Pith. sign in

REVIEW 3 major objections 4 minor 44 references

Noise bias in randomized measurements follows an exponential size law.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 17:03 UTC pith:JICIXKYY

load-bearing objection A plausible new light-cone damping law for noisy shadows, numerically well-supported, but the central extensivity proof is deferred to the SM. the 3 major comments →

arxiv 2607.17740 v1 pith:JICIXKYY submitted 2026-07-20 quant-ph

Light-Cone Scaling of In-Circuit Noise in Randomized Measurements

Pan Yu , Yan He , Yadong Wu This is my paper
classification quant-ph
keywords randomized measurementsclassical shadowsnoise biaslocally scrambled circuitsPauli twirlinglight coneoperator spreadingquantum error mitigation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper establishes that, in one-dimensional shallow locally scrambled circuits, gate noise does not act as a flat channel error: a local noise event contributes only when it overlaps the backward Heisenberg evolution of the measured Pauli operator. For a contiguous string of k Pauli operators, the activated light cone contains O(kd) bulk gates and O(d^2) boundary gates, so the logarithm of the damping ratio is linear in k, log eta_k = -alpha(d, Lambda) k - beta(d, Lambda). The authors derive this exponential law from a path-average formula and verify it numerically for random Clifford and locally scrambled iSWAP circuits with uniform, spatially fluctuating, and temporally drifting noise. The practical payoff is a small-string calibration protocol: measuring a few short strings on a product state fixes alpha and beta, and the law then predicts the damping of larger strings with the same geometry without learning the full noisy measurement channel. A sympathetic reader would care because this connects noise bias in shadow tomography to operator-spreading dynamics and offers a cheap, geometry-resolved error-mitigation strategy.

Core claim

For a locally scrambled shallow circuit with independent local twirling, the noisy measurement channel is Pauli-diagonal, and the damping ratio of a contiguous size-k Pauli string is eta_k = E_nu e^{-X_k}, where X_k sums local stochastic Pauli damping rates over space-time locations that overlap the Heisenberg support of the evolving operator. Because the light-cone bulk is extensive in k while the two fronts contribute only O(d^2), each cumulant of X_k is k a_n(d) + b_n(d); exponentiation of the cumulant expansion yields log eta_k = -alpha(d, Lambda) k - beta(d, Lambda). The paper verifies this exponential scaling for two-qubit random Clifford and locally scrambled iSWAP brick-wall circuits

What carries the argument

The key object is the activated path-average formula (Eq. 4): w_Lambda(P) = sum_gamma Pr(gamma|P, E_U) 3^{-m(gamma)} exp(-sum_a I_a(gamma) lambda_a(gamma)). Local twirling reduces CPTP noise to Pauli damping, and a noise location a is 'activated' (I_a = 1) only when it overlaps the instantaneous Heisenberg support of the measured operator. Dividing by the ideal path average defines the measurement-weighted path measure nu(gamma|P), which favors small final supports and is the distribution over which the noise is averaged. The supporting identity is the light-cone volume count: for a contiguous string away from boundaries, the bulk contains O(kd) gates and the two fronts O(d^2), giving linear

Load-bearing premise

The exponential law rests on the assertion that every cumulant of the activated noise is linear in the string length k, i.e., that noise events at different light-cone locations are effectively independent under the measurement-weighted path measure; if gates in a layer share a common-mode fluctuation, this linearity fails.

