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REVIEW 4 major objections 4 minor 22 references

Computed models of natural radiation backgrounds in qubits and superconducting detectors

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

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Pith's one-line read Natural radiation backgrounds in superconducting qubit substrates are summarized by a single scaling law that gives three event rates to roughly ±25 percent accuracy.

desk verdict Useful compact background-rate formulas, but the elevation scale height λ is never given, so Eq. 1 cannot be evaluated at nonzero altitude as published. read the letter →

arxiv 2411.16974 v1 pith:FCF5BD2J submitted 2024-11-25 quant-ph physics.ins-det

Joseph Fowler , Ian Fogarty Florang , Nathan Nakamura , Daniel Swetz , Paul Szypryt , Joel Ullom This is my paper · ORCID
classification quant-phphysics.ins-det
keywords superconductingqubitsnaturalradiationbackgroundscosmicraysterrestrialgammaenergydepositioninsubstratesscalinglawskineticinductancedetectorMonteCarloparticletransport
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

Natural radioactivity in concrete and cosmic rays from the atmosphere deposit energy in the silicon and compound-semiconductor substrates that hold superconducting qubits, producing the correlated decoherence events that currently limit large processors. This paper tries to turn that problem into a small set of numbers: it claims that, for any substrate area, thickness, material, ceiling thickness, and laboratory elevation, three rates---the rate of any energy-depositing event $R$, the rate of events above 1 MeV $M$, and the total deposited power $P$---are captured by one analytic formula, Eq. (1), with the parameters listed in Table II. If the formula is right, a researcher can estimate background levels to roughly $\pm25\%$ over realistic conditions without running Monte Carlo simulations. The modeled rates and energy-deposited spectrum are consistent with the authors' earlier measurement with a silicon thermal kinetic-inductance detector at 1640 m elevation.

What carries the argument

The load-bearing object is the three-rate summary and the closed-form scaling law of Eq. (1). The rates are: $R$, the rate of any event depositing energy $E>0$; $M$, the rate of events depositing $E>1$ MeV, chosen because it marks the transition from terrestrial-gamma and charged-particle events to proton and neutron events and corresponds to roughly one event per hour in the nominal substrate; and $P$, the total power deposited. Equation (1) is separable: an area factor $A/(100\,\mathrm{mm}^2)$, a sum over six background sources of terms $c_s+g_s\tau$, $p_s\tau^{\beta_s}$, and $m_s\tau^{\alpha_s}$, and corrections $\kappa_c\kappa_{sh}\kappa_\rho$ for ceiling, shape, and density. This factorization is what carries the argument, because it reduces a six-dimensional simulation space to a table of constants that any user can evaluate directly.

What would settle it

Measure the three rates $R$, $M$, and $P$ in a 10 mm by 10 mm, 500 µm silicon substrate at sea level under roughly 20 cm of concrete, using a detector that records the full deposited-energy spectrum; if the measured values differ from Eq. (1) with Table II by more than the claimed ±25 percent, or if the thickness dependence of $M$ departs from the quoted power laws, the central claim would be undercut.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that all simulated background behavior in superconducting-circuit substrates can be summarized by Eq. (1), which expresses the three summary rates as products of the wafer area, a sum over six source terms (the $^{40}$K, $^{232}$Th, and $^{238}$U decay chains, with the uranium and thorium chains each split into two halves around radon, plus cosmic rays), and three multiplicative corrections for ceiling, shape, and density. Each source term is proportional to its relative activity $\tilde a_s$; the gamma contributions scale with thickness $\tau \equiv t/500\,\mu\text{m}$ as $c_s + g_s\tau$ for $R$ and as power laws $\tau^{\beta_s}$ and $\tau^{\alpha_s}$ for $P$ and $M$, while the cosmic-ray contribution grows with elevation as $\exp(H/\lambda)$ with different scale heights for different particle species. The paper states that the absolute rates are accurate to about $\pm25\%$ over a range of realistic conditions, and that the model reproduces the measured event-rate spectrum of the authors' thermal kinetic-inductance detector.

Load-bearing premise

The fragile premise is that the simulated backgrounds, built from standard cosmic-ray spectra and typical European concrete, match real labs closely enough that the scaling formulas hold to the claimed 25 percent for any substrate and elevation, even though only one measurement is used for comparison.

