Pith. sign in

REVIEW 3 major objections 5 minor 46 references

Identifying slow relaxation in many-body quantum systems through state-graph geometry and state-graph heterogeneity

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

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

Pith's one-line read The paper claims that the spectral radius of the Hamiltonian-induced hitting matrix, normalized by $CD^2$, is a universal geometric indicator of the onset of slow relaxation, jumping from about $1.01$ to more than $10^{40}$ across the…

desk verdict A genuinely new diagnostic with a clean late-time limit, but the universality claim currently rests on unverified threshold behavior in two of the three models. read the letter →

arxiv 2608.05298 v1 pith:YOVOAMK7 submitted 2026-08-05 quant-ph cond-mat.quant-gascond-mat.stat-mech

Heiko Georg Menzler , Tom Ben-Ami This is my paper · ORCID
classification quant-phcond-mat.quant-gascond-mat.stat-mech
keywords quantumhittingtimesstategraphgeometryslowrelaxationspectralradiusEastmodelRosenzweig-Portertriangularlatticegaseigenstateoverlap
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the onset of slow relaxation in a quantum many-body system is visible as a sharp change in the geometry of its state graph, the graph whose nodes are basis states and whose dynamics is generated by the Hamiltonian. The authors define distances on this graph through quantum hitting times, the earliest time at which the cumulative probability to pass from one basis state to another reaches a threshold, and show that at late times these distances are fixed by eigenstate overlaps alone. They find that the distribution of hitting times and, most sensitively, the spectral radius of the hitting matrix change by many orders of magnitude when the system crosses into a slow-dynamics regime, with a characteristic crossing point in finite-size scaling. Because the same signature appears in three models with different microscopic origins of slowness, the paper proposes the normalized spectral radius as a unified geometric diagnostic for slow relaxation that requires no choice of observable.

What carries the argument

The central object is the quantum hitting matrix $\hat h$, whose entry $h_{ij}$ is the earliest time $T$ at which the time-integrated transition probability $k_{ij}(T)=\int_0^T |\langle i|e^{-i\hat H t}|j\rangle|^2\,dt$ reaches a threshold $C$. Expanding in eigenstates gives $k_{ij}(T)\sim \alpha_{ij}T$ at late times with $\alpha_{ij}=\sum_\nu |\langle i|\nu\rangle|^2|\langle j|\nu\rangle|^2$, so $h_{ij}=C/\alpha_{ij}$ and the normalized matrix $h_{ij}/C$ no longer depends on the threshold. The machinery does two jobs: the distribution of $h_{ij}$ measures how differently basis states communicate, and the spectral radius $\rho(\hat h)$ converts that heterogeneity into a single number that, for a uniform matrix, equals $D^2$ by Perron-Frobenius and rises sharply when the graph geometry becomes heterogeneous.

What would settle it

Compute $k_{ij}(T)$ for the quantum East and triangular lattice gas models at the parameter values used in the paper and check whether $k_{ij}(T)\approx \alpha_{ij}T$ holds at the time when it crosses $C=1$; if a substantial fraction of pairs have not yet entered the linear ramp, the crossing points in $\rho(\hat h)$ will shift when $C$ is varied, disproving the claim that the geometry is a long-time property.

Watch

Extended reading notes

Core claim

The central claim is that the spectral radius $\rho(\hat h)$ of the Hamiltonian-induced hitting matrix, normalized by $CD^2$, is a universal geometric indicator of the onset of slow relaxation. In the fast-dynamics regime the hitting matrix is nearly uniform, so by the Perron-Frobenius theorem its largest eigenvalue is close to the minimal value $D^2$, and $\rho(\hat h)/CD^2\to 1$ with increasing system size. At the transition the normalized spectral radius develops a crossing point in finite-size scaling and then grows dramatically with system size in the slow regime; the paper's most striking example is the quantum East model at $L=12$, where $\rho(\hat h)/CD^2$ goes from about $1.01$ at $s=-2$ to more than $10^{40}$ at $s=2$. The paper argues that these signatures are shared by all three models, the Rosenzweig-Porter model, the quantum East model, and the triangular lattice gas model, even though their microscopic slow-dynamics mechanisms differ, and interprets the slow phase as a breakdown of homogeneity in the geometry of the state graph.

