Pith. sign in

REVIEW 2 major objections 4 minor 110 references

Universal energy-space localization and stable quantum phases against time-dependent perturbations

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read For any q-local Hamiltonian driven with bounded total variation, an initial eigenstate stays exponentially concentrated in a narrow instantaneous-energy window.

desk verdict A genuinely new and largely rigorous energy-space localization theorem for time-dependent q-local Hamiltonians, with LDPC applications solid and the optimization application conditional on an explicitly unproven clustering input. read the letter →

arxiv 2510.14160 v2 pith:5KIXQNRK submitted 2025-10-15 quant-ph cond-mat.dis-nncond-mat.stat-mech

Hongye Yu , Tzu-Chieh Wei This is my paper
classification quant-phcond-mat.dis-nncond-mat.stat-mech MSC 81Q1081P7082B44
keywords energy-spacelocalizationtime-dependentperturbationsq-localHamiltonianstotalvariationinstantaneouseigenstatesclusteringpropertyLDPCcodesoptimizationbarriers
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 claims that a universal phenomenon, energy-space localization, governs the dynamics of generic local quantum systems: start in an eigenstate and drive with any time-dependent q-local perturbation whose total variation is at most λn; then, at every instant, the state's weight on instantaneous eigenstates whose energy differs by more than d n (for d>λ) is at most e^{-c n}, with c = (λ/Δ)(d/λ −1 − ln(d/λ)) up to o(1), where Δ=2qM. This is qualitatively stronger than the naive Markov bound λ/d and gives back classical energy conservation in the thermodynamic limit. The paper then shows that when the energy landscape has a clustering property with a linear energy barrier — as in good LDPC codes and some spin-glass optimization problems — energy-space localization becomes genuine configuration-space localization: information stays near its original codeword, and Hamiltonian algorithms get trapped in local minima. If correct, this would establish a broad class of provably stable quantum phases against generic time-dependent perturbations, and would reshape expectations for quantum memories and adiabatic optimization.

What carries the argument

The engine is a moment estimate. Define g_k(t)=||(H(t)-E0)^k|ψ(t)⟩||; the paper bounds the growth of these 2k-th central moments in the instantaneous eigenbasis and uses Markov's inequality with a carefully chosen order k≈(d−λt)n/Δ to turn moment growth into exponential leakage decay. The growth is controlled by nested commutators ad^m_H(t)(H′(t)): for q-local H these scale at most factorially, m!Δ^m, while for a commuting core with [V,H′]=0 they scale exponentially, Δ^m, giving the improved bound. The second ingredient is the clustering property: if eigenstates below an energy E_B split into clusters with intra-cluster distance ≤ν1 and inter-cluster distance ≥ν2 ∼ n, then exponential energy

What would settle it

Evolve an n-qubit system under H(t) with random all-to-all 2-local Pauli terms, local norm bounded by M=1, and drive it with a perturbation of total variation λn (say λ=0.1) for a time where d/λ=2. Numerically compute the weight on instantaneous eigenstates with energy |E−E0|≥2λn; if this weight decays slower than exp[-n(λ/Δ)(2−1−ln 2)] for n=10,…,20 after optimizing the o(1) correction, the theorem's exponential rate is violated or its constant is not tight at accessible sizes.

Watch

Extended reading notes

Core claim

The central discovery is a rigorous exponential leakage bound, not just an energy-expectation bound. For an initial eigenstate of H(0) evolving under a q-local H(t) with local norm ≤M and total variation ∫||dH||_X ≤λn, the probability that the state is found in instantaneous eigenstates with |E(t)−E0| ≥ d n is exponentially small in n for any d>λ; the rate is governed by λ/Δ times (d/λ −1 − ln(d/λ)), with Δ=2qM. When H can be split into a mutually commuting core plus a perturbation that commutes with H′, the bound improves to exp[-n (λ/Δ)((d/λ) ln(d/λ) − (d/λ −1))]. This is proven by controlling all 2k-th central moments of H(t)−E0 and applying Markov's inequality with an optimized moment or

Load-bearing premise

The stability applications all hinge on the clustering property holding in a linear energy window: the low-energy eigenstates must be cleanly separated into clusters whose mutual distance is O(n), enforced by an energy barrier of height B ∼ O(n) that grows linearly with system size; the paper cites linear soundness for LDPC codes but only a demonstrated-not-proved clustering for the spin-glass optimization example, and without such a barrier energy-space localization does not

