pith. the verified trust layer for science. sign in

arxiv: 2604.05973 · v1 · submitted 2026-04-07 · 🪐 quant-ph

Distributions of Noisy Expectation Values over Sets of Measurement Operators

Matthew Duschenes , Roger G. Melko , Juan Carrasquilla , Raymond Laflamme This is my paper

Pith reviewed 2026-05-10 19:55 UTC · model grok-4.3

classification 🪐 quant-ph
keywords expectation value distributionsdepolarizing noisequantum circuitsmeasurement operatorsrandom mixed statescombinatorial momentsbrickwork circuitseffective noise models
0
0 comments X p. Extension

The pith

Distributions of expectation values under local noise in quantum circuits can be approximated by a simpler effective global depolarizing model.

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

The paper extends prior results on the random distribution of expectation values to the case of multiple measurement operators acting on mixed states that live in environments of adjustable size. It derives exact expressions for the moments of these distributions through direct combinatorial counting. Numerical experiments then generate empirical distributions by evolving Haar-random states through brickwork circuits subject to local depolarizing noise. These histograms are compared against a proposed effective model that treats the noise as global depolarization whose strength and effective environment dimension are allowed to vary. The effective model reproduces the location and height of the central peak for a wide range of depths, noise rates, and system sizes, while the tails deviate in ways attributable to the locality of the noise.

Core claim

For random mixed states in variable-sized environments, the moments of the joint distribution of expectation values over any fixed set of measurement operators are given by explicit combinatorial formulas. When the same states are instead produced by noisy brickwork circuits, the resulting histograms are well captured near their peaks by an effective global-depolarizing distribution whose noise parameter and environment dimension are fitted to the data; the same effective model yields uni-modal histograms for symmetric operator sets and multi-modal histograms for non-symmetric sets.

What carries the argument

The effective global-depolarizing-like model whose noise scale and environment dimension are fitted to reproduce the peak of the observed distribution of expectation values under local depolarizing noise.

If this is right

  • The combinatorial moment formulas give exact, parameter-free predictions for the mean, variance, and higher moments of expectation-value distributions for any chosen set of operators.
  • The effective noise parameters extracted from the fit change smoothly and monotonically with circuit depth and with the physical noise strength.
  • Symmetric measurement operators produce single-peaked distributions while non-symmetric operators produce distinct multi-peaked distributions.
  • The tail deviations that remain after fitting are directly traceable to the spatially local character of the noise.
  • The same fitting procedure can be repeated for any circuit depth and noise scale without requiring new combinatorial derivations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the effective model collapses the entire local-noise dynamics onto a few global parameters, it may allow faster classical sampling of typical measurement statistics in large noisy circuits.
  • Multi-modal distributions for non-symmetric operators suggest that certain sets of measurements remain hard to simulate classically even when the underlying state distribution is simple.
  • The smooth dependence of the fitted parameters on depth offers a practical way to extrapolate noise effects to circuit sizes beyond direct simulation.

Load-bearing premise

That an effective global depolarizing model with adjustable noise strength and environment size can still match the main features of the distribution even when the actual noise acts only locally on each qubit.

What would settle it

A controlled simulation in which the fitted effective model is forced to use the true local noise rate and environment size, after which the peak mismatch is measured as a function of circuit depth; if the mismatch grows systematically rather than remaining bounded, the approximation fails.

Figures

Figures reproduced from arXiv: 2604.05973 by Juan Carrasquilla, Matthew Duschenes, Raymond Laflamme, Roger G. Melko.

Figure 1
Figure 1. Figure 1: FIG. 1: Analytical distributions for [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Circuit with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Empirical SIC-POVM (a,b) and NON-SIC (c,d) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a): Empirical Kolmogorov–Smirnov metric [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Empirical SIC-POVM probability histograms [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Sampling-dependence of empirical Kolmogorov–Smirnov metric [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Optimized parameters of effective noise ˜γ [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
read the original abstract

