Pith. sign in

REVIEW 3 major objections 5 minor 89 references

Classical Shadows with Improved Median-of-Means Estimation

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that swapping in a modified median-of-means estimator lowers classical-shadow error for global Clifford measurements.

desk verdict A useful, honest benchmark of Minsker's median-of-means estimator for classical shadows, but the headline Clifford advantage is confounded by unmatched k values and needs a controlled comparison before it carries weight. read the letter →

arxiv 2412.03381 v1 pith:5HGRX5AU submitted 2024-12-04 quant-ph cond-mat.stat-mechcs.LGstat.ML

Winston Fu , Dax Enshan Koh , Siong Thye Goh , Jian Feng Kong This is my paper · ORCID
classification quant-phcond-mat.stat-mechcs.LGstat.ML MSC 81P68 PACS 03.67.-a
keywords classicalshadowsmedian-of-meansestimatorU-statisticsincompleteCliffordmeasurementsPauliGHZstateshotcomplexity
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 tries to establish that the practical shot count of the classical shadows protocol can be reduced by replacing its standard median-of-means estimator with a modified estimator carrying tighter concentration constants. It supplies two efficient implementations of the modified estimator based on incomplete U-statistics, one using random sampling of subset averages and one using cyclically permuted sampling, and benchmarks them against the original estimator numerically. The central reported result is a setting-dependent ranking: for Pauli measurements on a 50-qubit Ising chain the original estimator remains the best, but for global Clifford measurements used to estimate the fidelity of noisy GHZ states the modified estimators achieve lower error and variance at the same shadow sizes. If that comparison holds, switching estimators in the classical post-processing step would lower measurement overhead for Clifford-based shadow estimation without changing the experiment.

What carries the argument

The argument runs on the modified median-of-means estimator, which splits the shadow data into groups, forms averages, then takes the median of averages over all (or a sampled subset of) $l$-element combinations of the group means, a U-statistic step. This changes the constant in the tail bound from $8e^2$ or $\sqrt{\pi}+o(1)$ for the standard estimator to $\sqrt{2}+o(1)$, which is what lowers the predicted shot count. Because taking all combinations is computationally expensive, the paper implements two incomplete U-statistic designs: a random sampling design and a cyclic-permutation design whose offsets form a Golomb ruler, and it uses variance formulas for incomplete U-statistics to compare them.

What would settle it

Run the GHZ fidelity benchmark with all estimators forced to use the same number of groups $k$ at each shadow size; if the modified estimators no longer beat the original, the claimed advantage is a $k$ artifact.

Watch

Extended reading notes

Core claim

With global Clifford measurements, the modified median-of-means estimators outperform the standard median-of-means estimator: in simulations estimating the fidelity of noisy GHZ states, the modified estimators had lower average error and variance than the original estimator at equal shadow sizes, and their 3.3-sigma errors stayed under the new tail bound while the original estimator's exceeded it. The paper interprets this as evidence that the tighter constants in the modified estimator produce a real practical advantage for Clifford measurements. With Pauli measurements the order reverses: the original estimator performs best, and the paper's appendix attributes the difference mainly to the number of groups $k$ rather than to the estimator type, because each group has fewer hits on a given Pauli observable.

Load-bearing premise

The comparison assumes the observed Clifford-measurement advantage comes from the estimator type and not from the different number of groups $k$ each estimator was run with, since the appendix itself shows $k$, not estimator type, drives the Pauli-measurement results.

Editorial extensions

If this is right

  • If the comparison holds, global-Clifford shadow estimation can use fewer shots for the same target fidelity by switching the post-processing estimator to one of the modified versions.
  • The improvement is purely classical, so existing classical-shadow datasets collected with Clifford measurements could be reanalyzed without new quantum experiments.
  • For Pauli measurements, the modified estimators are not a safe swap: the original estimator had the lowest error, and the group-size effect in the appendix warns against using them there.
  • The new tail bound is reported to be tight for Clifford measurements, which means the theoretical shot-count reduction is not just an artifact of a loose bound.

