Pith. sign in

REVIEW 3 major objections 5 minor 80 references

Quantum cellular automata for word statistics facilitated by quantum correlations

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read This paper claims that postselected random quantum circuits with effective two-site gates steer product states toward a target entangled state, with distinct gate-density thresholds for state update and information scrambling.

desk verdict Novel postselected corpus-state circuit with plausible numerics, but the percolation threshold and search speedup are both under-supported. read the letter →

arxiv 2504.14453 v1 pith:EVWF3UJ2 submitted 2025-04-20 quant-ph

Guanhua Chen , Yao Yao This is my paper · ORCID
classification quant-ph PACS 03.65.Ud03.67.Ac03.67.Mn05.50.+q
keywords quantumcellularautomata2-grammodelpostselectionrandomunitarycircuitsinformationscramblingbondpercolationentanglementasymmetrytripartitemutual
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

Adding effective two-site gates to a postselected random quantum circuit makes the search converge faster to a target entangled 'corpus state' built from a 2-gram word model, and the dynamics exhibits two distinct gate-density thresholds. The authors identify Sts = 3 as the point where the final-state distance to the corpus state stops improving, and Sts = 2 as the point where the tripartite mutual information vanishes, marking the critical density for information scrambling. The algorithm therefore provides a way to tune between local state refinement and global information spreading by changing only the probability of sampling two-site gates. If the results hold, they point to a postselection-driven protocol for directed preparation of entangled states that is governed by the structure of a corpus rather than by a Hamiltonian.

What carries the argument

The load-bearing object is the corpus state |CS⟩ = (2|you⟩|are⟩ + |are⟩|here⟩ + |here⟩|you⟩)/√6, a superposition of the allowed adjacent word pairs in the two-sentence corpus, together with the postselection rule: a random circuit proposes an update via one-site and two-site gates, and the update is kept only if the Frobenius distance D (averaged over nearest-neighbor pairs) between ρ_{j,j+1} and |CS⟩⟨CS| decreases. The circuit itself is a four-layer brickwork of staggered two-site gates on alternating bonds plus one-site gates, with gate entropies Sos and Sts controlling how often effective operations are sampled. The thresholds for update saturation (Sts = 3) and scrambling (Sts = 2) are extracted from the final-state distance and the tripartite mutual information respectively.

What would settle it

Simulate or implement the same update rule on a system with L = 8 or larger and d = 3, sampling gates from the stated distributions, and measure the final-state tripartite mutual information as a function of Sts: if the vanishing above Sts = 2 does not persist, or if the same step-function appears in a classical simulation that stores the full state without any quantum measurement, then the bond-percolation identification for information scrambling is not established.

Watch

Extended reading notes

Core claim

The paper's central claim is that effective two-site gates 'directionally optimize initial states to the target many-body entangled state' and that the resulting dynamics is organized by a bond-percolation threshold in gate sampling. With two-site gates, the average Frobenius distance D between the updating state's nearest-neighbor reduced density matrices and the corpus state decays faster than logarithmic, following a power law with an exponent controlled by the gate entropy; without them, the decay is essentially logarithmic. The final-state D saturates once the two-site gate entropy Sts exceeds 3, interpreted as the two-site bond-percolation threshold for the state-update process. In parallel, the tripartite mutual information of the final state vanishes for Sts > 2, interpreted as the critical point of information scrambling; the separation between the two thresholds shows that two-site gates can still update local blocks while long-range quantum information is localized. A modified indicator that groups two sites at a time suppresses scrambling entirely, which the authors take as evidence that the two-site interaction channel is what transmits quantum correlations.

Load-bearing premise

The whole construction rests on being able to check at every step whether a proposed random update moves the local state closer to the target, and to accept or reject the update on that basis; if that check cannot be done without destroying or measuring the state, the claimed quantum search protocol does not run as described.