What would settle it

Measure log eta_k versus k for a shallow 1D circuit in which the same noise value is applied to every gate in a layer (common-mode noise). If the curve exhibits curvature or a k-independent offset that breaks the linear fit within the predicted window, the extensivity of cumulants fails. Alternatively, on hardware, fit alpha and beta from small strings on a product state, then prepare a state with known large-string expectation values (e.g., a cluster state) and compare predicted versus measured damping at larger k; a systematic mismatch would falsify the transfer.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

If this is right

  • For any observable whose light cone is a single contiguous string, the damping ratio is determined by two scalar parameters alpha(d, Lambda) and beta(d, Lambda) instead of the full noisy measurement channel.
  • Small-string measurements on a known product state suffice to fit these parameters; the same fitted law predicts the coefficient of larger strings with matching support geometry.
  • The calibration corrects the bias of noisy shadow estimates, as demonstrated for cluster-state (ZXZ) strings, and the resulting sample variance follows the noisy shadow norm eta^{-2} w_0(P).
  • The slope alpha carries information about operator spreading: the heuristic equilibrium estimate alpha_eq = 15 d lambda / 32 does not reproduce finite-depth slopes, showing that activation has not relaxed to equilibrium and that Jensen's inequality matters.
  • Noncontiguous observables and higher-dimensional circuits obey the same path-average principle but acquire different volume scalings, so they must be calibrated with reference observables of the same light-cone geometry.

Where Pith is reading between the lines

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

  • If the exponential law persists beyond the simulated ranges, hardware could calibrate large-string shadow estimates from a single product-state run plus a noiseless transfer-matrix computation, turning operator-spreading dynamics into a practical noise model.
  • The damping-slope extraction suggests a converse diagnostic: measuring how alpha varies with circuit depth and observable geometry could serve as an experimental probe of operator-spreading velocities in locally scrambled circuits.
  • The linear-law assumption is directly testable for correlated noise: adding a shared common-mode fluctuation to every gate in a layer should produce curvature in log eta_k versus k, providing a clean experimental signature of such correlated errors.
  • A similar light-cone volume argument may extend to other randomized-measurement families with known Heisenberg support evolution, such as locally entangled or dual-unitary circuits, where the same geometry-to-noise relation could hold.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a microscopic, path-integral description of how local in-circuit noise biases classical-shadow estimators based on shallow locally scrambled circuits. Under independent local twirling, noise reduces to stochastic Pauli damping, and a noise event contributes only when it intersects the Heisenberg support path of the measured Pauli operator. This leads to a measurement-weighted path-average formula for the noisy Pauli coefficient, Eq. (4). For contiguous Pauli strings in 1D shallow circuits, the light-cone volume argument yields an exponential damping law log η_k = -α(d,Λ)k - β(d,Λ), Eq. (7), whose slope and offset are determined by bulk and boundary cumulant contributions. The authors verify the law numerically for random Clifford and locally scrambled iSWAP circuits of depth d=2,4,6 under uniform, spatially fluctuating, and temporally drifting noise, and use it in a small-string calibration protocol to predict larger-string damping and to correct cluster-state expectation values.

Significance. If the central exponential law is correct, this work gives a conceptually clean and practically useful connection between operator spreading and noise exposure in randomized measurements. It avoids learning the full noisy measurement channel, replacing it with a few geometry-resolved damping parameters. The numerical transfer-matrix verification is a strength: it covers two different gate ensembles, anisotropic noise, spatial fluctuations, and temporal drift, with high reported R² values. The derivation of Eq. (7) from a cumulant expansion is also not a fit to the target result; it is a parameter-free prediction up to the cumulant-extensivity step, and the heuristic equilibrium estimate provides a useful benchmark. The main open point is the missing proof or precise condition for the cumulant extensivity assertion, which is load-bearing for the large-k extrapolation underlying the calibration protocol.