Reading between the lines

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

  • Because the appendix shows that the Pauli-measurement ranking is governed by the number of groups $k$, the Clifford advantage may also depend on $k$; a fair test would run all estimators with the same $k$, and until that is done the advantage should be treated as provisional.
  • If a same-$k$ test confirms the advantage, the practical rule would be estimator selection by measurement ensemble: standard median-of-means for Pauli observables, modified estimators for global Clifford observables.
  • The theory predicts the cyclic design has higher asymptotic relative efficiency than random sampling, yet the paper observes the random version performing at least as well in the Clifford benchmark; understanding this gap could reveal finite-sample effects that a follow-up could test.
Share X Bluesky LinkedIn Reddit HN

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 applies Minsker's improved median-of-means (MoM) estimator with sharp constants to the classical shadows protocol, introducing two practical incomplete U-statistic implementations (MomRand via random sampling, and MomCyc via cyclically permuted sampling) to make the estimator computationally feasible. It benchmarks these estimators against the standard median-of-means estimator for Pauli measurements on a transverse-field Ising chain and Clifford measurements on GHZ states, reporting that the modified estimators outperform the original for Clifford measurements while underperforming for Pauli measurements. The paper also compares numerical errors against the theoretical bounds of Minsker and Huang et al., and discusses the practical reduction in shadow size implied by the tighter constants.

Significance. If the Clifford-measurement advantage were established under controlled conditions, the paper would make a useful practical point: tighter constants in MoM bounds can reduce the required number of shots, and the choice of estimator should be tailored to the measurement ensemble. The two implementations of Minsker's estimator via incomplete U-statistics are a useful contribution, and the direct numerical comparisons provide concrete evidence. However, the central claim is currently confounded by the number of groups k, a confound the authors explicitly identify in Appendix C for the Pauli case but do not control in the Clifford case. The theoretical bounds are asymptotic, and the o(1) terms are ignored at the simulated sample sizes. The internal inconsistency between the theoretical ARE ordering and the empirical ordering of MomCyc versus MomRand further weakens the attribution of the observed advantage to the estimator type rather than to the hyperparameters.

major comments (3)
  1. [Section II.F / III / Appendix C] The central claim that MomRand and MomCyc outperform Mom for Clifford measurements is not controlled for the number of groups k. The paper's own Appendix C concludes for Pauli measurements that 'rather than the type of estimator used, the error depends most heavily on the number of groups k' and shows in Figure 13 that error increases with k. The Clifford experiments in Section II.F use the same k recipe as the Pauli experiments: Mom uses k=43 for all N, while MomRand and MomCyc use k in [65, 219] as N goes from 1000 to 50000 (per Appendix C). No equivalent k-sweep is provided for the Clifford fidelity estimation. Consequently, the observed advantage of MomRand/MomCyc over Mom in Figures 4 and 5 may be a hyperparameter artifact rather than evidence that Minsker's estimator is intrinsically better. This confound is load-bearing because the paper's conclusion in Section III explicitly attributes the gap to Minsker's estimator.
  2. [Section II.B / II.C] The theoretical bounds in Theorem 1 (Eq. 13) and Theorem 2 (Eq. 23) are asymptotic, with o(1) terms that the authors explicitly ignore ('Hence, we will ignore this factor in our subsequent analysis'). The paper then compares simulated errors at N between 1000 and 50000 against these asymptotic bounds (e.g., Figures 2, 4, 5, 11). At these finite sample sizes, with k ranging from 43 to 219 and l ~ log(N/k), the o(1) terms may not be negligible, particularly because t is chosen as k/log(k) or n/l^2 log(l), so the conditions for the o(1) terms to vanish are only marginally satisfied. This weakens the claim in Section III that the practical performance 'followed the tightened bounds closely' and the associated recommendation based on Table I.
  3. [Section II.D / Section III] The empirical ordering within the modified family contradicts the theoretical ARE comparison. Section II.D derives a higher ARE for MomCyc than MomRand (Eqs. 29 and 32), but Section III states: 'Despite the higher ARE of MomCyc compared to MomRand, the latter showed better performance for Clifford measurements.' This inconsistency indicates that the asymptotic relative efficiency does not predict the finite-sample behavior, so attributing the Mom-versus-MomRand gap to Minsker's estimator per se, rather than to the specific values of k, l, and m, is unsupported without a matched-hyperparameter comparison.
minor comments (5)
  1. [Section III] The sentence 'This shows that Minsker's estimator offers and advantage over the traditional MoM protocol for Clifford measurements' contains a typo: 'and advantage' should be 'an advantage'.
  2. [Algorithm 4] In Algorithm 4, the loop 'for i ← 0, m do' should likely be 'for i ← 0, m−1 do' for consistency with the other algorithms and to generate exactly m rounds.
  3. [Section II.B] The text 'Therefore, t = log(2M/δ). To satisfy the condition t ≪ k, we will choose t = k/log(k)' is confusing: k is a free parameter, and setting t = k/log(k) changes the failure probability from the union-bound value. Please clarify how k is chosen so that k/log(k) is consistent with log(2M/δ) for the reported M and δ.
  4. [Table I] Table I gives the number of samples required for 'an average error of ε = 0.1' but the caption does not state the values of M, δ, and the estimator parameters used. Please include these in the caption for reproducibility.
  5. [Appendix C / Figure 13] Figure 13 is only referenced in Appendix C; it would be helpful to refer to it also in the main text when discussing the k-dependence of the Pauli results, since it is central to the interpretation of the Clifford results.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the improved bounds and estimators are imported from external published theorems, and the central Clifford-measurement claim is a direct numerical benchmark, not a quantity fitted from or equivalent to its inputs.