Editorial extensions

If this is right

  • Replacing the corpus state with a more complex n-gram state gives a family of directed state-preparation protocols that target entangled states beyond the three-word example.
  • Varying the two-site gate density alone lets one move between a regime where local updates occur but global scrambling is suppressed and a regime where quantum information spreads through the chain.
  • If the percolation interpretation is correct, the scrambling threshold is controlled by the connectivity of two-site bonds rather than by the detailed form of the two-site unitaries.
  • Changing the postselection statistic from single-site pairs to two-site groupings (the D' modification) suppresses long-range correlation, showing that the geometry of the update check determines whether information scrambles.

Reading between the lines

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

  • A physical implementation would likely use mid-circuit measurements or weak measurements; if the thresholds survive that replacement, the model becomes an instance of measurement-induced criticality in a language-motivated setting.
  • The gap between the update threshold (Sts=3) and scrambling threshold (Sts=2) suggests a design principle: a protocol can keep refining local structure while leaving long-range correlations localized, which may be useful when one wants to prepare a state without first scrambling the whole system.
  • A quick test: replace the Frobenius distance with the trace distance or swap the two-site gate set; if the thresholds do not move, the phenomenon is tied to the postselection statistic, and if they do, the gate details matter.
  • Because the corpus state resembles a valence-bond entangled state, replacing it with a matrix-product target state should allow the same algorithm to be tested at larger system sizes and the thresholds checked for finite-size scaling.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 proposes an iterative, postselected search algorithm on a one-dimensional random quantum cellular automaton. The update rule (Eq. 2) accepts a random circuit step only if the average Frobenius distance D from the two-site reduced density matrices to a 2-gram 'Corpus State' |CS⟩ decreases; otherwise the previous state is kept. The authors study how adding two-site gates affects the evolution, reporting (i) a 'search speedup' characterized by the fitted exponent b in the power law D = a t^b + c (Eq. 8), (ii) block-diagonalization of the reduced density matrices quantified by an entanglement asymmetry, and (iii) two distinct 'bond percolation' thresholds: Sts = 3 for the update rate (Fig. 3(b)) and Sts = 2 for the tripartite mutual information (Fig. 4(c)). The central claims are that the two-site gates directionally optimize initial states toward the target entangled state and that information scrambling undergoes a percolation transition.

Significance. If the claims were fully supported, the paper would offer a novel connection between n-gram-inspired cooperative interactions and quantum many-body dynamics, and a potentially useful postselected search paradigm. The authors provide a concrete numerical protocol, explicit gate sets, and averaged simulations over 500 configurations; the raw decrease of D under the accept-if-closer rule is a direct consequence of the rule and is reproduced in the simulations. Nevertheless, the two headline findings — the speedup and the bond-percolation transition — are currently supported only by fitted or visual evidence, and the physical implementability of the postselection rule is not addressed. The paper is therefore of moderate significance as a proposal, but its quantitative claims need substantial revision or re-scoping before they can be accepted.