major comments (3)
  1. [Light-cone volume law, after Eq. (6)] The central result Eq. (7) rests on the assertion that every cumulant of X_k is extensive in k: κ_n(X_k) = k a_n(d) + b_n(d). This is not a trivial consequence of the light-cone volume O(kd). The measurement-weighted path measure ν(γ|P_k) is non-product because of the 3^{-m(γ)} final-support tilt, and the indicators I_a are correlated along the support path. Connected cumulants of the sum over the light cone are affine in k only if the tilted path ensemble has sufficiently short-ranged spatial correlations; that condition is neither stated nor proved in the main text and is deferred to [40]. Since the calibration protocol extrapolates from fitted small-k parameters to large-k observables, this step is load-bearing. Please provide the proof or a precise mixing/clustering condition, and show that it holds for the Clifford and iSWAP ensembles considered. The paper's own caveat that a layer-
  2. [Figs. 2 and 3] The numerical evidence for Eq. (7) consists of linear fits over short even-k windows: k=4..14 for Clifford and k=6..14 for iSWAP, with reported R² in (0.99,0.99999). No error bars, residuals, or k-window stability analysis are given. Because the predictive content of the protocol is exactly the extrapolation of the fitted line to larger k, the reader needs to see (i) how α_p and β_p change with the fitting window, (ii) whether the linear form holds for k>14 in the transfer-matrix data, and (iii) the statistical uncertainty of the slope and intercept. Please add this analysis or explicitly state that the fits are single representative realizations.
  3. [Fig. 4(a)] The calibration demonstration uses a single 10^7-sample realization with circuit-global p~N(0.03,0.005^2). There is no sample-to-sample spread, and the error bars on the fitted α_p,β_p are not given. Since the final claim is that small-string fits predict larger strings, the uncertainty in that prediction should be quantified. A single realization cannot establish the reliability of the extrapolation; please report statistics over multiple noise realizations or over bootstrap resamples of the calibration data.
minor comments (4)
  1. [Abstract and Fig. 2 caption] Typos: 'iSW AP' should be 'iSWAP' in the abstract and in the Fig. 2 caption.
  2. [Calibration section] Notation is inconsistent: 'Eq. (2)' appears as 'eq. (2)' in the text, and the noiseless coefficient is written w_EU,0, w0, and w_0 in different places; please unify.
  3. [Fig. 4(b)] The definition of the cluster-state string O_ZXZ uses 'Z_9 Y_10 X_11 ...' but earlier text describes it as 'Z_1 Y_2 X_3 ...'; please make the indexing consistent or explain the shift.
  4. [General notation] The symbol d is used for circuit depth and appears in the same context as the operator size k; also the 'ZXZ state' is not defined. A brief definition would help.

Circularity Check

0 steps flagged

No significant circularity: the linear damping law is derived from an explicit cumulant-extensivity premise, and the calibration protocol extrapolates beyond its fitting window rather than predicting its own inputs.

full rationale

The paper's central result, log eta_k = -alpha(d,Lambda)k - beta(d,Lambda) (Eq. 7), is obtained from Eq. (6) together with the stated structural premise that every cumulant of the activated-noise variable X_k is affine in k: kappa_n(X_k) = k a_n(d) + b_n(d). That premise is stronger than, not equivalent to, the target linear law: log eta_k is an alternating sum over all cumulants, so the premise genuinely implies the conclusion rather than merely restating it. No parameter in Eq. (7) is fitted before the equation is derived; alpha and beta are defined as sums of the cumulant coefficients. The small-string calibration of Fig. 4 fits alpha_p and beta_p to small-k data and then extrapolates to larger strings that are not part of the fit, so the prediction is not statistically forced by the input data. The paper also explicitly delimits the regime of validity, noting that a random fluctuation shared by every gate in a layer correlates an entire light-cone slice and generates nonlinear corrections in k; this is a stated limitation of the premise, not evidence of circularity. The only same-author citation ([20], Y. Wu et al.) appears in a general list of shallow/locally scrambled ensembles and is not load-bearing for the derivation. The deferred proof of cumulant extensivity in the Supplemental Material is a support gap (correctness risk), but no equation or parameter in the paper reduces to its own input by construction, so no circular step is exhibited.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The central derivation rests on standard local-scrambling and weak-noise assumptions, plus the cumulant-extensivity hypothesis. There are no invented physical entities; the only fitted quantities are the slope and intercept of the damping law used in the calibration protocol.