Load-bearing premise

The load-bearing premise is that $C=1$ falls inside the late-time linear regime $k_{ij}(T)\sim \alpha_{ij}T$ for all three models, with the check shown only for the Rosenzweig-Porter model; otherwise the hitting matrix and its spectral radius depend on the arbitrary threshold $C$.

Editorial extensions

If this is right

  • The normalized spectral radius $\rho(\hat h)/CD^2$ can be used as a single-parameter diagnostic: the slow-dynamics transition is located by the crossing of this quantity with system size, without choosing a specific autocorrelation function.
  • Because $h_{ij}=C/\alpha_{ij}$ depends only on eigenstate overlaps, the geometric probe ties slow relaxation directly to eigenstate structure, so regimes with fractal or localized eigenstates should show up as broad hitting-time distributions.
  • The finite-size scaling of $\rho(\hat h)$ gives concrete estimates for transition parameters that have resisted precise determination: the paper extracts $s\approx 0.2$ for the quantum East model and $V/t_0\approx 1.05$ for the triangular lattice gas model.
  • The width of the hitting-time distribution controls the late-time value of autocorrelation functions of projectors, so a broad distribution permits heterogeneous decay between observables and explains long-lived memory effects.

Reading between the lines

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

  • The paper leaves open whether the same normalized spectral radius identifies other ergodicity-breaking phenomena such as Hilbert-space fragmentation or many-body scar subspaces; applying the crossing analysis to models with independently known transitions would test that extension.
  • The discrete-time kernel defined in the End Matter extends the geometry to Floquet systems and random unitary circuits, so the spectral-radius diagnostic could detect slow relaxation in driven systems, although no such simulation is reported.
  • Because the late-time geometry depends only on eigenstate overlaps, the spectral radius might be accessible from time-averaged correlation or Loschmidt-echo data without full state tomography, a practical route the paper only sketches as an outlook.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes a geometric probe of slow relaxation in many-body quantum systems: the pairwise hitting time hij, defined as the earliest time at which the time-integrated transition probability kij(T) reaches a threshold C. After deriving the late-time slope alpha_ij = sum_nu |<i|nu>|^2 |<j|nu>|^2, the authors replace hij by C/alpha_ij and study the distribution of matrix elements and the spectral radius rho(h) of the hitting matrix. They apply this to the Rosenzweig-Porter model, the quantum East model, and the triangular lattice gas model, and report that rho(h)/CD^2 sharply increases at the onset of slow dynamics, with a crossing point in a finite-size scaling exponent k. The central claim is that this normalized spectral radius is a universal geometric indicator of the onset of slow relaxation.

Significance. If the central claim holds, the paper offers a genuinely new diagnostic: a single graph-theoretic quantity that detects slow dynamics across three different mechanisms (fractal eigenstates, kinetic constraints, and metastable plateaus) and that is connected, through Eq. (4), to experimentally accessible time-averaged autocorrelation functions. The late-time derivation of hij = C/alpha_ij is straightforward and correct under the stated assumption of a threshold crossed in the linear regime, and the link to autocorrelation functions in the supplementary material is a useful conceptual contribution. The manuscript is also transparent about its own limitations, notably the severe finite-size effects in the QEM and TLGM scaling analysis, and it makes data publicly available on Zenodo. The main reservations are that the threshold assumption is verified only for the RPM, not for the other two models, and that the RPM results rest on single disorder realizations without error bars.