major comments (3)
  1. [Setup, Eq. (2)] The postselection rule requires evaluating D, which needs all nearest-neighbor two-site reduced density matrices of the full state at every step. The paper gives no measurement protocol that would obtain these reduced density matrices without destroying the state or otherwise paying a cost; on actual hardware such measurements generally collapse the state, while on a classical simulator the updating 'state' is a classical data structure and the phrase 'quantum search speedup' is not justified. The authors should either provide a non-demolition measurement/feedback scheme that makes the rule physically implementable, or explicitly state that the algorithm is analyzed as a classical postprocessing protocol on a simulated state and revise the quantum-search framing accordingly.
  2. [Results, Figs. 3(b) and 4(c)] The claimed bond-percolation thresholds, Sts = 3 in Fig. 3(b) and Sts = 2 in Fig. 4(c), are inferred from visual plateaus in the final-state distance D and the tripartite mutual information TMI. No percolation observable is computed: there is no spanning probability, cluster-size distribution, or finite-size scaling, and no mapping is given between Sts and a bond-occupation probability on a known percolation lattice. For L = 6 and t ≈ 5 × 10^3, the plateaus could be finite-size or saturation effects. Moreover, the text assigns two different threshold values to 'bond percolation' without explaining how one connectivity transition produces two critical points. Please provide a genuine percolation analysis with finite-size scaling, or relabel these observations as crossover or plateau features.
  3. [Results, Eq. (8) and Fig. 2(a)] The 'search speedup' is quantified by the fitted exponent b in D = a t^b + c. With c ≈ 1.1, D approaches a nonzero floor rather than converging to the target |CS⟩, so the fitted exponent describes a transient approach to a finite residual distance. No error bars or goodness-of-fit measures are reported for a, b, or c, and b is fitted to the same quantity D that defines the acceptance rule, making the 'speedup' a fitted characterization of the rule's own dynamics rather than an independent prediction. Please provide confidence intervals, model comparison (e.g., against a logarithmic or exponential decay), and a threshold-based definition of convergence (for example, time to reach a fixed small D) before claiming a quantum search speedup.
minor comments (5)
  1. [Fig. 2 caption] The sentence 'the the evolution of D is a smooth concave function' contains a duplicated article 'the'; please remove the duplicate.
  2. [Fig. 2(b) caption] The caption lists 'Sts = log10(1), log10(2), log10(3) and 1', which gives values 0, 0.301, 0.477, and 1, yet the text later discusses thresholds at Sts = 2 and Sts = 3. Please clarify the scale and the meaning of the plotted Sts values, presumably Sts = -log10(P) with P the sampling probability of effective two-site gates.
  3. [All figures] All results are averaged over 500 configurations, but no error bars or standard deviations are shown; please add error bars or state that they are smaller than the marker size.
  4. [Results, paragraph near Fig. 3(b)] The statement 'the update for Sts > 3 is totally determined by one-site gates' is too strong, because two-site gates are still sampled with nonzero probability in that regime; please rephrase as 'dominated by one-site gates' or give a precise quantitative criterion.
  5. [Abstract and Introduction] The phrase 'information scrambling dependent of gate sampling' should be 'information scrambling dependent on gate sampling'; please correct the grammar.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the postselected search results are numerical outputs of a defined update rule, and the speedup/percolation claims are descriptive fits and analogies rather than inputs renamed as predictions.

full rationale

The paper defines a postselected random-circuit dynamics: Eq. (2) defines the Frobenius distance D to the corpus state, and the update rule explicitly keeps only moves that decrease D. The resulting non-increasing behavior of D is therefore the definition of the algorithm, not a hidden prediction. The quantitative results presented as findings—entanglement asymmetry (Fig. 2(b)), three-site correlation C (Fig. 4(a)), and tripartite mutual information (Fig. 4(b,c))—are computed on the accepted trajectory and are not used in the accept/reject rule, so they provide independent information about the evolved state. The 'search speedup' is inferred from the fitted power law D = at^b + c: the exponent b is a descriptive fit to the same observable used in the acceptance rule, but the comparison between circuits with and without two-site gates is an empirical property of the stochastic process, not an equality forced by construction, and the paper does not predict an independent observable from that fit. The 'bond percolation' thresholds (Sts = 3 and Sts = 2) are identified from plateaus in the final-state distance and TMI, respectively; this is an interpretive analogy to percolation theory rather than a circular derivation, and whether the identification is sufficiently supported is a correctness/statistical question, not a circularity. There are no load-bearing self-citations, no imported uniqueness theorems from the authors, and no ansatz smuggled in by citation; the algorithm and all reported observables are self-contained numerical experiments. The absence of a Supplemental Material file for the cross-referenced numerical details is a completeness issue, but it does not make the derivation circular.