full rationale

The paper's derivation chain is self-contained in the relevant sense. Its starting points are Minsker's published median-of-means bounds (refs. 82, 83, 86) and Lee's incomplete U-statistic analysis (ref. 84), both external to this paper; they are not re-derived from, or defined by, the paper's numerical results. The modified estimators MomRand and MomCyc are implementations of those external constructions, not quantities fitted to the benchmark data. The central claim for Clifford measurements is stated as an observed comparison: 'Mom now had higher error and variance than MomRand and MomCyc. This shows that Minsker's estimator offers and advantage over the traditional MoM protocol for Clifford measurements.' This is a direct numerical observation, not a prediction obtained by fitting parameters and then reading the fitted value back off as a result. The constants C = 8e^2, sqrt(pi), and sqrt(2) come from the cited theorems, and the hyperparameter choices (k, l, m, t) come from stated formulas rather than from optimizing against the simulated fidelity data. The authors' self-citations (refs. 23, 68, 71) appear only in literature-review lists and are not load-bearing for any derivation or uniqueness claim. The admitted k-dependence in Appendix C ('rather than the type of estimator used, the error depends most heavily on the number of groups k') is a genuine external-validity concern, because Mom and the modified estimators are not compared at matched k; however, this is a possible confound or alternative explanation, not a reduction of the Clifford claim to its own inputs by construction. Under the hard-rule standard requiring an exhibit of the specific reduction, no circular step is present.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper contributes two sampling algorithms for incomplete U-statistics, but the core bounds and variance formulas come from Minsker and Lee. The only hand-set quantities are the estimator hyperparameters (k, l, m), which are not fitted to data but are chosen differently for each estimator, creating a confound.

free parameters (3)
  • k (number of groups) = 43 for Mom; 65 to 219 for MomRand/MomCyc in Ising runs
    Chosen from t = log(2M/delta) and t = k/log(k); differs across estimators and confounds the comparison.
  • l (combination size) = log(N/k)
    Chosen heuristically; affects the variance of the median-of-combinations estimator.
  • m (number of sampled combinations) = 10kl
    Chosen heuristically; larger m increases computational cost and reduces variance.
assumptions (4)
  • standard math Minsker's Theorem 1 and Theorem 2 bound the failure probability of MoM and modified MoM estimators (Eqs. 13 and 23).
    Imported from Minsker; the paper relies on these for the improved constants and the shot-count comparison in Table I.
  • domain assumption Lee's variance formulas for incomplete U-statistics (Eqs. 26 and 31) apply to the proposed median-of-combinations estimators.
    The paper uses these to claim MomCyc has higher ARE than MomRand, but the estimator is a median of sampled U-statistics, not the U-statistic itself; the variance transfer is not proved.
  • domain assumption The o(1) terms in Minsker's bounds are negligible at the simulated shadow sizes (N = 1000 to 50000).
    With k around 43 to 219 and N/k around 15 to 1163, the asymptotic o(1) may be significant at the lower end; the paper ignores it.
  • domain assumption Normality of the mean estimator for N >= 1000 (Appendix B).
    Used to justify the mean's strong empirical performance; not central to the main estimator comparison.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Classical Shadows with Improved Median-of-Means Estimation." pith.science (2026). https://pith.science/paper/5HGRX5AU

@misc{pith2026241203381,
  author       = {Pith},
  title        = {Pith review of: Classical Shadows with Improved Median-of-Means Estimation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5HGRX5AU}},
  note         = {Machine review of arXiv:2412.03381}
}
abstract

The classical shadows protocol, introduced by Huang et al. [Nat. Phys. 16, 1050 (2020)], makes use of the median-of-means (MoM) estimator to efficiently estimate the expectation values of $M$ observables with failure probability $\delta$ using only $\mathcal{O}(\log(M/\delta))$ measurements. In their analysis, Huang et al. used loose constants in their asymptotic performance bounds for simplicity. However, the specific values of these constants can significantly affect the number of shots used in practical implementations. To address this, we studied a modified MoM estimator proposed by Minsker [PMLR 195, 5925 (2023)] that uses optimal constants and involves a U-statistic over the data set. For efficient estimation, we implemented two types of incomplete U-statistics estimators, the first based on random sampling and the second based on cyclically permuted sampling. We compared the performance of the original and modified estimators when used with the classical shadows protocol with single-qubit Clifford unitaries (Pauli measurements) for an Ising spin chain, and global Clifford unitaries (Clifford measurements) for the Greenberger-Horne-Zeilinger (GHZ) state. While the original estimator outperformed the modified estimators for Pauli measurements, the modified estimators showed improved performance over the original estimator for Clifford measurements. Our findings highlight the importance of tailoring estimators to specific measurement settings to optimize the performance of the classical shadows protocol in practical applications.