Expectation values of measurement operators, interpreted as measurement probabilities, arise frequently throughout quantum algorithms. When quantum states are randomly distributed, their expectation values are also randomly distributed. In this work, with the goal of understanding non-unitary dynamics, we generalize previous derivations for distributions of expectation values (Campos Venuti and Zanardi, Physics Letters A (377), 2013) to the case of sets of measurement operators and random mixed quantum states within variable sized environments. Using combinatorics approaches, we derive expressions for their moments. We proceed to construct empirical distributions of simulated Haar random brickwork quantum circuits with local depolarizing noise, and compare their form to a proposed effective global-depolarizing-like model with variable effective noise scales and environment dimensions. The fitted effective distributions reproduce peak behaviour across circuit depths, noise scales, and system sizes, while deviations in the distribution tails arise from local noise effects. The fit effective model parameters are also shown to vary smoothly and consistently with circuit depth and noise scale. Finally, sets of non-symmetric measurement operators are shown to exhibit distinct multi-modal distributions relative to uni-modal distributions for symmetric measurement operators, opening up questions about their simulability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper generalizes prior derivations of expectation-value distributions to sets of measurement operators and random mixed states in variable-sized environments. Using combinatorial methods, it derives expressions for the moments of these distributions. It then generates empirical histograms from Haar-random brickwork circuits subject to local depolarizing noise and compares them to an effective global-depolarizing model whose noise scale and environment dimension are fitted to the same data; the fitted model is reported to reproduce peak locations across depths, noise strengths, and sizes, with tail deviations ascribed to locality, while non-symmetric operators produce multi-modal distributions.

Significance. The generalization to sets of operators and mixed states, together with the combinatorial moment expressions, would constitute a modest technical advance if the derivations are made fully explicit. The empirical observation that non-symmetric measurements yield multi-modal histograms is potentially useful for assessing simulability. However, because the effective-model parameters are obtained by direct fitting to the simulation histograms, the reported peak agreement does not constitute an independent test of the model's utility; quantitative distance metrics and predictive use of the moments are absent, limiting the strength of the central modeling claim.

major comments (3)
  1. [Theoretical moment derivations] The combinatorial derivations of the moment expressions are asserted but not supplied in sufficient detail (no explicit counting arguments, generating functions, or intermediate steps are shown), preventing verification and blocking any use of those moments to predict or constrain the effective-model parameters.
  2. [Empirical distributions and effective-model comparison] The effective global-depolarizing-like model is fitted directly to the same circuit-simulation histograms whose peaks it is claimed to reproduce; this renders the agreement tautological. No quantitative goodness-of-fit statistic (Kolmogorov-Smirnov, Wasserstein, total-variation distance, etc.) is reported, and the acknowledged tail deviations are left unquantified.
  3. [Discussion of fitted parameters and model utility] The derived moment expressions are never connected to the effective model; the noise scale and environment dimension are chosen by post-hoc fitting rather than by matching moments or by any other independent procedure, so the model remains descriptive rather than predictive.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction should more sharply separate the combinatorial theoretical results from the subsequent phenomenological fitting exercise.
  2. [Numerical methods] Simulation parameters (brickwork depth, local noise rate, system size, number of samples per histogram) should be tabulated for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism. The comments correctly identify places where the manuscript is insufficiently explicit or quantitative. We will revise the paper to supply the missing derivation details, add quantitative goodness-of-fit metrics, quantify tail discrepancies, and clarify the relationship (or lack thereof) between the combinatorial moments and the fitted effective model. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Theoretical moment derivations] The combinatorial derivations of the moment expressions are asserted but not supplied in sufficient detail (no explicit counting arguments, generating functions, or intermediate steps are shown), preventing verification and blocking any use of those moments to predict or constrain the effective-model parameters.

    Authors: We agree that the combinatorial derivations are presented at too high a level. In the revised manuscript we will expand the relevant section to include explicit counting arguments for the moments of expectation values over sets of measurement operators acting on random mixed states of variable environment dimension. We will also display the generating-function approach used to obtain the closed-form moment expressions and show the first few intermediate steps for both symmetric and non-symmetric operator sets. This will make the results verifiable and will allow readers to assess whether the moments can be used to constrain the effective-model parameters. revision: yes

  2. Referee: [Empirical distributions and effective-model comparison] The effective global-depolarizing-like model is fitted directly to the same circuit-simulation histograms whose peaks it is claimed to reproduce; this renders the agreement tautological. No quantitative goodness-of-fit statistic (Kolmogorov-Smirnov, Wasserstein, total-variation distance, etc.) is reported, and the acknowledged tail deviations are left unquantified.