Assumptions & free parameters 1 free parameters · 4 assumptions · 1 invented entities

The model depends on several choices that are not derived from first principles: the specific corpus and the resulting |CS⟩, the greedy distance-decrease acceptance rule, the choice of d=3 as minimal, and the specific gate sets. None of these have independent empirical support; they define the model rather than follow from it. The only fitted parameters are the power-law coefficients used to quantify the speedup. The invented entity is the corpus state itself, which has no external falsifiable handle.

free parameters (1)
  • Power-law coefficients a, b, c in D = a t^b + c (Eq. 8) = b decreases when two-site gates are added; c = 1.1 for the minimal D in L=6; a is the initial distance
    The claimed search speedup is the decrease of the fitted exponent b; the floor c=1.1 is fitted to the data and is interpreted as the minimum achievable distance. These fitted coefficients, not a derivation, support the speedup claim.
assumptions (4)
  • ad hoc to paper The 2-gram corpus state |CS⟩, Eq. (1), is a valid target and interaction for a quantum search.
    The corpus 'you are here' and 'here you are' is chosen by hand; the analogy to AKLT states is noted but no independent justification is given for why this target should be searched for.
  • domain assumption The postselection rule (accept only if D decreases, Eq. 2) is a legitimate update mechanism.
    Introduced in the Setup; it is a greedy Metropolis-like rule with no temperature and no discussion of measurement cost or backaction on the quantum state.
  • domain assumption d=3 is the minimal dimension for interesting cooperative interactions.
    Stated in Setup: 'the qutrit with d = 3 is the minimal local dimension to consider intriguing cooperative interactions.' No proof is given.
  • ad hoc to paper The specific gate sets {X, Z_you, Z_are, Z_here} and {CF_you, CF_are, CF_here} (Eqs. 4-7) are a fair representative sampling for studying correlation and scrambling.
    These operators are constructed for the three words; no argument shows they are generic or that thresholds are independent of this choice.
invented entities (1)
  • Corpus State |CS⟩
    purpose: Acts as the target state and the 'interaction' defining the 2-gram cooperative update; D measures distance to it.
    Introduced in Eq. (1) as a superposition of adjacent word pairs. It has no experimentally measurable handle independent of the paper's own model.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum cellular automata for word statistics facilitated by quantum correlations." pith.science (2026). https://pith.science/paper/EVWF3UJ2

@misc{pith2026250414453,
  author       = {Pith},
  title        = {Pith review of: Quantum cellular automata for word statistics facilitated by quantum correlations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVWF3UJ2}},
  note         = {Machine review of arXiv:2504.14453}
}
read the original abstract

We propose an iterative algorithm to investigate the cooperative evolution dominated by information encoded within state spaces in a random quantum cellular automaton. Inspired by the 2-gram model in statistical linguistics, the updates of quantum states are determined by a given corpus, which serves as the interactions to induce quantum correlations. These local cooperative interactions lead to block-diagonal evolution quantified by the entanglement asymmetry. We evaluate the influence of two-site gates on the iteration process and reveal the associated search speedup. Crucially, we demonstrate the information scrambling dependent of gate sampling, uncovering an intriguing bond percolation. Our results provide an adaptive paradigm for quantum search algorithms and random many-body dynamics.

Figures

Figures reproduced from arXiv: 2504.14453 by the authors.

Figure 1
Figure 1. FIG. 1. (a) Sentence diagram for ‘ [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Update processes with chain [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Final-state average Frobenius distance [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of (a) three-site correlation function [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

80 extracted references · 69 canonical work pages

  1. [1]

    Gong, M.-C

    M. Gong, M.-C. Chen, Y. Zheng, S. Wang, C. Zha, H. Deng, Z. Y an, H. Rong, Y. Wu, S. Li, F. Chen, Y. Zhao, F. Liang, J. Lin, Y. Xu, C. Guo, L. Sun, A. D. Ca stellano, H. Wang, C. Peng, C.-Y. Lu, X. Zhu, and J.-W. Pan, Phys. Rev. Lett. 122, 110501 (2019)

    work page 2019

  2. [2]