Figures

Figures reproduced from arXiv: 2412.03381 by the authors.

Figure 1
Figure 1. FIG. 1: Predicted and exact values for the 2-point [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (Left) Average error of each correlator over 10 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: (Left) Average error for the fidelity over 100 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figures from the paper (8 more)
Figure 5
Figure 5. Figure 5: FIG. 5: (Top half): Error in fidelity with the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Error and variance of predictions with varying [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Predicted purity for the 2-qubits GHZ state [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Error and variance vs shadow size for the Ising model using Pauli measurements when estimating the [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Error and variance vs shadow size for the 2-qubit GHZ state using Clifford measurements when estimating [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Error and variance vs shadow size for the 5, 10, 15, 20-qubit GHZ states using Clifford measurements when [PITH_FULL_IMAGE:figures/full_fig_p014_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Error and variance vs shadow size for the 2-qubit GHZ state using Clifford measurements when estimating [PITH_FULL_IMAGE:figures/full_fig_p015_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Error vs group size [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

89 extracted references · 60 canonical work pages

  1. [1]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measure- ments, Nature Physics 16, 1050 (2020)

    2020

  2. [2]

    S. Aaronson, Shadow tomography of quantum states, in Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing , STOC 2018 (Association for Computing Machinery, New York, NY, USA, 2018) p. 325–338

    2018

  3. [3]

    Gut ¸˘ a, J

    M. Gut ¸˘ a, J. Kahn, R. Kueng, and J. A. Tropp, Fast state tomography with optimal error bounds, Journal of Physics A: Mathematical and Theoretical 53, 204001 (2020)

    2020

  4. [4]

    Sugiyama, P

    T. Sugiyama, P. S. Turner, and M. Murao, Precision- Guaranteed Quantum Tomography, Physical Review Let- ters 111, 160406 (2013)

    2013

  5. [5]

    J. Haah, A. Harrow, Z. Ji, X. Wu, and N. Yu, Sample- Optimal Tomography of Quantum States, IEEE Transac- tions on Information Theory 63, 5628 (2017)

    2017

  6. [6]

    O’Donnell and J

    R. O’Donnell and J. Wright, Efficient quantum tomog- raphy, in Proceedings of the forty-eighth annual ACM symposium on Theory of Computing (2016) pp. 899–912

    2016

  7. [7]

    J. M. Lukens, K. J. Law, and R. S. Bennink, A Bayesian analysis of classical shadows, npj Quantum Information 7, 113 (2021)

    2021

  8. [8]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Efficient Estima- tion of Pauli Observables by Derandomization, Physical Review Letters 127, 030503 (2021)

    2021

Show all 89 references
  1. [9]

    Hadfield, Adaptive Pauli shadows for energy estima- tion, arXiv preprint 2105.12207 (2021)

    C. Hadfield, Adaptive Pauli shadows for energy estima- tion, arXiv preprint 2105.12207 (2021)

    2021 arXiv

  2. [10]

    A. Zhao, N. C. Rubin, and A. Miyake, Fermionic par- tial tomography via classical shadows, Physical Review Letters 127, 110504 (2021)

    2021

  3. [11]

    Hadfield, S

    C. Hadfield, S. Bravyi, R. Raymond, and A. Mezzacapo, Measurements of Quantum Hamiltonians with Locally- Biased Classical Shadows, Communications in Mathemat- 10 ical Physics 391, 951 (2022)

    2022

  4. [12]

    Zhou and Z

    Y. Zhou and Z. Liu, A hybrid framework for estimat- ing nonlinear functions of quantum states, npj Quantum Information 10, 62 (2024)

    2024

  5. [14]

    H.-Y. Hu, S. Choi, and Y.-Z. You, Classical shadow tomog- raphy with locally scrambled quantum dynamics, Physical Review Research 5, 023027 (2023)

    2023

  6. [15]

    Shivam, C

    S. Shivam, C. W. von Keyserlingk, and S. L. Sondhi, On classical and hybrid shadows of quantum states, SciPost Phys. 14, 094 (2023)

    2023

  7. [16]