major comments (3)
  1. [Emergent Geometry; Eqs. (1)-(3) and Fig. 2] Equation (3) and the subsequent replacement hij = C/alpha_ij are only valid if, for every pair (i,j), the threshold C in Eq. (2) is crossed after kij(T) has entered its linear late-time regime. The manuscript verifies this for the RPM in Fig. 2, but provides no equivalent check for the QEM or TLGM; the statement that C=1 is 'often sufficient' is not a substitute. In the slow-dynamics regimes of those models, near-degeneracies or long-lived precursors can delay the linear ramp, so C=1 may cut in during the transient. In that case hij, its distribution, and rho(h)/CD^2 all depend on the arbitrary scale C, and the reported crossing point is not the long-time geometric quantity the paper claims. Please provide representative kij(T) curves for QEM and TLGM in both fast and slow regimes, with the horizontal line C=1, or define hij directly from the fitted asymptotic slope alpha_ij and demonstrate that the normalized results are insensitive to the choice of C within the linear regime.
  2. [Geometric Signatures; Fig. 3(d) and caption] The RPM data in Fig. 3(d) are described as 'single realizations', with each (gamma, L) being an independent disorder sample. The spectral radius is an extreme statistic and the RPM is a random ensemble; a single realization per parameter point gives no handle on sample-to-sample fluctuations. Since Fig. 4 extracts the RPM crossing point from these single-realization values, disorder fluctuations could shift or create the apparent crossing, especially because different L use different realizations. Please repeat the RPM analysis over an ensemble of realizations and report the mean and spread of ln[rho(h)]/CD^2, for example as standard errors or quantiles, and confirm that the crossing at gamma near 1 is stable against disorder averaging.
  3. [End Matter, 'Extracting positions of crossing points'; Fig. 4] The finite-size scaling ansatz ln[rho(h)] = a L^k is fitted to at most six points per parameter value, with no confidence intervals on k and no goodness-of-fit measure. The text itself states that QEM and TLGM 'suffer from strong finite size effects' and that away from the crossing the data points systematically deviate from the linear fit, so the ansatz is incomplete. Because the crossing point is defined as the value where k changes sign, these systematic deviations directly affect the reported values s ≈ 0.2 and V/t0 ≈ 1.05. Please provide error bars on the fitted exponents and on the interpolated zero crossing, show residual plots, and quantify how the crossing moves when the largest system sizes are excluded (the lighter-color fits in Fig. 4 are a useful start but are not quantified).
minor comments (5)
  1. [State Graph Geometry, Eq. (2) and following text] The text first says 'C acts as a scale parameter' and later says 'C carries no physical information'; these statements should be reconciled once the linear-regime condition is formally stated.
  2. [Emergent Geometry, Eq. (3)] The non-degeneracy assumption is only parenthetical in the main text; the supplementary material shows that degenerate contributions can be absorbed into alpha_ij. A short main-text sentence to this effect would prevent a false impression that exact degeneracies invalidate the result.
  3. [Supplementary Material, S9] There are several typos: 'realizes on of the most simple constraints' should be 'realizes one of'; 'Triangular Lattice Gase model' should be 'Triangular Lattice Gas model'; and in the main text 'asympototic limit' should be 'asymptotic limit'.
  4. [Fig. 1] The colorbar cutoff with orange overlay for values beyond the cutoff makes the dynamic range hard to read; a log-scale color map or an explicit stated maximum value would improve clarity.
  5. [Outlook] The proposed operator reformulation via Tr(O_i O_j(t)) is only sketched; a brief derivation showing how coarse-grained sector projectors map onto kij would strengthen the claimed experimental relevance.