    S. Cao, B. Wu, F. Chen, M. Gong, Y. Wu, Y. Ye, C. Zha, H. Qian, C. Ying, S. Guo, Q. Zhu, H.-L. Huang, Y. Zhao, S. Li, S. Wang, J. Yu, D. Fan, D. Wu, H. Su, H. Deng, H. Rong, Y. Li, K. Zhang, T.-H. Chung, F. Liang, J. Lin, Y. Xu, L. Sun, C. Guo, N . Li, Y.-H. Huo, C.-Z. Peng, C.-Y. Lu, X. Yuan, X. Zhu, and J.-W. Pan, Nature 619, 738 (2023)

    work page 2023

  3. [3]

    Adesso, T

    G. Adesso, T. R. Bromley, and M. Cianciaruso, Journal of P hysics A: Mathematical and Theoretical 49, 473001

    work page

  4. [4]

    Fr´ erot, M

    I. Fr´ erot, M. Fadel, and M. Lewenstein, Reports on Progr ess in Physics 86, 114001 (2023)

    work page 2023

  5. [5]

    Islam, R

    R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Risp oli, and M. Greiner, Nature 528, 77 (2015)

    work page 2015

  6. [6]

    J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. R ey, M. Foss-Feig, and J. J. Bollinger, Science 352, 1297 (2016)

    work page 2016

  7. [7]

    Zhu, Z.-H

    Q. Zhu, Z.-H. Sun, M. Gong, F. Chen, Y.-R. Zhang, Y. Wu, Y. Y e, C. Zha, S. Li, S. Guo, H. Qian, H.-L. Huang, J. Yu, H. Deng, H. Rong, J. Lin, Y. Xu, L. S un, C. Guo, N. Li, F. Liang, C.-Z. Peng, H. Fan, X. Zhu, and J.-W. Pan, Phys. Rev. Lett. 128, 160502 (2022)

    work page 2022

  8. [8]

    Yuan, S.-Y

    D. Yuan, S.-Y. Zhang, Y. Wang, L.-M. Duan, and D.-L. Deng, Phys. Rev. Res. 4, 023095 (2022)

    work page 2022

Show all 80 references
  1. [9]

    A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko , P. M. Preiss, and M. Greiner, Science 353, 794 (2016)

    2016

  2. [10]

    Mathew, S

    G. Mathew, S. L. L. Silva, A. Jain, A. Mohan, D. T. Adroja, V. G. Sakai, C. V. Tomy, A. Banerjee, R. Goreti, A. V. N., R. Singh, and D. Ja iswal-Nagar, Phys. Rev. Res. 2, 043329 (2020). 12

    2020

  3. [11]

    K. J. Satzinger, Y.-J. Liu, A. Smith, C. Knapp, M. Newman , C. Jones, Z. Chen, C. Quintana, X. Mi, A. Dunsworth, C. Gidney, I. Aleiner, F. Arute, K. Arya, J. Atalaya, R. Babbush, J. C. Bardin, R. Barends, J. Basso, A. Bengtsson, A. Bilmes, M . Broughton, B. B. Buck- ley, D. A....

    2021

  4. [12]

    Hirschberg and C

    J. Hirschberg and C. D. Manning, Science 349, 261 (2015)

    2015

  5. [13]

    Gangal, A

    A. Gangal, A. Shrivastava, N. M. Hussien, P. Singh, M. Di wakar, K. Joshi, S. Bisht, and N. C. Joshi, AIP Conference Proceedings 2771, 020010 (2023), https://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0153960/18105894/020010 1 5.0153960.pdf

    2023 doi

  6. [14]

    Guarasci, G

    R. Guarasci, G. De Pietro, and M. Esposito, Applied Scie nces 12 (2022), 10.3390/app12115651

    2022 doi

  7. [15]

    Zhang, Z

    P. Zhang, Z. Su, L. Zhang, B. Wang, and D. Song, in Proceedings of the 27th ACM International Conference on Informa tion and Knowledge Management , CIKM ’18 (Association for Computing Machinery, New York, NY , USA, 2018) p. 1303–1312