    A. A. Akhtar, H.-Y. Hu, and Y.-Z. You, Scalable and Flex- ible Classical Shadow Tomography with Tensor Networks, Quantum 7, 1026 (2023)

    2023

  8. [17]

    K. Wan, W. J. Huggins, J. Lee, and R. Babbush, Match- gate shadows for fermionic quantum simulation, Commu- nications in Mathematical Physics 404, 629 (2023)

    2023

  9. [18]

    Innocenti, S

    L. Innocenti, S. Lorenzo, I. Palmisano, F. Albarelli, A. Fer- raro, M. Paternostro, and G. M. Palma, Shadow tomog- raphy on general measurement frames, PRX Quantum 4, 040328 (2023)

    2023

  10. [19]

    Ippoliti, Y

    M. Ippoliti, Y. Li, T. Rakovszky, and V. Khemani, Opera- tor relaxation and the optimal depth of classical shadows, Phys. Rev. Lett. 130, 230403 (2023)

    2023

  11. [20]

    Helsen and M

    J. Helsen and M. Walter, Thrifty shadow estimation: Reusing quantum circuits and bounding tails, Phys. Rev. Lett. 131, 240602 (2023)

    2023

  12. [21]

    Grier, H

    D. Grier, H. Pashayan, and L. Schaeffer, Principal eigen- state classical shadows, in Proceedings of Thirty Seventh Conference on Learning Theory , Proceedings of Machine Learning Research, Vol. 247, edited by S. Agrawal and A. Roth (PMLR, 2024) pp. 2122–2165

    2024

  13. [22]

    Sauvage and M

    F. Sauvage and M. Larocca, Classical shadows with sym- metries, arXiv preprint arXiv:2408.05279 (2024)

    2024 arXiv

  14. [23]

    K. Bu, D. E. Koh, R. J. Garcia, and A. Jaffe, Classi- cal shadows with Pauli-invariant unitary ensembles, npj Quantum Information 10, 6 (2024)

    2024

  15. [24]

    Grier, H

    D. Grier, H. Pashayan, and L. Schaeffer, Sample-optimal classical shadows for pure states, Quantum 8, 1373 (2024)

    2024

  16. [25]

    M. West, A. A. Mele, M. Larocca, and M. Cerezo, Real classical shadows, arXiv preprint arXiv:2410.23481 (2024)

    2024 arXiv

  17. [26]

    Ippoliti, Classical shadows based on locally-entangled measurements, Quantum 8, 1293 (2024)

    M. Ippoliti, Classical shadows based on locally-entangled measurements, Quantum 8, 1293 (2024)

    2024

  18. [27]

    R. King, D. Gosset, R. Kothari, and R. Babbush, Triply efficient shadow tomography, arXiv preprint arXiv:2404.19211 (2024)

    2024 arXiv

  19. [28]

    De Palma, T

    G. De Palma, T. Klein, and D. Pastorello, Classical shad- ows meet quantum optimal mass transport, Journal of Mathematical Physics 65, 092201 (2024)

    2024

  20. [29]

    Becker, N

    S. Becker, N. Datta, L. Lami, and C. Rouze, Classical shadow tomography for continuous variables quantum systems, IEEE Transactions on Information Theory 70, 3427 (2024)

    2024

  21. [30]

    S. N. Hearth, M. O. Flynn, A. Chandran, and C. R. Laumann, Efficient local classical shadow tomography with number conservation, Phys. Rev. Lett. 133, 060802 (2024)

    2024

  22. [31]

    Z. Cai, A. Chapman, H. Jnane, and B. Koczor, Biased estimator channels for classical shadows, arXiv preprint arXiv:2402.09511 (2024)

    2024 arXiv

  23. [32]

    D. T. S. Chen, Z. H. Saleem, and M. A. Perlin, Quantum circuit cutting for classical shadows, ACM Transactions on Quantum Computing 5(2), 13 (2024)

    2024

  24. [33]

    Heyraud, H

    V. Heyraud, H. Chomet, and J. Tilly, Unified frame- work for matchgate classical shadows, arXiv preprint arXiv:2409.03836 (2024)

    2024 arXiv

  25. [34]

    A. A. Akhtar, N. Anand, J. Marshall, and Y.-Z. You, Dual-unitary classical shadow tomography, arXiv preprint arXiv:2404.01068 (2024)

    2024 arXiv

  26. [35]

    Wang, Quantum advantage via efficient post- processing on qudit shadow tomography, arXiv preprint arXiv:2408.16244 (2024)

    Y. Wang, Quantum advantage via efficient post- processing on qudit shadow tomography, arXiv preprint arXiv:2408.16244 (2024)

    2024

  27. [36]