Editorial extensions

If this is right

  • Any lab can compute approximate background rates for its own substrate geometry, material, shielding, and elevation by evaluating Eq. (1) with Table II, without specialized simulation expertise.
  • Thinning the substrate is an effective lever: the high-energy event rate scales as $M_\gamma \propto \tau^5$ for gamma rays and as $\tau^{1.8}$ for cosmic rays, so a thinner wafer suppresses the rare damaging events far more than it suppresses the total rate.
  • Adding concrete ceiling reduces cosmic-ray $R$ and $P$ by only about 2% per 10 cm, so overhead shielding is not a strong mitigation against cosmic-ray backgrounds.
  • The $M$ rate in the nominal 500 µm silicon substrate is about once per hour, the level flagged as a concern for quantum error correction.
  • Denser and gallium-containing substrates have higher rates; for gallium the gamma-ray rates scale as if the density were increased by a source-dependent amount, and high-energy events scale roughly as $\tilde\rho^{2.7}$.

Reading between the lines

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

  • Extension: if Eq. (1) generalizes as claimed, a lab can identify its dominant background term without simulation---at sea level terrestrial gamma rays set the low-energy rate, while the once-per-hour MeV events are mostly cosmic-ray protons and neutrons---so the relative benefit of underground siting versus additional local shielding follows directly from the two scale heights.
  • Extension: the gamma-ray $M_\gamma \propto \tau^5$ exponent is estimated from very few simulated events; a testable consequence is that measuring MeV events in substrates of thickness 30, 100, and 500 µm should reveal this steep dependence, and if it does not, the exponent needs revision.
  • Extension: because the nominal concrete activities are European averages, labs using unusual aggregates or with elevated radon should expect systematic offsets; assaying local concrete and rescaling via $\tilde a_s$ is a natural extension of the formula.
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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

4 major / 4 minor

Summary. The paper uses Geant4/TOPAS with PARMA cosmic-ray spectra to simulate energy deposition from natural radiation (terrestrial gamma rays and cosmic rays) in silicon and other substrates typical of superconducting qubits and detectors. It distills the simulation output into three rates—R (any energy event), M (events >1 MeV), and P (total deposited power)—and packages them in Eq. (1) with parameters in Table II, intended to let experimenters estimate background rates as functions of substrate area, thickness, material, elevation, and ceiling thickness. The authors report consistency with their earlier TKID measurement at 1640 m altitude and claim roughly ±25% accuracy over realistic conditions.

Significance. If Eq. (1) and Table II are reliable, the paper provides a genuinely useful practical tool: experimenters could obtain approximate background rates for superconducting qubit and sensor substrates without running dedicated Monte Carlo simulations. The two-step re-aiming Monte Carlo scheme and the explicit three-rate parametrization are sensible and clearly described. However, the central formula as printed is incomplete because the cosmic-ray elevation scale height λ is never given, and the claimed accuracy is not backed by quantitative uncertainties or a quantitative validation. The paper's practical value therefore depends on addressing these gaps.