    2018

  8. [16]

    Y. Chen, Y. Pan, and D. Dong, IEEE Transactions on Cybern etics 53, 3467 (2023)

    2023

  9. [17]

    Hibat-Allah, M

    M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko, an d J. Carrasquilla, Phys. Rev. Res. 2, 023358 (2020)

    2020

  10. [18]

    Zhang and M

    Y.-H. Zhang and M. Di Ventra, Phys. Rev. B 107, 075147 (2023)

    2023

  11. [19]

    L. L. Viteritti, R. Rende, and F. Becca, Phys. Rev. Lett. 130, 236401 (2023)

    2023

  12. [20]

    G. J. Stephens and W. Bialek, Phys. Rev. E 81, 066119 (2010)

    2010

  13. [21]

    Corral and M

    l. Corral and M. Garc ´ ıa del Muro, Entropy 22 (2020), 10.3390/e22020179

    2020 doi

  14. [22]

    Jurafsky and J

    D. Jurafsky and J. H. Martin, Speech and Language Processing: An Introduction to Natural L anguage Processing, 3rd ed. (2024) online manuscript released August 20, 2024. 13

    2024

  15. [23]

    G. A. Fink, Markov models for pattern recognition: from theory to appli cations (Springer Science & Business Media, 2014)

    2014

  16. [24]

    X. He, T. Huang, S. Bai, and X. Bai, in 2019 IEEE/CVF International Conference on Computer Vision (ICCV) (2019) pp. 7514–7523

    2019

  17. [25]

    Lopez-Gazpio, M

    I. Lopez-Gazpio, M. Maritxalar, M. Lapata, and E. Agirr e, Expert Systems with Applications 132, 1 (2019)

    2019

  18. [26]

    L. K. Grover, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theor y of Computing , STOC ’96 (Association for Computing Machinery, New York, NY , USA, 1996) p. 212–219

    1996

  19. [27]

    I. L. Chuang, N. Gershenfeld, and M. Kubinec, Phys. Rev. Lett. 80, 3408 (1998)

    1998

  20. [28]

    P. R. Giri and V. E. Korepin, Quantum Information Proces sing 16, 315 (2017)

    2017

  21. [29]

    Mandviwalla, K

    A. Mandviwalla, K. Ohshiro, and B. Ji, in 2018 IEEE International Conference on Big Data (Big Data) (2018) pp. 2531–2537

    2018

  22. [30]

    Wang and P

    Y. Wang and P. S. Krstic, Phys. Rev. A 102, 042609 (2020)

    2020

  23. [31]

    Niedermeier, J

    M. Niedermeier, J. L. Lado, and C. Flindt, Phys. Rev. Res . 6, 033325 (2024)

    2024

  24. [32]

    Cui and H

    J. Cui and H. Fan, Journal of Physics A: Mathematical and Theoretical 43, 045305 (2010)

    2010

  25. [33]

    A. Dunn, Q. Wang, A. Ganose, D. Dopp, and A. Jain, npj Computational Materials 6, 138 (2020)

    2020

  26. [34]

    Li, H.-P

    C.-N. Li, H.-P. Liang, X. Zhang, Z. Lin, and S.-H. Wei, npj Computational Materials 9, 176 (2023)

    2023

  27. [35]

    J. Wang, H. Gao, Y. Han, C. Ding, S. Pan, Y. Wang, Q. Jia, H. -T. Wang, D. Xing, and J. Sun, National Science Review 10, nwad128 (2023), https://academic.oup.com/nsr/article-pdf/10/7/nwad128/50709989/nwad128.pdf

    2023

  28. [36]

    L. Ci, L. Song, C. Jin, D. Jariwala, D. Wu, Y. Li, A. Srivas tava, Z. F. Wang, K. Storr, L. Balicas, F. Liu, and P. M. Ajayan, Nature Materials 9, 430 (2010)

    2010

  29. [37]