    Y. Wu, C. Wang, J. Yao, H. Zhai, Y.-Z. You, and P. Zhang, Contractive unitary and classical shadow tomography, arXiv preprint arXiv:2412.01850 (2024)

    2024 arXiv

  28. [37]

    Zhou and P

    T.-G. Zhou and P. Zhang, Efficient Classical Shadow Tomography through Many-body Localization Dynamics, Quantum 8, 1467 (2024)

    2024

  29. [38]

    Cioli, E

    R. Cioli, E. Ercolessi, M. Ippoliti, X. Turkeshi, and L. Piroli, Approximate inverse measurement channel for shallow shadows, arXiv preprint arXiv:2407.11813 (2024)

    2024 arXiv

  30. [39]

    Q. Liu, Z. Li, X. Yuan, H. Zhu, and Y. Zhou, Auxiliary-free replica shadow estimation, arXiv preprint arXiv:2407.20865 (2024)

    2024

  31. [40]

    Bertoni, J

    C. Bertoni, J. Haferkamp, M. Hinsche, M. Ioannou, J. Eis- ert, and H. Pashayan, Shallow shadows: Expectation es- timation using low-depth random Clifford circuits, Phys. Rev. Lett. 133, 020602 (2024)

    2024

  32. [41]

    G.-L. R. Anselmetti, M. Degroote, N. Moll, R. Santa- gati, and M. Streif, Classical optimisation of reduced density matrix estimations with classical shadows using N-representability conditions under shot noise considera- tions, arXiv preprint arXiv:2411.18430 (2024)

    2024

  33. [42]

    C. Mao, C. Yi, and H. Zhu, The magic in qudit shadow estimation based on the Clifford group, arXiv preprint arXiv:2410.13572 (2024)

    2024 arXiv

  34. [43]

    Caprotti, J

    A. Caprotti, J. Morris, and B. Daki´ c, Optimizing quantum tomography via shadow inversion, Phys. Rev. Res. 6, 033301 (2024)

    2024

  35. [44]

    Zhang, K

    Z.-J. Zhang, K. Nakaji, M. Choi, and A. Aspuru-Guzik, A composite measurement scheme for efficient quantum observable estimation, arXiv preprint arXiv:2305.02439 (2023)

    2023 arXiv

  36. [45]

    Huang, Y.-T

    B. Huang, Y.-T. Chen, B. Gupt, M. Suchara, A. Tran, S. McArdle, and G. Galli, Evaluating a quantum-classical quantum Monte Carlo algorithm with matchgate shadows, Phys. Rev. Res. 6, 043063 (2024)

    2024

  37. [46]

    Avdic and D

    I. Avdic and D. A. Mazziotti, Enhanced shadow tomog- raphy of molecular excited states via the enforcement of N -representability conditions by semidefinite program- ming, Phys. Rev. A 110, 052407 (2024)

    2024

  38. [47]

    Y. Wang, I. Avdic, and D. A. Mazziotti, Shadow ansatz for the many-fermion wave function in scalable molec- ular simulations on quantum computing devices, arXiv preprint arXiv:2408.11026 (2024)

    2024 arXiv

  39. [48]

    Avdic and D

    I. Avdic and D. A. Mazziotti, Fewer measurements from shadow tomography with N -representability conditions, Phys. Rev. Lett. 132, 220802 (2024)

    2024

  40. [49]

    Huang, R

    H.-Y. Huang, R. Kueng, G. Torlai, V. V. Albert, and J. Preskill, Provably efficient machine learning for quan- tum many-body problems, Science 377, eabk3333 (2022)

    2022

  41. [50]

    Lewis, H.-Y

    L. Lewis, H.-Y. Huang, V. T. Tran, S. Lehner, R. Kueng, and J. Preskill, Improved machine learning algorithm for predicting ground state properties, Nature Communica- 11 tions 15, 895 (2024)

    2024

  42. [51]

    Jerbi, C

    S. Jerbi, C. Gyurik, S. C. Marshall, R. Molteni, and V. Dunjko, Shadows of quantum machine learning, Nature Communications 15, 5676 (2024)

    2024

  43. [52]

    Zhao and A

    A. Zhao and A. Miyake, Group-theoretic error mitiga- tion enabled by classical shadows and symmetries, npj Quantum Information 10, 57 (2024)

    2024

  44. [53]

    Seif, Z.-P

    A. Seif, Z.-P. Cian, S. Zhou, S. Chen, and L. Jiang, Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors, PRX Quan- tum 4, 010303 (2023)

    2023

  45. [54]