Circularity Check

0 steps flagged · score 1.0 of 10

No load-bearing circularity: the hitting-time probe and spectral-radius metric are derived from the unitary dynamics, and the crossing points are extracted by scaling analysis rather than imposed.

full rationale

The derivation chain is self-contained. k_ij(T) is defined from |<i|U(t)|j>|^2 (Eq. 1), and the asymptotic identity h_ij/C = alpha_ij^{-1} follows by expanding Eq. (1) in the eigenbasis (Eq. 3) and taking the late-time limit; no parameter is fitted to make this identity hold. The normalized spectral radius rho(h)/CD^2 is computed from the resulting matrix, and the fast-regime value unity is derived from the Perron-Frobenius theorem, proven in the Supplemental Material, not imposed. The finite-size scaling ln rho = a L^k is a standard extraction protocol: the crossing point is defined by the sign change of the fitted exponent k, and for the RPM it independently reproduces the known transition gamma=1, while for the QEM and TLGM the extracted values (s approximately 0.2, V/t0 approximately 1.05) are presented as estimates with acknowledged finite-size limitations. The paper's self-citations ([19], [24]) are contextual and non-load-bearing: no theorem, dataset, or fitted value from those papers is required for the central result. The main caveat -- that C=1 is assumed to lie in the linear regime for the QEM and TLGM, verified only for the RPM -- is an unverified validity condition on the asymptotic replacement h_ij/C = alpha_ij^{-1}, not a circular reduction of the prediction to an input. Accordingly, no circular step meets the evidentiary bar.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are posited; the hitting matrix is a derived observable computed directly from the Hamiltonian and unitary evolution. The main extra elements are the chosen threshold C, the scaling ansatz, and the assumptions listed above.

free parameters (3)
  • C (hitting threshold) = 1
    Threshold in the hitting time definition of Eq. (2). The authors claim C carries no physical information in the late-time linear regime and set C=1, but this is only verified for the Rosenzweig-Porter model in Fig. 2.
  • a (scaling prefactor) = fit per parameter value and model
    Prefactor in the finite-size scaling ansatz ln[rho(h)] = a L^k used to extract crossing points in Fig. 4.
  • k (scaling exponent) = fit per parameter value and model
    Exponent in the finite-size scaling ansatz; the crossing point is defined where k crosses zero. The ansatz is acknowledged to be incomplete for the quantum East and triangular lattice gas models away from the crossing.
assumptions (5)
  • standard math Perron-Frobenius theorem for symmetric nonnegative matrices
    Used in the Supplemental to show the spectral radius is minimal for a uniform matrix and that the all-ones vector is the Perron-Frobenius eigenvector.
  • domain assumption Random eigenstate assumption in the fast dynamics regime, |<i|nu>|^2 ~ 1/D for all i, nu
    Used to derive the uniform limit of the hitting matrix and the value rho(h)/CD^2 approaching unity in the ergodic regime.
  • ad hoc to paper C=1 lies in the linear late-time regime of k_ij(T) for all relevant pairs
    The entire analysis uses hij = C/alpha_ij; this holds only if k_ij is in its linear ramp. The authors state that C=1 is 'often sufficient' based on Fig. 2 for the Rosenzweig-Porter model.
  • ad hoc to paper Finite-size scaling ansatz ln[rho(h)] = a L^k
    No rigorous scaling theory exists; the crossing point extraction relies on this ansatz, which the authors note has strong corrections for the quantum East and triangular lattice gas models.
  • domain assumption The state graph within the studied symmetry sector is effectively irreducible, or infinite hitting times are handled without bias
    The triangular lattice gas is particle conserving, and the quantum East model with added boundary terms is argued to be symmetry-free; the numerical treatment of disconnected pairs in fragmented graphs is not described.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Identifying slow relaxation in many-body quantum systems through state-graph geometry and state-graph heterogeneity." pith.science (2026). https://pith.science/paper/YOVOAMK7

@misc{pith2026260805298,
  author       = {Pith},
  title        = {Pith review of: Identifying slow relaxation in many-body quantum systems through state-graph geometry and state-graph heterogeneity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YOVOAMK7}},
  note         = {Machine review of arXiv:2608.05298}
}
read the original abstract