free parameters (1)
  • Damping slope alpha_p and intercept beta_p (per parity branch)
    In the calibration protocol these are fitted to small-string data and then used to predict larger strings. Their closed-form values are not derived from first principles; the heuristic alpha_eq fails to match numerical fits.
axioms (4)
  • domain assumption The random circuit ensemble is locally scrambled, so independent local sign flips remove off-diagonal Pauli components of the noisy channel.
    Used to justify the Pauli-diagonal form of the measurement channel and the factor 3^{-m(gamma)} in Eq. (3).
  • domain assumption Noise is weak, local, CPTP, gate-independent, with independent local twirls between circuit layers and no leakage.
    Stated in the introduction and used to reduce each local noise channel to a stochastic Pauli damping factor exp(-lambda_a) in Eq. (4).
  • standard math Operator spreading in 1D shallow circuits has a light cone with O(kd) bulk and O(d^2) front gates (from refs [38,39]).
    Used to conclude that each cumulant of the activated noise is extensive in k, giving the linear law Eq. (7).
  • domain assumption The measurement-weighted path measure nu and noise variables satisfy a cluster property so that kappa_n(X_k) = k a_n(d) + b_n(d).
    This is the key step from Eq. (6) to Eq. (7); it is asserted in the main text and deferred to the Supplemental Material. The paper notes that common-mode noise can break it.

pith-pipeline@v1.3.0-alltime-deepseek · 13058 in / 13851 out tokens · 158547 ms · 2026-08-01T17:03:32.556319+00:00 · methodology

0 comments
Cite this review

Pith. "Pith review of Light-Cone Scaling of In-Circuit Noise in Randomized Measurements." pith.science (2026). https://pith.science/paper/JICIXKYY

@misc{pith2026260717740,
  author       = {Pith},
  title        = {Pith review of: Light-Cone Scaling of In-Circuit Noise in Randomized Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JICIXKYY}},
  note         = {Machine review of arXiv:2607.17740}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Randomized measurements provide an efficient way to extract physical properties of an unknown quantum state from limited data. On near-term hardware, gate and readout errors bias the reconstructed observables. Here we develop a microscopic description of this bias for locally scrambled shallow circuits. Independent local twirling reduces local implementation noise to stochastic Pauli damping, and a noise event contributes only when it overlaps the Heisenberg evolution of the measured Pauli operator. This gives an activated path-average formula for the noisy Pauli coefficient. In one-dimensional shallow circuits, the activated noise volume grows linearly with the size of a contiguous observable, leading to an exponential damping ratio. We verify this scaling for two-qubit random Clifford and locally scrambled iSWAP circuits with two-qubit Pauli noise, including spatial fluctuations and temporal drift. The scaling supports a small-string calibration protocol that predicts larger string observables without learning the full noisy measurement channel. Our result relates the noise bias of shallow-shadow protocols directly to operator-evolving dynamics.

Figures

Figures reproduced from arXiv: 2607.17740 by Pan Yu, Yadong Wu, Yan He.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) uses 107 samples with a circuit-global two￾qubit depolarizing probability pcircuit ∼ N (0.03, 0.0052 ) to extract the damping law from the calibration strings Zˆ 9 · · ·Zˆ 8+k, which have the same light-cone geometry as the target observable. The scaling parameters fit￾ted at small k accurately extrapolate the damping fac￾tor to larger operator sizes. We then apply the fit￾ted damping law to the cluste… view at source ↗

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

44 extracted references · 3 linked inside Pith

  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]

    Aaronson, Shadow tomography of quantum states (2018), arXiv:1711.01053 [quant-ph]

    S. Aaronson, Shadow tomography of quantum states (2018), arXiv:1711.01053 [quant-ph]

    Pith/arXiv arXiv 2018

  3. [3]

    Elben, B

    A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, R´ enyi entropies from random quenches in atomic hubbard and spin models, Phys. Rev. Lett. 120, 050406 (2018)

    2018

  4. [4]

