking 100 kHz traces, whose photon numbers are reliably assigned by an inner-product method, and overlapping them to synthesize traces at the target repetition rate; a KNN classifier then labels new high-rate traces. The unsupervised method projects traces onto their first two principal components and clusters them with HDBSCAN, which works well for squeezed-light thermal distributions even at 800 kHz. Tomographic reconstruction of the detector's POVM confirms that the KNN method's diagonal terms remain accurate up to at least five photons at 800 kHz, and the authors state the fidelity of the reconstructed POVM is on par with the low-rate reference.

Load-bearing premise

The supervised method's training assumes the TES response is linear and additive, so that overlapping 100 kHz voltage traces faithfully emulates the detector's response at 800 kHz; the paper's own supplementary data show that at 900 kHz and 1 MHz the traces flatten, indicating this assumption breaks down at higher rates.

Editorial extensions

If this is right

TES-based experiments that currently demultiplex or discard pulses to stay within the thermal recovery window could instead run at 800 kHz without adding hardware overhead.
The methods are hardware-agnostic, so they can be combined with faster TES designs that trade photon-number resolution for speed, potentially pushing the useful rate further.
Accurate PNR at 800 kHz enables fast active feed-forward of detection outcomes, a requirement for non-Gaussian state preparation and other conditional quantum operations.
The unsupervised HDBSCAN method provides training-free classification, which is useful when calibration data at low rate is unavailable or noisy, at the cost of leaving some traces unclassified.
Reproducible code is released with the paper, so other groups can test the classifiers on their own TES data.

Reading between the lines

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

The training-emulation assumption—that overlapping low-rate traces reproduces high-rate responses—sets a practical ceiling: the paper itself notes that at 900 kHz and 1 MHz the traces flatten, suggesting nonlinear thermal saturation begins around 800 kHz; a similar approach might work for other slow thermal detectors (e.g., kinetic inductance detectors) if the same additivity holds.
The reported 93.3% detection efficiency is calibrated against an uncalibrated power meter, so absolute photon-number accuracy inherits that uncertainty; the POVM comparison against the 100 kHz reference sidesteps this, but absolute efficiency claims should be treated with caution.
If the method holds for other TES geometries and materials, it could increase the data-acquisition rate in photon-starved applications such as bio-imaging and astrophysics, where the detector's photon-number resolution is currently rate-limited.

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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

Let me give you the quick version. This paper shows a practical way to speed up TES photon-number-resolving detection by a factor of four (to 800 kHz) using machine learning on overlapping pulses. The supervised KNN method, trained on overlapped low-rate traces, works for coherent states up to about five photons; the unsupervised HDBSCAN method works for thermal/squeezed states. The central claim is believable, and the benchmarking is more careful than most: POVM reconstruction with bootstrap error bars, heralded Fock state tests, and honest discussion of where each method fails.

What's genuinely new is the specific combination: using overlapped 100 kHz traces as a synthetic training set to emulate higher rates, then applying KNN for coherent and HDBSCAN for thermal states. The paper also compares several classifiers (RF, SVM, GBDT, CNN) and reports timings. That's useful practical information for anyone building fast PNR systems.

The soft spots are real but not fatal. The training-set construction assumes linear superposition of traces. The supplement shows trace flattening at 900 kHz and 1 MHz, and attributes it to accumulated heat driving the film normal. The question is whether that effect begins to bite already at 800 kHz for the higher photon numbers in the n≤5 claim. The TVD benchmark in Fig. 4 has no error bars, so we don't know the uncertainty on the 800 kHz point. The POVM diagonals have bootstrap errors, but those errors reflect the fit, not the training-assumption mismatch. There's also a mild circularity: the KNN is trained on IP labels at 100 kHz and compared against the IP at 100 kHz as reference, so any IP bias is inherited. That's a calibration choice, not an invalid loop, but it means the reported accuracy is relative to the IP method, not absolute.

The abstract oversells the unsupervised method slightly: HDBSCAN fails for coherent light at 800 kHz, and the general statement 'extend the TES operation rate to 800 kHz, maintaining accurate assignment up to at least five photons' is really only true for the supervised method on coherent states and for the unsupervised on thermal states. The body is clear about this; the abstract is a bit too compressed.

Data are not public, only the code on GitHub without a commit hash. That limits reproducibility but doesn't undermine the claims.

Bottom line: this is a solid, useful paper for anyone working with TES detectors or photonic experiments that need fast PNR. The core result is credible, the limitations are mostly disclosed, and the presentation is honest. It deserves a proper peer review; I'd send it out with a request to add error bars to the TVD plots and to comment explicitly on the linearity assumption at 800 kHz. I'd also suggest slightly softening the abstract about the unsupervised method. But these are revisions, not rejection reasons.