    J. Lu, K. Zhang, X. Feng Liu, H. Zhang, T. Chien Sum, A. H. C astro Neto, and K. P. Loh, Nature Communications 4, 2681 (2013)

    2013

  30. [38]

    Z. Liu, X. Zhong, H. Yan, and R.-Z. Wang, Phys. Chem. Chem . Phys. 18, 974 (2016)

    2016

  31. [39]

    Meakin, P

    P. Meakin, P. Ramanlal, L. M. Sander, and R. C. Ball, Phys . Rev. A 34, 5091 (1986)

    1986

  32. [40]

    Tang and S

    C. Tang and S. Liang, Phys. Rev. Lett. 71, 2769 (1993)

    1993

  33. [41]

    Nahum, J

    A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Phys. Rev. X 7, 031016 (2017). 14

    2017

  34. [42]

    The kardar-parisi-zhang equation and univ ersality class,

    I. Corwin, “The kardar-parisi-zhang equation and univ ersality class,” (2011), arXiv:1106.1596 [math.PR]

    2011 arXiv

  35. [43]

    Lambek, Journal of Logic, Language and Information 16, 303 (2007)

    J. Lambek, Journal of Logic, Language and Information 16, 303 (2007)

    2007

  36. [44]

    Coecke, M

    B. Coecke, M. Sadrzadeh, and S. Clark, “Mathematical fo undations for a compositional distributional model (2010), arXiv:1003.4394 [cs.CL]

    2010 arXiv

  37. [45]

    Zeng and B

    W. Zeng and B. Coecke, Electronic Proceedings in Theore tical Computer Science 221, 67–75 (2016)

    2016

  38. [46]

    Meichanetzidis, A

    K. Meichanetzidis, A. Toumi, G. de Felice, and B. Coecke , Quantum Machine Intelligence 5, 10 (2023)

    2023

  39. [47]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev . Lett. 59, 799 (1987)

    1987

  40. [48]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Communications in Mathematical Physics 115, 477 (1988)

    1988

  41. [49]

    Moudgalya, S

    S. Moudgalya, S. Rachel, B. A. Bernevig, and N. Regnault , Phys. Rev. B 98, 235155 (2018)

    2018

  42. [50]

    Fagotti and F

    M. Fagotti and F. H. L. Essler, Phys. Rev. B 87, 245107 (2013)

    2013

  43. [51]

    Farrelly, Quantum 4, 368 (2020)

    T. Farrelly, Quantum 4, 368 (2020)

    2020

  44. [52]

    Pozsgay, Journal of Physics A: Mathematical and Theo retical 54, 384001 (2021)

    B. Pozsgay, Journal of Physics A: Mathematical and Theo retical 54, 384001 (2021)

    2021

  45. [53]

    Brighi, M

    P. Brighi, M. Ljubotina, and M. Serbyn, SciPost Phys. 15, 093 (2023)

    2023

  46. [54]

    M. H. Khan and M. A. Perkowski, Journal of Systems Archit ecture 53, 453 (2007)

    2007

  47. [55]

    Haghparast, R

    M. Haghparast, R. Wille, and A. T. Monfared, Quantum Inf ormation Processing 16, 284 (2017)

    2017

  48. [56]

    Walter and G

    J.-C. Walter and G. Barkema, Physica A: Statistical Mec hanics and its Applications 418, 78 (2015), proceedings of the 13th International Summer School on Fund amental Problems in Statistical Physics

    2015

  49. [57]

    Z. Ma, C. Han, Y. Meir, and E. Sela, Phys. Rev. A 105, 042416 (2022)

    2022

  50. [58]

    F. Ares, S. Murciano, and P. Calabrese, Nature Communic ations 14, 2036 (2023)

    2023

  51. [59]

    B. J. J. Khor, D. M. K¨ urk¸ c¨ uoglu, T. J. Hobbs, G. N. Perd ue, and I. Klich, Quantum 8, 1462 (2024)

    2024

  52. [60]