    Authors: The referee is correct that fitting to the same histograms renders the peak agreement non-independent. In the revision we will report quantitative distances (Wasserstein-1 distance, Kolmogorov-Smirnov statistic, and total-variation distance) between the empirical histograms and the fitted effective distributions for all depths, noise strengths, and system sizes shown. We will also quantify the tail deviations by computing the integrated absolute error beyond the 95th percentile and by reporting the ratio of empirical to model tail probabilities. These additions will make the comparison objective rather than visual. revision: yes

  3. Referee: [Discussion of fitted parameters and model utility] The derived moment expressions are never connected to the effective model; the noise scale and environment dimension are chosen by post-hoc fitting rather than by matching moments or by any other independent procedure, so the model remains descriptive rather than predictive.

    Authors: We acknowledge that the manuscript does not link the combinatorial moments to the choice of effective noise scale or environment dimension. In the revision we will (i) compute the first two moments of the effective global-depolarizing model analytically, (ii) compare them numerically to the combinatorially derived moments for the same operator sets, and (iii) discuss the extent to which moment matching is possible given the locality of the underlying noise. Where direct matching fails we will state explicitly that the effective model is phenomenological and motivated by the observed peak locations rather than by moment equality, thereby clarifying its descriptive versus predictive status. revision: yes

Circularity Check

0 steps flagged

No significant circularity: combinatorial derivations independent of post-hoc fitting for comparison

full rationale

The paper first derives closed-form expressions for the moments of expectation-value distributions over sets of measurement operators acting on random mixed states in variable environments, using combinatorics. This step is self-contained and does not rely on the later numerical work. The manuscript then generates empirical histograms from explicit simulations of brickwork circuits under local depolarizing noise and compares their shape to an explicitly proposed effective global-depolarizing model whose two parameters (noise scale and environment dimension) are fitted to those same histograms. The text presents the comparison as an empirical check that reproduces peak location while noting tail deviations due to locality; it does not claim the fitted model as a first-principles prediction or derive the effective parameters from the combinatorial moments. Because the fitting is openly post-hoc and the central analytic result stands apart, no load-bearing step reduces to self-definition, fitted-input renaming, or self-citation chains.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The central claims rest on the assumption that random mixed states can be modeled via Haar measure and that an effective global noise model with fitted parameters adequately captures local noise statistics; two free parameters are introduced via fitting.

free parameters (2)
  • effective noise scale = variable
    Variable parameter fitted to reproduce peak behavior in circuit simulations across depths and system sizes
  • effective environment dimension = variable
    Variable parameter adjusted to match distributions for different system sizes
axioms (1)
  • domain assumption Quantum states are randomly distributed according to the Haar measure on mixed states in variable-sized environments
    Invoked to generalize the 2013 derivation and to generate empirical distributions
invented entities (1)
  • effective global-depolarizing-like model no independent evidence
    purpose: To approximate the distributions arising from local depolarizing noise in brickwork circuits
    Introduced as a simplified model whose parameters are fitted to simulation data

pith-pipeline@v0.9.0 · 5510 in / 1510 out tokens · 50907 ms · 2026-05-10T19:55:58.764549+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

104 extracted references · 104 canonical work pages

  1. [1]

    D. M. Appleby, ˚A. Ericsson, and C. A. Fuchs, Properties of QBist State Spaces, Foundations of Physics41, 564 (2009)

    work page 2009

  2. [2]

    Appleby, C

    M. Appleby, C. A. Fuchs, B. C. Stacey, and H. Zhu, Intro- ducing the Qplex: A Novel Arena for Quantum Theory, European Physical Journal D71(2016)

    work page 2016

  3. [3]

    A. J. Scott, Tight informationally complete quantum measurements, Journal of Physics A: Mathematical and General39, 13507 (2006)

    work page 2006

  4. [4]

    V. I. Yashin, E. O. Kiktenko, A. S. Mastiukova, and A. K. Fedorov, Minimal informationally complete mea- surements for probability representation of quantum dy- namics, New Journal of Physics22, 103026 (2020)

    work page 2020

  5. [5]

    J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, Symmetric Informationally Complete Quantum Measurements, Journal of Mathematical Physics45, 2171 (2003)

    work page 2003

  6. [6]

    H. J. D. Miller, Statistical Mechanics of Random Mixed State Ensembles with Fixed Energy, arXiv preprint arXiv:2508.00809 (2025)

    work page arXiv 2025

  7. [7]

    Zyczkowski and H.-J

    K. Zyczkowski and H.-J. Sommers, Average fidelity be- tween random quantum states, Physical Review A71, 032313 (2003)

    work page 2003

  8. [8]