Referee Report

3 major / 4 minor

Summary. This manuscript reports two machine-learning post-processing methods for photon-number assignment from transition-edge sensor (TES) voltage traces at repetition rates above the detector's thermal recovery limit. The supervised method trains a K-nearest-neighbour classifier on synthetic high-rate traces formed by overlapping 100 kHz calibration traces labelled by the inner-product method; the unsupervised method applies HDBSCAN clustering to the first two PCA factor scores. The authors benchmark against coherent states using total variation distance and POVM reconstruction, and against two-mode squeezed vacuum light using heralded Fock-state preparation, claiming a four-fold increase in operation rate to 800 kHz while preserving accurate photon-number assignment up to at least five photons.

Significance. If the 800 kHz claim holds, the result is practically valuable: it offers a hardware-agnostic way to extend the rate of TES-based photon-number-resolving detection without modifying the detector, which could benefit photonic quantum information experiments that require fast feed-forward. The paper has clear strengths: the code is publicly available, the POVM reconstruction includes bootstrap error bars in Fig. 5, and the squeezed-light benchmark provides an independent test on a different photon-number distribution. The methods themselves are standard ML tools, so the novelty is mainly in the application and the careful benchmarking rather than in new algorithmic ideas, but the demonstrated four-fold rate improvement is a useful experimental contribution.

major comments (3)

[Section 3.1 and Supplement SII] The central 800 kHz claim rests on the training-set construction in Section 3.1, where 100 kHz voltage traces are overlapped to emulate the detector response at the target repetition rate. This assumes linear, additive detector response. Supplement SII explicitly states that at 900 kHz and 1 MHz the voltage traces flatten and that the training step does not account for this change, and it identifies accumulated heat as a possible cause. The authors should provide direct evidence that this effect is absent or negligible at 800 kHz for the photon-number range n<=5. A concrete test would be to compare the PCA distribution of synthetic training traces with the PCA distribution of real 800 kHz traces, or to report a quantitative mismatch metric. Without such a check, the reconstructed POVM diagonals at 800 kHz could be optimistic if nonlinear thermal saturation already begins at that rate.
[Section 4.1.1, Fig. 4] The TVD curves in Fig. 4 are the primary quantitative evidence that the supervised methods maintain accuracy up to 800 kHz, yet they are plotted without error bars, confidence intervals, or bootstrap estimates. The differences between methods at a given rate are therefore difficult to judge: a TVD value of 0.05 may be within statistical uncertainty of a value of 0.10 if the sample sizes are not reported. The authors should add error bars or state the number of independent repetitions used to generate each curve, especially for the KNN and IP curves where the improvement is claimed to be significant.
[Section 4.1.2, Eq. (3)] The fidelity metric in Eq. (3) compares the reconstructed POVM at high repetition rates against a reference POVM obtained from the same inner-product method at 100 kHz that also supplies the KNN training labels. This means the fidelity is partly a self-consistency check between the classifier and the IP labelling procedure, not an absolute calibration of photon-number assignment accuracy. The paper acknowledges this choice, but the limitation should be stated explicitly in the main text. In addition, the diagonal POVM terms in Fig. 5(a-c) show only the reconstructed conditional probabilities; reporting the associated posterior assignment probabilities or a separate absolute calibration with a calibrated power meter would strengthen the claim.

minor comments (4)

[Fig. 4 caption] The caption contains a typo: the third panel is labelled '(a) mu=5.29' instead of '(c) mu=5.29'. Please correct this.
[Fig. 3 caption] The caption states that KNN classifies traces up to 13 photons at 500 kHz and 12 photons at 800 kHz, but the POVM reconstruction in Section 4.1.2 claims accurate assignment only up to five photons at 800 kHz. The distinction between visible cluster separation and calibrated POVM accuracy should be clarified in the caption or the main text.
[Section 5, Conclusions] The sentence 'This is verified against the tomographic reconstruction of the TES's POVM at 100 kHz' is slightly ambiguous because the verification is against the 100 kHz POVM used as a reference, not an independent absolute POVM. Rephrasing to 'This is benchmarked against the tomographic reconstruction obtained at 100 kHz' would be clearer.
[Abstract] The abstract states 'at least a four-fold improvement' without specifying the baseline. The introduction clarifies that the intrinsic thermal recovery limit is about 200 kHz, so 800 kHz is four-fold, but stating this baseline in the abstract would make the claim self-contained.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the shared IP labeling source between training and benchmark creates a mild validation dependency, but the central claim is not reduced to its inputs.