    Brydges, A

    T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, and C. F. Roos, Probing r´ enyi entanglement entropy via randomized measurements, Science 364, 260 (2019), https://www.science.org/doi/pdf/10.1126/science.aau4963

    doi:10.1126/science.aau4963 2019

  5. [5]

    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

  6. [6]

    Huang, R

    H.-Y. Huang, R. Kueng, and J. Preskill, Efficient estima- tion of pauli observables by derandomization, Phys. Rev. Lett. 127, 030503 (2021)

    2021

  7. [7]

    Hadfield, S

    C. Hadfield, S. Bravyi, R. Raymond, and A. Mezzacapo, Measurements of quantum hamiltonians with locally- biased classical shadows, Communications in Mathemat- ical Physics 391, 951 (2022)

    2022

  8. [8]

    Huang, Learning quantum states from their classi- cal shadows, Nature Reviews Physics 4, 81 (2022)

    H.-Y. Huang, Learning quantum states from their classi- cal shadows, Nature Reviews Physics 4, 81 (2022)

    2022

  9. [9]

    Elben, S

    A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The random- ized measurement toolbox, Nature Reviews Physics 5, 9 (2023)

    2023

  10. [10]

    Hu and Y.-Z

    H.-Y. Hu and Y.-Z. You, Hamiltonian-driven shadow to- mography of quantum states, Phys. Rev. Res. 4, 013054 (2022)

    2022

  11. [11]

    H.-Y. Hu, S. Choi, and Y.-Z. You, Classical shadow tomography with locally scrambled quantum dynamics, Phys. Rev. Res. 5, 023027 (2023)

    2023

  12. [12]

    Ippoliti, Y

    M. Ippoliti, Y. Li, T. Rakovszky, and V. Khemani, Oper- ator relaxation and the optimal depth of classical shad- ows, Phys. Rev. Lett. 130, 230403 (2023)

    2023

  13. [13]

    A. A. Akhtar, H.-Y. Hu, and Y.-Z. You, Scalable and Flexible Classical Shadow Tomography with Tensor Net- works, Quantum 7, 1026 (2023)

    2023

  14. [14]

    Zhang, X

    S. Zhang, X. Feng, M. Ippoliti, and Y.-Z. You, Holographic classical shadow tomography (2024), arXiv:2406.11788 [quant-ph]

    arXiv 2024

  15. [15]

    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

  16. [16]

    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

  17. [17]

    Bertoni, J

    C. Bertoni, J. Haferkamp, M. Hinsche, M. Ioannou, J. Eisert, and H. Pashayan, Shallow shadows: Expecta- tion estimation using low-depth random clifford circuits, Phys. Rev. Lett. 133, 020602 (2024)

    2024

  18. [18]

    Zhou and P

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

    2024

  19. [19]

    A. A. Akhtar, N. Anand, J. Marshall, and Y.-Z. You, Dual-unitary shadow tomography, Quantum 9, 1816 (2025)

    2025

  20. [20]

    Y. Wu, C. Wang, J. Yao, H. Zhai, Y.-Z. You, and P. Zhang, Contractive unitary and classical shadow to- mography, npj Quantum Information 12, 86 (2026)

    2026

  21. [21]

    Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018)

    J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018)

    2018

  22. [22]

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

    2021

  23. [23]

    H.-Y. Hu, R. LaRose, Y.-Z. You, E. Rieffel, and Z. Wang, Logical shadow tomography: Efficient estimation of error-mitigated observables (2022), arXiv:2203.07263 [quant-ph]

    Pith/arXiv arXiv 2022

  24. [24]

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

    2022

  25. [25]

    Seif, Z.-P

    A. Seif, Z.-P. Cian, S. Zhou, S. Chen, and L. Jiang, Shadow distillation: Quantum error mitigation with clas- sical shadows for near-term quantum processors, PRX Quantum 4, 010303 (2023)

    2023

  26. [26]

    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

  27. [27]