    Sommers and K

    H. Sommers and K. Zyczkowski, Bures volume of the set of mixed quantum states, Journal of Physics A: Mathe- matical and General36, 10083 (2005)

    work page 2005

  9. [9]

    J. J. Meyer, Fisher Information in Noisy Intermediate- Scale Quantum Applications, Quantum5, 539 (2021)

    work page 2021

  10. [10]

    Y. Zhou, M. Stoudenmire, and X. Waintal, What Limits the Simulation of Quantum Computers?, Physical Re- view X10, 041038 (2020)

    work page 2020

  11. [11]

    Sommers and K

    H.-J. Sommers and K. Zyczkowski, Statistical properties of random density matrices, Journal of Physics A: Math- ematical and General37, 8457 (2004)

    work page 2004

  12. [12]

    Fefferman, S

    B. Fefferman, S. Ghosh, M. Gullans, K. Kuroiwa, and K. Sharma, Effect of Nonunital Noise on Random-Circuit Sampling, PRX Quantum , 030317 (2024)

    work page 2024

  13. [13]

    Holmes, K

    Z. Holmes, K. Sharma, M. Cerezo, and P. J. Coles, Con- necting ansatz expressibility to gradient magnitudes and barren plateaus, PRX Quantum3, 010313 (2022)

    work page 2022

  14. [14]

    Duschenes, D

    M. Duschenes, D. Garc´ ıa-Mart´ ın, Z. Holmes, and M. Cerezo, Moments of quantum channel ensembles, arXiv preprint arXiv:2511.12700 (2025)

    work page arXiv 2025

  15. [15]

    Emerson, R

    J. Emerson, R. Alicki, and K. Zyczkowski, Scalable Noise Estimation with Random Unitary Operators, Journal of Optics B7(2005)

    work page 2005

  16. [16]

    Y. Quek, D. Stilck Fran¸ ca, S. Khatri, J. J. Meyer, and J. Eisert, Exponentially tighter bounds on limitations of quantum error mitigation, Nature Physics20, 1648 (2024)

    work page 2024

  17. [17]

    S. Wang, P. Czarnik, A. Arrasmith, M. Cerezo, L. Cincio, and P. J. Coles, Can Error Mitigation Improve Trainabil- ity of Noisy Variational Quantum Algorithms?, Quantum 8, 1287 (2024)

    work page 2024

  18. [18]

    Emerson, M

    J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Lafor- est, J. Baugh, D. G. Cory, and R. Laflamme, Sym- metrized characterization of noisy quantum processes, Science317, 1893 (2007)

    work page 2007

  19. [19]

    Srednicki, Chaos and quantum thermalization, Phys

    M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E50, 888 (1994)

    work page 1994

  20. [20]

    Zhang, C

    M. Zhang, C. Wang, and Y. Han, Noisy Random Quan- tum Circuit Sampling and its Classical Simulation, Ad- vanced Quantum Technologies6, 2300030 (2023)

    work page 2023

  21. [21]

    M. P. Fisher, V. Khemani, A. Nahum, and S. Vijay, Ran- dom Quantum Circuits, Annual Review of Condensed Matter Physics14, 335 (2023)

    work page 2023

  22. [22]

    Cheng and M

    Z. Cheng and M. Ippoliti, Efficient Sampling of Noisy Shallow Circuits Via Monitored Unraveling, PRX Quan- tum4, 040326 (2023)

    work page 2023

  23. [23]

    Czischek, G

    S. Czischek, G. Torlai, S. Ray, R. Islam, and R. G. Melko, Simulating a measurement-induced phase tran- sition for trapped ion circuits, Physical Review A104, 062405 (2021)

    work page 2021

  24. [24]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S...

    work page 2019

  25. [25]

    Boixo, S

    S. Boixo, S. V. Isakov, V. N. Smelyanskiy, R. Babbush, N. Ding, Z. Jiang, M. J. Bremner, J. M. Martinis, and H. Neven, Characterizing quantum supremacy in near- term devices, Nature Physics14, 595 (2018)

    work page 2018

  26. [26]

    LaRose, A brief history of quantum vs classical computa- tional advantage, (2024), arXiv:2412.14703

    R. LaRose, A brief history of quantum vs classical com- putational advantage, arXiv preprint arXiv:2412.14703 (2024)

    work page arXiv 2024

  27. [27]