    L. K. Joshi, J. Franke, A. Rath, F. Ares, S. Murciano, F. K ranzl, R. Blatt, P. Zoller, B. Ver- mersch, P. Calabrese, C. F. Roos, and M. K. Joshi, Phys. Rev. L ett. 133, 010402 (2024)

    2024

  53. [61]

    P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollm ann, Phys. Rev. X 10, 011047 (2020)

    2020

  54. [62]

    Feng and B

    X. Feng and B. Skinner, Phys. Rev. Res. 4, 013053 (2022). 15

    2022

  55. [63]

    Moudgalya, B

    S. Moudgalya, B. A. Bernevig, and N. Regnault, Reports on Progress in Physics 85, 086501 (2022)

    2022

  56. [64]

    Kesten, Communications in Mathematical Physics 74, 41 (1980)

    H. Kesten, Communications in Mathematical Physics 74, 41 (1980)

    1980

  57. [65]

    A. A. Saberi, Physics Reports 578, 1 (2015), recent advances in percolation theory and its applications

    2015

  58. [66]

    X. Meng, J. Gao, and S. Havlin, Phys. Rev. Lett. 126, 170501 (2021)

    2021

  59. [67]

    Rota, Journal of High Energy Physics 2016, 75 (2016)

    M. Rota, Journal of High Energy Physics 2016, 75 (2016)

    2016

  60. [68]

    Hosur, X.-L

    P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Journal of High Energy Physics 2016, 4 (2016)

    2016

  61. [69]

    Seshadri, V

    A. Seshadri, V. Madhok, and A. Lakshminarayan, Phys. Re v. E 98, 052205 (2018)

    2018

  62. [70]

    Kitaev and J

    A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006)

    2006

  63. [71]

    Iyoda and T

    E. Iyoda and T. Sagawa, Phys. Rev. A 97, 042330 (2018)

    2018

  64. [72]

    Schnaack, N

    O. Schnaack, N. B¨ olter, S. Paeckel, S. R. Manmana, S. Ke hrein, and M. Schmitt, Phys. Rev. B 100, 224302 (2019)

    2019

  65. [73]

    Xu and B

    S. Xu and B. Swingle, PRX Quantum 5, 010201 (2024)

    2024

  66. [74]

    Schollw¨ ock, Annals of Physics 326, 96 (2011), january 2011 Special Issue

    U. Schollw¨ ock, Annals of Physics 326, 96 (2011), january 2011 Special Issue

    2011

  67. [75]

    J. I. Cirac, D. P´ erez-Garc ´ ıa, N. Schuch, and F. Verstr aete, Rev. Mod. Phys. 93, 045003 (2021)

    2021

  68. [76]

    L. J. O’Riordan, M. Doyle, F. Baruffa, and V. Kannan, Machine Learning: Science and Technology 2, 015011 (2020)

    2020

  69. [77]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omra n, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Nature 551, 579 (2017)

    2017

  70. [78]

    Barends, L

    R. Barends, L. Lamata, J. Kelly, L. Garc ´ ıa-´Alvarez, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen, Z. Chen, B. C hiaro, A. Dunsworth, I.- C. Hoi, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, A . Vainsencher, J. Wenner...

    2015

  71. [79]

    Shi, Z.-H

    Y.-H. Shi, Z.-H. Sun, Y.-Y. Wang, Z.-A. Wang, Y.-R. Zhan g, W.-G. Ma, H.-T. Liu, K. Zhao, J.-C. Song, G.-H. Liang, J.-C. Mei, Zheng-Yangand Z hang, H. Li, C.-T. Chen, X. Song, J. Wang, G. Xue, H. Yu, K. Huang, Z. Xiang, K. Xu, D. Zhe ng, and H. Fan, Nature Communications 15, 7...

    2024

  72. [80]

    R. G. Melko and J. Carrasquilla, Nature Computational S cience 4, 11 (2024). 16

    2024

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.