    Onorati, J

    E. Onorati, J. Kitzinger, J. Helsen, M. Ioannou, A. H. Werner, I. Roth, and J. Eisert, Noise-mitigated random- ized measurements and self-calibrating shadow estima- tion (2024), arXiv:2403.04751 [quant-ph]

    Pith/arXiv arXiv 2024

  28. [28]

    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). 6

    2024

  29. [29]

    Rozon, N

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

    2024

  30. [30]

    H.-Y. Hu, A. Gu, S. Majumder, H. Ren, Y. Zhang, D. S. Wang, Y.-Z. You, Z. Minev, S. F. Yelin, and A. Seif, Demonstration of robust and efficient quantum property learning with shallow shadows, Nature Communications 16, 2943 (2025)

    2025

  31. [31]

    Brieger, M

    R. Brieger, M. Heinrich, I. Roth, and M. Kliesch, Sta- bility of classical shadows under gate-dependent noise, Phys. Rev. Lett. 134, 090801 (2025)

    2025

  32. [32]

    W.-T. Kuo, A. A. Akhtar, D. P. Arovas, and Y.-Z. You, Markovian entanglement dynamics under locally scrambled quantum evolution, Phys. Rev. B 101, 224202 (2020)

    2020

  33. [33]

    A. A. Akhtar and Y.-Z. You, Multiregion entanglement in locally scrambled quantum dynamics, Phys. Rev. B 102, 134203 (2020)

    2020

  34. [34]

    C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlocality without entanglement, Phys. Rev. A 59, 1070 (1999)

    1999

  35. [35]

    J. J. Wallman and J. Emerson, Noise tailoring for scalable quantum computation via randomized compiling, Phys. Rev. A 94, 052325 (2016)

    2016

  36. [36]

    Cai and S

    Z. Cai and S. C. Benjamin, Constructing smaller pauli twirling sets for arbitrary error channels, Scientific Re- ports 9, 11281 (2019)

    2019

  37. [37]

    S. Chen, Y. Liu, M. Otten, A. Seif, B. Fefferman, and L. Jiang, The learnability of pauli noise, Nature Com- munications 14, 52 (2023)

    2023

  38. [38]

    C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, otocs, and en- tanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018)

    2018

  39. [39]

    Nahum, S

    A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018)

    2018

  40. [40]

    See supplemental material for details of the locally scram- bled noisy-shadow protocol, the local twirling reduction of cptp noise to stochastic pauli damping, the activated path-average formula, the cumulant expansion and light- cone scaling, transfer-matrix calculations for random clif- ford and locally scrambled iswap circuits, anisotropic- noise and r...

  41. [41]

    X. Mi, P. Roushan, C. Quintana, S. Mandr` a, J. Marshall, C. Neill, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, J. Basso, A. Bengtsson, S. Boixo, A. Bourassa, M. Broughton, B. B. Buck- ley, D. A. Buell, B. Burkett, N. Bushnell, Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. R. Derk, A. Dunsworth, D. Eppens, C. Erickson...

    2021

  42. [42]

    McGinley, S

    M. McGinley, S. Leontica, S. J. Garratt, J. Jovanovic, and S. H. Simon, Quantifying information scrambling via classical shadow tomography on programmable quantum simulators, Phys. Rev. A 106, 012441 (2022)

    2022

  43. [43]

    Harris, B

    J. Harris, B. Yan, and N. A. Sinitsyn, Benchmarking information scrambling, Phys. Rev. Lett. 129, 050602 (2022)

    2022

  44. [44]

    Wang, T.-Q

    J.-H. Wang, T.-Q. Cai, X.-Y. Han, Y.-W. Ma, Z.-L. Wang, Z.-H. Bao, Y. Li, H.-Y. Wang, H.-Y. Zhang, L.-Y. Sun, Y.-K. Wu, Y.-P. Song, and L.-M. Duan, Informa- tion scrambling dynamics in a fully controllable quantum simulator, Phys. Rev. Res. 4, 043141 (2022)

    2022