    Y. Kim, A. Eddins, S. Anand, K. X. Wei, E. van den Berg, S. Rosenblatt, H. Nayfeh, Y. Wu, M. Zaletel, K. Temme, and A. Kandala, Evidence for the utility of quantum computing before fault tolerance, Nature618, 9 500 (2023)

    work page 2023

  28. [28]

    Q. Zhu, S. Cao, F. Chen, M. C. Chen, X. Chen, T. H. Chung, H. Deng, Y. Du, D. Fan, M. Gong, C. Guo, C. Guo, S. Guo, L. Han, L. Hong, H. L. Huang, Y. H. Huo, L. Li, N. Li, S. Li, Y. Li, F. Liang, C. Lin, J. Lin, H. Qian, D. Qiao, H. Rong, H. Su, L. Sun, L. Wang, S. Wang, D. Wu, Y. Wu, Y. Xu, K. Yan, W. Yang, Y. Yang, Y. Ye, J. Yin, C. Ying, J. Yu, C. Zha, ...

    work page 2022

  29. [29]

    Torlai, C

    G. Torlai, C. J. Wood, A. Acharya, G. Carleo, J. Car- rasquilla, and L. Aolita, Quantum process tomography with unsupervised learning and tensor networks, Nature Communications14, 2858 (2023)

    work page 2023

  30. [30]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measure- ments, Nature Physics16, 1050 (2020)

    work page 2020

  31. [31]

    Carrasquilla, G

    J. Carrasquilla, G. Torlai, R. G. Melko, and L. Aolita, Reconstructing quantum states with generative models, Nature Machine Intelligence1, 155 (2019)

    work page 2019

  32. [32]

    H. Hu, S. Choi, and Y. You, Classical shadow tomogra- phy with locally scrambled quantum dynamics, Physical Review Research5, 023027 (2023)

    work page 2023

  33. [33]

    F. Hu, G. Angelatos, S. A. Khan, M. Vives, E. T¨ ureci, L. Bello, G. E. Rowlands, G. J. Ribeill, and H. E. T¨ ureci, Tackling Sampling Noise in Physical Systems for Machine Learning Applications: Fundamental Limits and Eigen- tasks, Physical Review X13, 041020 (2023)

    work page 2023

  34. [34]

    Larocca, S

    M. Larocca, S. Thanasilp, S. Wang, K. Sharma, J. Bia- monte, P. J. Coles, L. Cincio, J. R. McClean, Z. Holmes, and M. Cerezo, Barren plateaus in variational quantum computing, Nature Reviews Physics7, 174–189 (2025)

    work page 2025

  35. [35]

    V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsiout- sios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. De- voret, Real-time quantum error correction beyond break- even, Nature616, 50 (2023)

    work page 2023

  36. [36]

    Duschenes, J

    M. Duschenes, J. Carrasquilla, and R. Laflamme, Charac- terization of overparametrization in the simulation of re- alistic quantum systems, Physical Review A109, 062607 (2024)

    work page 2024

  37. [37]

    Georgopoulos, C

    K. Georgopoulos, C. Emary, and P. Zuliani, Modelling and Simulating the Noisy Behaviour of Near-term Quan- tum Computers, Physical Review A104, 062432 (2021)

    work page 2021

  38. [38]

    X. Ge, R. B. Wu, and H. Rabitz, The optimization landscape of hybrid quantum–classical algorithms: From quantum control to NISQ applications, Annu. Rev. Con- trol54, 314 (2022)

    work page 2022

  39. [39]

    Tsubouchi, Y

    K. Tsubouchi, Y. Mitsuhashi, K. Sharma, and N. Yosh- ioka, Symmetric Clifford twirling for cost-optimal quan- tum error mitigation in early FTQC regime, npj Quan- tum Information11(2025)

    work page 2025

  40. [40]

    Jnane, J

    H. Jnane, J. Steinberg, Z. Cai, H. C. Nguyen, and B. Koc- zor, Quantum Error Mitigated Classical Shadows, PRX Quantum5, 010324 (2024)

    work page 2024

  41. [41]

    V. B. Bulchandani, S. L. Sondhi, and J. T. Chalker, Random-matrix models of monitored quantum circuits, Journal of Statistical Physics191(2024)

    work page 2024

  42. [42]

    Skinner, J

    B. Skinner, J. Ruhman, and A. Nahum, Measurement- Induced Phase Transitions in the Dynamics of Entangle- ment, Phys. Rev. X9, 031009 (2019)

    work page 2019

  43. [43]