We adapt tools from the theory of quantum random walks to investigate slow relaxation dynamics through the many-body state graph. Specifically, we construct a probe of heterogeneity between basis states defined using hitting times derived from the unitary time-evolution operator. We find that the state-graph geometry, encoded by the pairwise hitting time of basis states, is a highly sensitive indicator of slow relaxation dynamics in a variety of systems. We study three paradigmatic models: the Rosenzweig-Porter model, the quantum East model, and the triangular lattice gas model, exhibiting a sudden onset of slow dynamics upon tuning of a control parameter. As a global characterization of the graph geometry, we analyze the spectral radius of the hitting matrix. We find that it increases sharply at the onset of slow dynamics, spanning many orders of magnitude, with a characteristic crossing point at the transition. Our work provides a geometric framework for describing and identifying phases with slow relaxation using a unified graph-theoretic formalism.

Figures

Figures reproduced from arXiv: 2608.05298 by the authors.

Figure 1
Figure 1. Hitting matrix ˆh in the QEM with L = 10 (D = 210) in the (a) fast-dynamics (s = −0.2) and (b) slow dynamics (s = 1) regime. Large values beyond the colorbar-cutoff up to hij/D ≈ 1015 are shown in orange. (c) Normalized distribution P˜(hij ) of matrix elements for (a) (blue) and (b) (orange) such that the maximum of both distribution is the same. arXiv:2608.05298v1 [quant-ph] 5 Aug 2026 [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 2
Figure 2. Kernel function kij (T) for a representative state |i⟩ and other states |j⟩ in the computational basis in the RPM for (a) ergodic extended phase γ = 0.5, (b) non￾ergodic extended phase γ = 1.5 and (c) localized phase γ = 2.5. (d) combines data from the three other subplots, showing linear behavior of kij (T) at large T. Horizontal dashed lines indicate a cutoff value C = 1. HˆTLG = P ⟨ℓ,ℓ′⟩ Cˆ ℓℓ′ [PITH_FULL_IMAGE:… view at source ↗
Figure 3
Figure 3. Characterization of heterogeneous relaxation through [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Finite-size scaling analysis of ρ( ˆh) for the RPM [(a)-(b)], QEM [(c)-(d)], and TLGM [(e)-(f)]. The crossing point is determined from the finite-size scaling exponent. The exponent k is obtained from a linear fit in log-log space, as ln[ρ( ˆh)] ∝ L k . The left column…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

46 extracted references · 24 canonical work pages

  1. [1]

    Farhi and S

    E. Farhi and S. Gutmann, Quantum computation and decision trees, Phys. Rev. A58, 915 (1998)

    1998

  2. [2]

    Kempe, Discrete quantum walks hit exponentially faster, Probab

    J. Kempe, Discrete quantum walks hit exponentially faster, Probab. Theory Relat. Fields133, 215–235 (2005)

    work page 2005

  3. [3]

    S. E. Venegas-Andraca, Quantum walks: A comprehensive review, Quantum Inf. Process.11, 1015–1106 (2012)

    work page 2012

  4. [4]

    Aharonov, A

    D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, Quantum walks on graphs, inProceedings of the thirty- third annual ACM symposium on Theory of computing, STOC01 (ACM, 2001) p. 50–59

    work page 2001

  5. [5]

    Mülken and A

    O. Mülken and A. Blumen, Continuous-time quantum walks: Models for coherent transport on complex net- works, Physics Reports502, 37–87 (2011)

    work page 2011

  6. [6]

    Rigol, V

    M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature452, 854–858 (2008)

    2008

  7. [7]

    J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A43, 2046 (1991)

    1991

  8. [8]

    Srednicki, Chaos and quantum thermalization, Phys

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

    1994

Show all 46 references
  1. [9]

    D’Alessio, Y

    L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Adv. Phys. 65, 239 (2016)

    2016

  2. [10]

    J. M. Deutsch, Eigenstate thermalization hypothesis, Rep. Prog. Phys.81, 082001 (2018)

    2018

  3. [11]

    D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Metal– insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys. 321, 1126 (2006)