major comments (4)
  1. [Section VI, Eq. (1), Table II] The cosmic-ray elevation factor is defined as ãCR ≡ exp(H/λ) with "an appropriate scale height λ," but no value of λ is given anywhere in the paper: Table II has no λ entry, and Section V reports only species-dependent scale heights (about 5 km for muons, 1 km for nuclear particles, and intermediate values for e±, γ, and the overall rate). This is a load-bearing omission because Eq. (1) cannot be evaluated at any nonzero elevation without λ, including the 1640 m altitude of the paper's own validation measurement. Moreover, the Section V observation that scale heights differ by particle species directly undermines the single-exponential form for all three rates; at 1640 m, exp(1.64/1) ≈ 5.1 versus exp(1.64/3) ≈ 1.7, a factor-of-three spread that dwarfs the claimed ±25% accuracy. The authors must either supply the λ value used for each rate or reformulate the elevation dependence to reflect the species-dependent composition changes.
  2. [Table II and Section VII] Table II lists fitted parameters cs, gs, ps, βs, ms, αs, and ρGa,s without any statistical or systematic uncertainties, yet Section VII claims the absolute rates are accurate to about ±25%. The uncertainty claim is therefore unsupported: the reader cannot tell whether 25% reflects Monte Carlo statistical errors, PARMA model uncertainty, concrete activity variation, or the re-aiming approximation. I ask the authors to provide uncertainties for the fitted parameters (at least for the dominant terms) or to weaken the accuracy claim to a qualitative estimate until such uncertainties are quantified.
  3. [Section IV, Mγ scaling] The >1 MeV terrestrial gamma rate Mγ is stated to scale approximately as t^5 (Section IV), but the paper acknowledges that this is "difficult to estimate from the few simulated events above that energy." No event counts, confidence intervals, or goodness-of-fit measures are given, and Section VI similarly states that too few MeV-scale events were generated to characterize any shape correction to M. A power-law exponent of 5 inferred from a handful of events is not a reliable interpolation or extrapolation basis, and it directly affects Eq. (1)'s M row. The authors should provide the number of simulated events above 1 MeV for the relevant thicknesses and a quantitative measure of the fit quality, or explicitly mark the Mγ scaling as preliminary.
  4. [Section VI, validation against Ref. [8]] The only validation is a qualitative consistency statement against the authors' own TKID measurement at 1640 m altitude: "A TKID-based spectroscopic measurement [8] is consistent with the models' results under nominal conditions." No quantitative comparison (e.g., fitted normalization, chi-square, ratio of modeled to measured rates) is shown, and the validation measurement is at an altitude where the model cannot currently be evaluated because λ is missing. Without a quantitative validation, the ±25% accuracy claim in Section VII cannot be checked. I recommend adding a direct comparison plot or table with the measured and modeled R, P, and energy spectra, including the elevation scaling with a specified λ.
minor comments (4)
  1. [Table II] The table's header row and last row (Cosmic rays) are hard to parse because the columns ns, cs, gs, ps, βs, ms, αs, ρGa,s are not all applicable to cosmic rays; consider using separate sub-tables or explicit placeholders (e.g., em dashes) with a footnote explaining which coefficients are used for the cosmic-ray term in Eq. (1).
  2. [Section VI] The sentence "Each term in the event rate R depends linearly on τ, with both constants depending on the source" is confusing because Eq. (1) shows R depending on τ through gs τ only, with cs independent of τ; consider rewording to "the gamma-ray contribution to R scales linearly with τ, while the charged-particle contribution cs is thickness-independent."
  3. [Section V, Figure 3] The y-axis of Figure 3 is labeled "Event rate (s−1)" but the axis values appear to span 10−3 to 10−1; please clarify the units (per substrate? per cm2?) and ensure the axis label matches the text's description of rates in the nominal substrate.
  4. [Section IV, Figure 2] Panel (b) is labeled "P/t (keV s−1 µm−1)", but the text discusses P growing as t^1.12; please define whether P is the total deposited power or power per unit thickness, and keep the notation consistent with Eq. (1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. 1 is a fit to Geant4/PARMA simulations, not to the validation measurement; the only self-citation is a consistency check.

full rationale

The paper's central equations (Eq. 1 and Table II) are obtained by fitting analytic scaling forms to Geant4 simulation outputs driven by PARMA cosmic-ray spectra and nominal European concrete activities. None of the constants in Table II is fitted to the TKID measurement of Ref. [8]; that measurement appears only as a consistency check ('consistent with our earlier measurement'), so it is not an input to the model. The acknowledgment that co-authors of Ref. [8] helped 'set the direction' does not amount to tuning parameters. The elevation factor aCR = exp(H/λ) is an ansatz, and λ is not tabulated, which is an incompleteness/robustness problem rather than a circular one. Likewise, the sparse Mγ proportional to t^5 extrapolation is a statistical weakness. No load-bearing step in the derivation is equivalent by construction to its inputs, and no uniqueness or ansatz is imported from a self-citation. Score 0.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central formulas in Eq. 1 rest on external simulation frameworks treated as ground truth, on a set of fitted coefficients calibrated to those simulations, and on a validation against one measurement from the same group. These are model assumptions and fitting parameters rather than derived first principles, and the absence of uncertainty estimates makes the implied precision a matter of judgment.