    Y. Li, X. Chen, and M. P. Fisher, Quantum Zeno ef- fect and the many-body entanglement transition, Physi- cal Review B98, 205136 (2018)

    work page 2018

  44. [44]

    Heinosaari, M

    T. Heinosaari, M. A. Jivulescu, and I. Nechita, Random positive operator valued measures, Journal of Mathemat- ical Physics61, 042202 (2019)

    work page 2019

  45. [45]

    Acharya, S

    A. Acharya, S. Saha, and A. M. Sengupta, Information- ally complete POVM-based shadow tomography, arXiv preprint arXiv:2105.05992 (2021)

    work page arXiv 2021

  46. [46]

    Anshu and S

    A. Anshu and S. Arunachalam, A survey on the complex- ity of learning quantum states, Nature Reviews Physics 6, 59 (2023)

    work page 2023

  47. [47]

    R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, Qudit quantum-state tomography, Physical Review A 66, 012303 (2002)

    work page 2002

  48. [48]

    Gross, Y

    D. Gross, Y. K. Liu, S. T. Flammia, S. Becker, and J. Eis- ert, Quantum state tomography via compressed sensing, Physical Review Letters105, 150401 (2010)

    work page 2010

  49. [49]

    Cramer, M

    M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y. K. Liu, Efficient quantum state tomography, Na- ture Communications1, 149 (2010)

    work page 2010

  50. [50]

    D. E. Koh and S. Grewal, Classical Shadows With Noise, Quantum6(2020)

    work page 2020

  51. [51]

    Sampling random quantum circuits: a pedestrian’s guide.arXiv preprint arXiv:2007.07872, 2020

    S. Mullane, Sampling random quantum circuits: a pedes- trian’s guide, arXiv preprint arXiv:2007.07872 (2020)

    work page arXiv 2007

  52. [52]

    P. W. Claeys and G. De Tomasi, Fock-space delocaliza- tion and the emergence of the Porter-Thomas distribu- tion from dual-unitary dynamics, Physical Review Let- ters134, 050405 (2025)

    work page 2025

  53. [53]

    Collins, S

    B. Collins, S. Matsumoto, and J. Novak, The Weingarten Calculus, Notices Of The American Mathematical Soci- ety69, 734 (2022)

    work page 2022

  54. [54]

    Kukulski, I

    R. Kukulski, I. Nechita, L. Pawela, Z. Puchala, and K. Zyczkowski, Generating random quantum channels, Journal of Mathematical Physics62, 062201 (2021)

    work page 2021

  55. [55]

    Belkin, J

    D. Belkin, J. Allen, S. Ghosh, C. Kang, S. Lin, J. Sud, F. T. Chong, B. Fefferman, and B. K. Clark, Approx- imate t-Designs in Generic Circuit Architectures, PRX Quantum5, 040344 (2024)

    work page 2024

  56. [56]

    Deshpande, P

    A. Deshpande, P. Niroula, O. Shtanko, A. V. Gorshkov, B. Fefferman, and M. J. Gullans, Tight bounds on the convergence of noisy random circuits to the uniform dis- tribution, PRX Quantum3(2022)

    work page 2022

  57. [57]

    M. Kus, J. Mostowski, and F. Haake, Universality of eigenvector statistics of kicked tops of different symme- tries, Journal of Physics A: Mathematical and General 21, L1073 (1988)

    work page 1988

  58. [58]

    A. M. Dalzell, N. Hunter-Jones, and F. G. S. L. Brand˜ ao, Random Quantum Circuits Anticoncentrate in Log Depth, PRX Quantum3, 010333 (2022)

    work page 2022

  59. [59]

    Campos Venuti and P

    L. Campos Venuti and P. Zanardi, Probability density of quantum expectation values, Physics Letters A377, 1854 (2013)

    work page 2013

  60. [60]

    C. F. Dunkl, P. Gawron, J. A. Holbrook, J. A. Miszczak, Z. Pucha la, and K. ˙Zyczkowski, Numerical shadow and geometry of quantum states, Journal of Physics A: Math- ematical and Theoretical44, 335301 (2011)

    work page 2011

  61. [61]

    Gutkin and K

    E. Gutkin and K. Zyczkowski, Joint numerical ranges, quantum maps, and joint numerical shadows, Linear Al- gebra and its Applications438, 2394 (2013)

    work page 2013

  62. [62]