    2006

  4. [12]

    P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Poll- mann, Ergodicity breaking arising from Hilbert space fragmentation in dipole-conserving Hamiltonians, Phys. Rev. X10, 011047 (2020)

    2020

  5. [13]

    Z.-C. Yang, F. Liu, A. V. Gorshkov, and T. Iadecola, Hilbert-spacefragmentationfromstrictconfinement,Phys. Rev. Lett.124, 207602 (2020)

    2020

  6. [14]

    D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Col- loquium: Many-body localization, thermalization, and entanglement, Rev. Mod. Phys.91, 021001 (2019)

    2019

  7. [15]

    Z. Lan, M. van Horssen, S. Powell, and J. P. Garrahan, Quantum slow relaxation and metastability due to dy- 6 namical constraints, Phys. Rev. Lett.121, 040603 (2018)

    2018

  8. [16]

    Pancotti, G

    N. Pancotti, G. Giudice, J. I. Cirac, J. P. Garrahan, and M. C. Bañuls, Quantum East model: Localization, nonthermal eigenstates, and slow dynamics, Phys. Rev. X10, 021051 (2020)

    2020

  9. [17]

    Brighi, M

    P. Brighi, M. Ljubotina, and M. Serbyn, Hilbert space fragmentation and slow dynamics in particle-conserving quantum East models, SciPost Phys.15, 093 (2023)

    2023

  10. [18]

    Rakovszky, B

    T. Rakovszky, B. Placke, N. P. Breuckmann, and V. Khe- mani, Bottlenecks in quantum channels and finite tem- perature phases of matter (2024), arXiv:2412.09598

    2024 arXiv

  11. [19]

    H. G. Menzler, M. C. Bañuls, and F. Heidrich-Meisner, Graph theory and tunable slow dynamics in quantum East Hamiltonians, Phys. Rev. B112, 115141 (2025)

    2025

  12. [20]

    Ben-Ami, M

    T. Ben-Ami, M. Heyl, and R. Moessner, Many-body cages: Disorder-free glassiness from flat bands in Fock space, and many-body Rabi oscillations (2025), arXiv:2504.13086

    2025 arXiv

  13. [21]

    Tan and Y.-P

    T.-L. Tan and Y.-P. Huang, Interference-caged quantum many-body scars: The Fock space topological localization and interference zeros (2025), arXiv:2504.07780

    2025 arXiv

  14. [22]

    Nicolau, M

    E. Nicolau, M. Ljubotina, and M. Serbyn, Fragmentation, zero modes, and collective bound states in constrained models, PRX Quantum7, 010352 (2026)

    2026

  15. [23]

    Jonay and F

    C. Jonay and F. Pollmann, Localized Fock space cages in kinetically constrained models, Phys. Rev. B113, 134313 (2026)

    2026

  16. [24]

    H. G. Menzler, R. Świętek, M. C. Bañuls, and F.Heidrich-Meisner,Graph-theorymeasurescaptureweak ergodicity breaking on large quantum systems (2026), arXiv:2605.00094

    2026 arXiv

  17. [25]

    Redner,A guide to first-passage processes(Cambridge university press, 2001)

    S. Redner,A guide to first-passage processes(Cambridge university press, 2001)

    2001

  18. [26]

    Aldous and J

    D. Aldous and J. A. Fill, Reversible Markov chains and random walks on graphs (2002), unfinished monograph, recompiled 2014, available atwww.stat.berkeley.edu/ ~aldous/RWG/book.html

    2002

  19. [27]

    Repulsion of energy levels

    N. Rosenzweig and C. E. Porter, “Repulsion of energy levels” in complex atomic spectra, Phys. Rev.120, 1698 (1960)

    1960

  20. [28]

    V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, and M. Amini, A random matrix model with localization and ergodic transitions, New J. Phys.17, 122002 (2015)