    Elben, R

    A. Elben, R. Kueng, H.-Y. R. Huang, R. van Bij- nen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller, and B. Vermersch, Mixed-state en- tanglement from local randomized measurements, Phys. Rev. Lett. 125, 200501 (2020)

    2020

  46. [55]

    Neven, J

    A. Neven, J. Carrasco, V. Vitale, C. Kokail, A. Elben, M. Dalmonte, P. Calabrese, P. Zoller, B. Vermersch, R. Kueng, et al. , Symmetry-resolved entanglement de- tection using partial transpose moments, npj Quantum Information 7, 152 (2021)

    2021

  47. [56]

    A. Rath, C. Branciard, A. Minguzzi, and B. Vermersch, Quantum Fisher information from randomized measure- ments, Phys. Rev. Lett. 127, 260501 (2021)

    2021

  48. [57]

    Boyd and B

    G. Boyd and B. Koczor, Training variational quantum cir- cuits with CoVaR: Covariance root finding with classical shadows, Phys. Rev. X 12, 041022 (2022)

    2022

  49. [58]

    S. H. Sack, R. A. Medina, A. A. Michailidis, R. Kueng, and M. Serbyn, Avoiding Barren Plateaus Using Classical Shadows, PRX Quantum 3, 020365 (2022)

    2022

  50. [59]

    Basheer, Y

    A. Basheer, Y. Feng, C. Ferrie, and S. Li, Alternating layered variational quantum circuits can be classically op- timized efficiently using classical shadows, in Proceedings of the AAAI Conference on Artificial Intelligence , Vol. 37 (2023) pp. 6770–6778

    2023

  51. [60]

    Nakaji, S

    K. Nakaji, S. Endo, Y. Matsuzaki, and H. Hakoshima, Measurement optimization of variational quantum simula- tion by classical shadow and derandomization, Quantum 7, 995 (2023)

    2023

  52. [61]

    R. J. Garcia, Y. Zhou, and A. Jaffe, Quantum scrambling with classical shadows, Phys. Rev. Res. 3, 033155 (2021)

    2021

  53. [62]

    Helsen, M

    J. Helsen, M. Ioannou, J. Kitzinger, E. Onorati, A. Werner, J. Eisert, and I. Roth, Shadow estimation of gate-set properties from random sequences, Nature Communications 14, 5039 (2023)

    2023

  54. [63]

    G. A. L. White, K. Modi, and C. D. Hill, Filtering crosstalk from bath non-Markovianity via spacetime clas- sical shadows, Phys. Rev. Lett. 130, 160401 (2023)

    2023

  55. [64]

    Ippoliti and V

    M. Ippoliti and V. Khemani, Learnability transitions in monitored quantum dynamics via eavesdropper’s classical shadows, PRX Quantum 5, 020304 (2024)

    2024

  56. [65]

    E. A. Ruiz Guzman and D. Lacroix, Restoring symme- tries in quantum computing using classical shadows, The European Physical Journal A 60, 1 (2024)

    2024

  57. [66]

    Conrad, J

    J. Conrad, J. Eisert, and S. T. Flammia, Chasing shad- ows with Gottesman-Kitaev-Preskill codes, arXiv preprint arXiv:2411.00235 (2024)

    2024

  58. [67]

    S. Chen, W. Yu, P. Zeng, and S. T. Flammia, Robust Shadow Estimation, PRX Quantum 2, 030348 (2021)

    2021

  59. [68]

    D. E. Koh and S. Grewal, Classical Shadows With Noise, Quantum 6, 776 (2022)

    2022

  60. [69]

    Brieger, M

    R. Brieger, M. Heinrich, I. Roth, and M. Kliesch, Stability of classical shadows under gate-dependent noise, arXiv preprint arXiv:2310.19947 (2023)

    2023 arXiv

  61. [70]

    Nguyen, Shadow tomography with noisy readouts, arXiv preprint arXiv:2310.17328 (2023)

    H.-C. Nguyen, Shadow tomography with noisy readouts, arXiv preprint arXiv:2310.17328 (2023)

    2023 arXiv

  62. [71]

    Wu and D

    B. Wu and D. E. Koh, Error-mitigated fermionic clas- sical shadows on noisy quantum devices, npj Quantum Information 10, 39 (2024)

    2024

  63. [72]

    Rozon, N

    P.-G. Rozon, N. Bao, and K. Agarwal, Optimal twirling depth for classical shadows in the presence of noise, Phys- ical Review Letters 133, 130803 (2024)

    2024

  64. [73]

    Jnane, J

    H. Jnane, J. Steinberg, Z. Cai, H. C. Nguyen, and B. Koc- zor, Quantum error mitigated classical shadows, PRX Quantum 5, 010324 (2024)

    2024

  65. [74]