    Zhang, S

    L. Zhang, S. Luo, S.-M. Fei, and J. Wu, Uncertainty re- gions of observables and state-independent uncertainty relations, Quantum Information Processing20(2021). 10

    work page 2021

  63. [63]

    Sauliere, B

    A. Sauliere, B. Magni, G. Lami, X. Turkeshi, and J. D. Nardis, Universality in the anticoncentration of chaotic quantum circuits, Physical Review B112, 134312 (2025)

    work page 2025

  64. [64]

    W. G. Brown and L. Viola, Convergence Rates for Arbi- trary Statistical Moments of Random Quantum Circuits, Phys. Rev. Lett.104, 250501 (2010)

    work page 2010

  65. [65]

    Sauliere, G

    A. Sauliere, G. Lami, C. Boyer, J. De Nardis, and A. De Luca, Universality in the Anticoncentration of Noisy Quantum Circuits at Finite Depths, arXiv preprint arXiv:2508.14975 (2025)

    work page arXiv 2025

  66. [66]

    L´ oio, G

    H. L´ oio, G. Cecile, S. Gopalakrishnan, G. Lami, and J. De Nardis, Correlations, spectra, and entanglement transitions in ensembles of matrix product states, Phys- ical Review B112, 035127 (2025)

    work page 2025

  67. [67]

    De Luca, C

    A. De Luca, C. Liu, A. Nahum, and T. Zhou, Univer- sality Classes for Purification in Nonunitary Quantum Processes, Physical Review X15, 041024 (2025)

    work page 2025

  68. [68]

    Collins and S

    B. Collins and S. Matsumoto, Weingarten calculus via orthogonality relations: new applications, arXiv preprint arXiv:1701.04493 (2017)

    work page arXiv 2017

  69. [69]

    Collins and P

    B. Collins and P. ´Sniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Communications in Mathematical Physics264, 773 (2006)

    work page 2006

  70. [70]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.-C. Kwek, and A. Aspuru-Guzik, Noisy intermediate-scale quantum algorithms, Rev. Mod. Phys.94, 015004 (2022)

    work page 2022

  71. [71]

    S. Lee, C. Oh, Y. Wong, S. Chen, and L. Jiang, Universal Spreading of Conditional Mutual Information in Noisy Random Circuits, Physical Review Letters133, 200402 (2024)

    work page 2024

  72. [72]

    S. Sang, Y. Zou, and T. H. Hsieh, Mixed-state Quan- tum Phases: Renormalization and Quantum Error Cor- rection, Physical Review X14, 031044 (2023)

    work page 2023

  73. [73]

    Zhang, C

    M. Zhang, C. Wang, S. Dong, H. Zhang, Y. Han, and L. He, Entanglement entropy scaling of noisy random quantum circuits in two dimensions, Physical Review A 106, 052430 (2022)

    work page 2022

  74. [74]

    Denzler, J

    J. Denzler, J. Carrasco, J. Eisert, and T. Guaita, Simu- lation of noisy quantum circuits using frame representa- tions, arXiv preprint arXiv:2601.05131 (2026)

    work page arXiv 2026

  75. [75]

    J. Bai, J. Wang, and Z. Yin, Primitivity for random quan- tum channels, Quantum Information Processing23, 1 (2024)

    work page 2024

  76. [76]

    Collins and I

    B. Collins and I. Nechita, Random matrix techniques in quantum information theory, Journal of Mathematical Physics57, 015215 (2016)

    work page 2016

  77. [77]

    Zyczkowski and H.-J

    K. Zyczkowski and H.-J. Sommers, Induced measures in the space of mixed quantum states, Journal of Physics A: Mathematical and General34, 7111 (2001)

    work page 2001

  78. [78]

    Collins and I

    B. Collins and I. Nechita, Random quantum channels I: graphical calculus and the Bell state phenomenon, Com- munications in Mathematical Physics297, 345 (2009)

    work page 2009

  79. [79]

    Zyczkowski, K

    K. Zyczkowski, K. A. Penson, I. Nechita, and B. Collins, Generating random density matrices, Journal of Mathe- matical Physics52, 062201 (2011)

    work page 2011

  80. [80]

    S. G. Bobkov and F. G¨ otze, Concentration of empiri- cal distribution functions with applications to non-i.i.d. models, Bernoulli16, 1385 (2010)

    work page 2010

Showing first 80 references.