    2015

  21. [29]

    van Horssen, E

    M. van Horssen, E. Levi, and J. P. Garrahan, Dynam- ics of many-body localization in a translation-invariant quantum glass model, Phys. Rev. B92, 100305 (2015)

    2015

  22. [30]

    Royen, S

    K. Royen, S. Mondal, F. Pollmann, and F. Heidrich- Meisner, Enhanced many-body localization in a kinetically constrained model, Phys. Rev. E109, 024136 (2024)

    2024

  23. [31]

    P. G. Doyle and J. L. Snell,Random walks and electric networks, Vol. 22 (American Mathematical Soc., 1984)

    1984

  24. [32]

    D. J. Klein and M. Randić, Resistance distance, J. Math. Chem.12, 81–95 (1993)

    1993

  25. [33]

    Lovász, Random walks on graphs: A survey, inCombi- natorics, Paul Erdös is eighty, Vol

    L. Lovász, Random walks on graphs: A survey, inCombi- natorics, Paul Erdös is eighty, Vol. 2 (1993) pp. 1–46

    1993

  26. [34]

    A. K. Chandra, P. Raghavan, W. L. Ruzzo, R. Smolensky, and P. Tiwari, The electrical resistance of a graph captures its commute and cover times, Comput. Complexity6, 312–340 (1996)

    1996

  27. [35]

    R. R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis21, 5 (2006)

    2006

  28. [36]

    Coutinho and K

    G. Coutinho and K. Guo, Selected open problems in continuous-time quantum walks, Special Matrices12, 20240025 (2024)

    2024

  29. [37]

    A. M. Childs, E. Farhi, and S. Gutmann, An example of the difference between quantum and classical random walks, Quantum Inf. Process.1, 35–43 (2002)

    2002

  30. [38]

    A. M. Childs and J. Goldstone, Spatial search by quantum walk, Phys. Rev. A70, 022314 (2004)

    2004

  31. [39]

    Szegedy, Quantum speed-up of Markov chain based al- gorithms, in45th Annual IEEE symposium on foundations of computer science(IEEE, 2004) pp

    M. Szegedy, Quantum speed-up of Markov chain based al- gorithms, in45th Annual IEEE symposium on foundations of computer science(IEEE, 2004) pp. 32–41

    2004

  32. [40]

    Magniez, A

    F. Magniez, A. Nayak, P. C. Richter, and M. Santha, On the hitting times of quantum versus random walks, Algorithmica63, 91 (2012)

    2012

  33. [41]

    Moore and A

    C. Moore and A. Russell, Quantum walks on the hyper- cube, inRandomization and Approximation Techniques in Computer Science(Springer Berlin Heidelberg, 2002) p. 164–178

    2002

  34. [42]

    Shenvi, J

    N. Shenvi, J. Kempe, and K. B. Whaley, Quantum random-walk search algorithm, Phys. Rev. A67, 052307 (2003)

    2003

  35. [43]

    See Supplemental Material for further details on the quan- tum East and Triangular Lattice Gas model Hamiltonians and details on the analytical calculations and proofs for the hitting matrix

  36. [44]

    Meyer,Matrix Analysis and Applied Linear Algebra (SIAM, 2000)

    C. Meyer,Matrix Analysis and Applied Linear Algebra (SIAM, 2000)

    2000

  37. [45]

    Karch, S

    S. Karch, S. Bandyopadhyay, Z.-H. Sun, A. Impertro, S. Huh, I. P. Rodríguez, J. F. Wienand, W. Ketterle, M. Heyl, A. Polkovnikov,et al., Probing quantum many- body dynamics using subsystem Loschmidt echos (2025), arXiv:2501.16995

    2025 arXiv

  38. [46]

    H. G. Menzler and T. Ben-Ami, Identifying slow relax- ation in many-body quantum systems through state-graph geometry and state-graph heterogeneity, 10.5281/zen- odo.20547595 (2026). 7 End matter Discrete-time evolution.—In many practical cases, the whole time-dependence of th...

    2026 doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.