    Farias, R

    R. Farias, R. D. Peddinti, I. Roth, and L. Aolita, Robust shallow shadows, arXiv preprint arXiv:2405.06022 (2024)

    2024 arXiv

  66. [75]

    Onorati, J

    E. Onorati, J. Kitzinger, J. Helsen, M. Ioannou, A. Werner, I. Roth, and J. Eisert, Noise-mitigated random- ized measurements and self-calibrating shadow estimation, arXiv preprint arXiv:2403.04751 (2024)

    2024 arXiv

  67. [76]

    Devroye, M

    L. Devroye, M. Lerasle, G. Lugosi, and R. I. Oliveira, Sub-Gaussian mean estimators, The Annals of Statistics 44, 2695 (2016)

    2016

  68. [77]

    Lugosi and S

    G. Lugosi and S. Mendelson, Mean Estimation and Re- gression Under Heavy-Tailed Distributions: A Survey, Foundations of Computational Mathematics 19, 1145 (2019)

    2019

  69. [78]

    Struchalin, Ya

    G. Struchalin, Ya. A. Zagorovskii, E. Kovlakov, S. Straupe, and S. Kulik, Experimental Estimation of Quantum State Properties from Classical Shadows, PRX Quantum 2, 010307 (2021)

    2021

  70. [79]

    Zhang, J

    T. Zhang, J. Sun, X.-X. Fang, X.-M. Zhang, X. Yuan, and H. Lu, Experimental Quantum State Measurement with Classical Shadows, Physical Review Letters 127, 200501 (2021)

    2021

  71. [80]

    R. Levy, D. Luo, and B. K. Clark, Classical shadows for quantum process tomography on near-term quantum computers, Physical Review Research 6, 013029 (2024)

    2024

  72. [81]

    A. Dutt, W. Kirby, R. Raymond, C. Hadfield, S. Sheldon, I. L. Chuang, and A. Mezzacapo, Practical Benchmark- ing of Randomized Measurement Methods for Quantum Chemistry Hamiltonians, arXiv preprint arXiv:2312.07497 (2023)

    2023 arXiv

  73. [82]

    Minsker, Efficient median of means estimator, in Pro- ceedings of Thirty Sixth Conference on Learning The- ory, Proceedings of Machine Learning Research, Vol

    S. Minsker, Efficient median of means estimator, in Pro- ceedings of Thirty Sixth Conference on Learning The- ory, Proceedings of Machine Learning Research, Vol. 195, edited by G. Neu and L. Rosasco (PMLR, 2023) pp. 5925– 5933

    2023

  74. [83]

    Minsker, U-statistics of growing order and sub-Gaussian mean estimators with sharp constants, Mathematical Statistics and Learning 7, 1 (2023)

    S. Minsker, U-statistics of growing order and sub-Gaussian mean estimators with sharp constants, Mathematical Statistics and Learning 7, 1 (2023)

    2023

  75. [84]

    to present and benchmark two possible methods of sampling: randomly, and using cyclic permutations. D. Incomplete Modified Median-of-Means Algorithm 3 Median-of-means estimator with random sampling (MomRand). Input: • x1, . . . , xN : Array of data. • k: Changes the number of ...

  76. [85]

    A. J. Lee, U-Statistics: Theory and Practice (Routledge, New York, 2019)

    2019

  77. [86]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond bell’s theorem, in Bell’s Theorem, Quantum The- ory and Conceptions of the Universe , edited by M. Kafatos (Springer Netherlands, Dordrecht, 1989) pp. 69–72

    1989

  78. [87]

    Minsker, Distributed statistical estimation and rates of convergence in normal approximation, Electronic Journal of Statistics 13, 5213 (2019)

    S. Minsker, Distributed statistical estimation and rates of convergence in normal approximation, Electronic Journal of Statistics 13, 5213 (2019)

    2019

  79. [88]

    Drakakis, A review of the available construction meth- ods for Golomb rulers., Advances in Mathematics of Com- munications 3, 235 (2009)

    K. Drakakis, A review of the available construction meth- ods for Golomb rulers., Advances in Mathematics of Com- munications 3, 235 (2009)

    2009

  80. [89]

    A. J. Ferris and G. Vidal, Perfect sampling with unitary tensor networks, Physical Review B 85, 165146 (2012)

    2012

  81. [90]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Efficient Estima- tion of Pauli Observables by Derandomization, Physical 12 Review Letters 127, 030503 (2021). Appendix A: MoM Bound We present a proof for the bound P [|ˆµN − E(X)| ≥ε] ≤ 2e−k/2, (A1) where ε = p 4e2σ2k/N and N is the to...

    2021

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.