free parameters (9)
  • cs, charged-particle event-rate coefficients for 40K, 232Th-a, 232Th-b, 238U-a, 238U-b, cosmic rays = 2.2, 0.6, 1.5, 0.02, 1.9, 40 (10^-3 s^-1)
    Fitted to Geant4 simulation outputs; they set the thickness-independent part of the total event rate R in Eq. 1.
  • gs, gamma-ray event-rate coefficients for the six sources = 6.8, 4.9, 6.6, 0.6, 11.7, 1.4 (10^-3 s^-1)
    Fitted to Geant4 gamma-ray simulations; they set the linear-in-thickness part of R.
  • ps, deposited-power coefficients for the six sources = 1.4, 0.5, 0.9, 0.03, 1.4, 8.0 (keV/s)
    Fitted to simulated power deposition; used in the P term of Eq. 1.
  • beta_s, thickness power-law exponents for P = 1.12 for gamma sources; 1.0 for cosmic rays
    Chosen to fit the thickness scaling of simulated deposited power.
  • ms, >1 MeV event-rate coefficients for the six sources = 15, 2, 20, 0, 13, 180 (10^-6 s^-1)
    Fitted to sparse high-energy simulated events; these are the most statistically fragile parameters in the model.
  • alpha_s, thickness power-law exponents for M = 5.0 for 40K, 4 for 232Th chains and 238U-b, 15 for 238U-a, 1.8 for cosmic rays
    Fitted from few simulated MeV-scale events; the authors state that M_gamma proportional to t^5 is approximate.
  • rho_Ga,s, effective density adjustments for gallium substrates = 2, 4, 4, 15, 4 g/cm^3 for the five gamma sources
    Ad hoc correction used to make gamma-ray rates for GaAs and GaN substrates match simulations, reflecting gallium's high x-ray photoabsorption cross section.
  • lambda, cosmic-ray elevation scale height = Not specified precisely; approximately 5 km for muons, 1 km for nuclear particles, and between these for the overall…
    Used in Eq. 1 through the term exp(H/lambda); the exact combined value is not given.
  • kappa_c, kappa_sh, kappa_rho, correction factors for ceiling, shape, and density = Approximately 2% per 10 cm concrete, up to 20% increase in R and 3% decrease in P for a 10x1 mm wafer, and M…
    These empirical corrections are fitted to simulation outputs and used as multiplicative factors in Eq. 1.
assumptions (6)
  • domain assumption Secular equilibrium within decay chains, with radon-gap half-chains treated as distinct sources
    Invoked in Section II to simplify the decay chains; if radon migration breaks this balance in a specific laboratory, the specific activities need adjustment.
  • domain assumption Nominal specific activities of European building materials represent typical laboratory concrete
    Section II adopts 400 Bq/kg for 40K, 30 Bq/kg for 232Th, and 40 Bq/kg for 238U chains; the paper acknowledges measured variation by factors of 5 to 10.
  • domain assumption PARMA accurately models ground-level cosmic-ray spectra by species and elevation
    Section V uses PARMA, a model fitted to air-shower simulations, as the source term; it is treated as ground truth without direct comparison to local measurements.
  • ad hoc to paper The two-step re-aimed simulation preserves the energy and direction distributions relevant to a small substrate
    Section III uses a geometric shift plus time rescaling to make particle transport onto a centimeter-scale substrate tractable; this transformation is not independently validated in the paper.
  • domain assumption Thin superconducting films can be ignored, and the substrate dominates energy deposition
    Section III states that the film is neglected; the connection between substrate energy deposition and actual qubit decoherence is not modeled.
  • domain assumption Energy deposited in the substrate is a sufficient proxy for qubit decoherence impact
    The model stops at deposited energy and does not propagate to quasiparticle densities, phonon losses, or qubit error rates, which is the quantity of ultimate interest.

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Cite this review

Pith. "Pith review of Computed models of natural radiation backgrounds in qubits and superconducting detectors." pith.science (2026). https://pith.science/paper/FCF5BD2J

@misc{pith2026241116974,
  author       = {Pith},
  title        = {Pith review of: Computed models of natural radiation backgrounds in qubits and superconducting detectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FCF5BD2J}},
  note         = {Machine review of arXiv:2411.16974}
}
read the original abstract

Naturally occurring radiation backgrounds cause correlated decoherence events in superconducting qubits. These backgrounds include both gamma rays produced by terrestrial radioisotopes and cosmic rays. We use the particle-transport code Geant4 and the PARMA summary of the cosmic-ray spectrum to model both sources of natural radiation and to study their effects in the typical substrates used in superconducting electronics. We focus especially on three rates that summarize radiation's effect on substrates. We give analytic expressions for these rates, and how they depend upon parameters including laboratory elevation, substrate material, ceiling thickness, and wafer area and thickness. The modeled rates and the distribution of event energies are consistent with our earlier measurement of radiation backgrounds using a silicon thermal kinetic-inductance detector.

Figures

Figures reproduced from arXiv: 2411.16974 by the authors.

Figure 1
Figure 1. Spectrum of terrestrial gamma rays and x rays (a) as emitted by perfectly isolated radionuclides. (b) The same gamma rays, emerging from a thick [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Elevation dependence of cosmic-ray event rate in the nominal [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 2
Figure 2. Dependence of terrestrial gamma-ray effects on substrate thickness [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Spectrum of energy deposited by cosmic rays (all particle species) [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

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