full rationale

The paper's core validation chain is not circular by construction. The supervised KNN classifier is trained on 100 kHz voltage traces whose labels are assigned by the inner-product (IP) method, and the main coherent-state benchmarks (TVD in Sec. 4.1.1 and POVM fidelity in Sec. 4.1.2) use the IP reconstruction at 100 kHz as the reference. This shared label source means the high-rate fidelity partly measures agreement with the IP labeling rule rather than with an independent photon-number ground truth, and any systematic IP bias at 100 kHz would be inherited by the classifier. However, the KNN predictions are evaluated on real 800 kHz traces, not on the synthetic overlapped training traces, and no parameter is fitted to the benchmark target. The unsupervised HDBSCAN demonstration on squeezed light and the heralded-Fock preparation benchmarks provide independent support, and the POVM reconstruction uses power-meter-calibrated coherent-state amplitudes as external ground truth. Supplement SII explicitly concedes that the linear-overlap training step does not account for the trace flattening observed at 900 kHz and 1 MHz; this is a correctness caveat at the margin of the 800 kHz claim, not a circularity. Citations to prior work by the same group (e.g., Refs. [22], [31], [46]) are standard methodological references and are not load-bearing uniqueness claims. Score 2 reflects the mild shared-label validation dependency, not circular derivation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the validity of IP labels at 100 kHz, on the linear-overlap assumption for synthesizing high-rate training data, and on calibration of the coherent state amplitudes via an uncalibrated power meter. No new physical entities are introduced.

free parameters (3)

KNN number of neighbours K = 5
Hand-chosen in Section 3.1; classification results depend on it.
HDBSCAN hyperparameters = not fully specified
Minimum cluster size, minimum number of neighbours, and minimum distance are tuned per dataset (Sections 3.2 and 4.1.1) but exact values are not given.
Number of retained PCA components = 2
PCA is truncated to D=2 for visualization and clustering; higher-order components are discarded.

assumptions (5)

standard math PCA/SVD decomposition of voltage traces captures the relevant clustering structure with D=2 components.
Section 2.2 invokes singular value decomposition and truncates the sum in Eq. 1 to two principal components.
domain assumption The inner-product (IP) method at 100 kHz assigns accurate photon-number labels.
IP labels at 100 kHz are used as the reference distribution and as training labels for the supervised method (Sections 2.1 and 3.1).
domain assumption Coherent states from the attenuated laser diode follow a Poissonian photon-number distribution with mean set by a calibrated power meter.
Used for POVM reconstruction in Section 4.1.2 and for computing ground-truth distributions.
ad hoc to paper Overlapping 100 kHz voltage traces linearly emulates the detector response at 500 and 800 kHz.
This is the training-data construction in Section 3.1; Supplement SII shows the assumption fails above 800 kHz due to signal flattening.
domain assumption The power meter reading uncertainty (2.6%) and linearity (0.5%) bound the calibration error, despite the meter not being recalibrated.
Supplement SI uses these manufacturer specifications and acknowledges the calibration uncertainty could be higher.

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Cite this review

Pith. "Pith review of Boosting Photon-Number-Resolved Detection Rates of Transition-Edge Sensors by Machine Learning." pith.science (2026). https://pith.science/paper/BLF5N6IR

@misc{pith2026241115360,
  author       = {Pith},
  title        = {Pith review of: Boosting Photon-Number-Resolved Detection Rates of Transition-Edge Sensors by Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLF5N6IR}},
  note         = {Machine review of arXiv:2411.15360}
}

read the original abstract

Transition-Edge Sensors (TESs) are very effective photon-number-resolving (PNR) detectors that have enabled many photonic quantum technologies. However, their relatively slow thermal recovery time severely limits their operation rate in experimental scenarios compared to leading non-PNR detectors. In this work, we develop an algorithmic approach that enables TESs to detect and accurately classify photon pulses without waiting for a full recovery time between detection events. We propose two machine-learning-based signal processing methods: one supervised learning method and one unsupervised clustering method. By benchmarking against data obtained using coherent states and squeezed states, we show that the methods extend the TES operation rate to 800 kHz, achieving at least a four-fold improvement, whilst maintaining accurate photon-number assignment up to at least five photons. Our algorithms will find utility in applications where high rates of PNR detection are required and in technologies which demand fast active feed-forward of PNR detection outcomes.

Figures

Figures reproduced from arXiv: 2411.15360 by the authors.

**Figure 2.** TES and PCA data for coherent states measured at rates 100 kHz (left panels), [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Classification of coherent light pulses at 500 kHz (which are shown in the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Detection of coherent light with varying mean photon number per pulse, (a) [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: The reconstructed confusion matrix, 𝚯 by the IP and KNN method at (a) 100 kHz, (b) 500 kHz, (c) 800 KHz. Bar plots of the diagonal elements are also plotted. The error bars are estimated by bootstrapping over the uncertainties of the power meter readings and Poissonian…

**Figure 6.** Figure 6: (a) Density scatter plot of the PCA factor scores for light pulses in a thermal [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: The photon-number distribution of the signal mode whilst heralding on (a) one, [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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