Editorial extensions

If this is right

  • LDPC codes with linear soundness protect an encoded state for exponentially long times against generic time-dependent (quasi-)q-local noise, as long as the perturbation's total variation stays below a constant fraction of the linear energy barrier.
  • For static perturbations of commuting LDPC Hamiltonians, the eigenstates are exponentially localized in the unperturbed spectrum; the bound extends to quasi-q-local perturbations, which prior tridiagonal-matrix methods did not cover.
  • Hamiltonian-based quantum algorithms for clustered optimization problems cannot escape their starting cluster in polynomial time when the driving has small total variation, so late-time evolution neither improves nor ruins near-optimal outputs.
  • Local Gibbs samplers for these LDPC Hamiltonians have exponentially long mixing times even with weak quasi-q-local perturbations, extending thermodynamic-stability results to a broader noise class.
  • In the thermodynamic limit, the exponentially small high-energy leakage means the quantum dynamics obeys the classical energy-conservation bound, giving a provable quantum-to-classical correspondence for driven local systems.

Reading between the lines

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

  • Editorial extension: the moment-based method suggests a direct numerical protocol — simulate n∈[8,20] qubits with all-to-all random q-local Pauli driving, measure the instantaneous-energy leakage as a function of d/λ, and check whether the rate approaches the predicted (λ/Δ)(d/λ−1−ln(d/λ)); deviations would reveal where the o(1) correction and finite-size effects matter.
  • Because the bound depends only on the total variation, rapid pulses and slow ramps with the same ∫||dH|| should produce the same leakage; this time-rescaling invariance is a crisp, testable signature that distinguishes energy-space localization from mechanisms tied to specific speed limits.
  • If a model lacks the clustering property, energy-space localization should coexist with thermalization inside the window; a natural next step is to quantify how the window's internal mixing rate depends on the density of states, which the current theorem leaves open.
  • The nested-commutator formulation connects naturally to operator-growth complexity; for models with slower-than-factorial growth of ad^m(H′), such as certain mean-field models, the same proof could produce tighter windows or exact rates.
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

2 major / 4 minor

Summary. The paper proves a universal energy-space localization theorem for q-local, time-dependent Hamiltonians H(t) with bounded local norm. Theorem 1 states that an initial eigenstate of H(0) evolves so that its weight on instantaneous eigenstates with energy difference at least dn from E0 is at most exp[-n(λ/Δ)(d/λ−1−ln(d/λ)+o(1))] for d>λ, where Δ=2qM and ∫||dH||_X≤λn. A sharper bound is given when H(t) has a commuting core (Case 3). The result is extended to static perturbations by a quench limit, yielding eigenstate localization results for LDPC codes, and to quasi-q-local perturbations. Applications are developed for LDPC codes with linear soundness (exponential dynamical localization, eigenstate localization, slow mixing) and for classical optimization problems whose low-energy landscape satisfies a clustering property, where the authors argue that Hamiltonian-based algorithms become trapped in local minima.

Significance. If the main theorem is correct, it is a significant and general contribution: it gives parameter-free, rigorous leakage bounds for arbitrary time-dependent local driving, with no assumption on driving speed, and it provides a new route from energy-space localization to stability in systems with extensive energy barriers. The proof in Appendix B is detailed and the moment recursion is internally consistent; the Stirling asymptotics reproduce Eq. (3). The LDPC applications are well-supported by the linear soundness property (Eq. (9)) from the cited literature. The optimization application is the conditional part: it depends on a clustering property that, for the explicit p-spin example, is not proved in the manuscript. The paper is honest about this gap in Sec. IV A, but the gap is load-bearing for the advertised algorithmic-trap claim.

major comments (2)
  1. [Sec. IV A, Eq. (16)] The optimization application depends on the clustering property (Definition 1) with a linear energy barrier B~Θ(n) and inter-cluster separation ν2~Θ(n). For the q-spin Hamiltonian (16), the paper states: "It was demonstrated, though not explicitly proved, in [62] that the model has the clustering property in a linear energy window." This is a load-bearing input for Proposition IV.1 and the algorithmic-trap claim. Theorem 1 proves only energy-space localization; converting it to configuration-space localization requires the clustering property. If the OGP window is sublinear or ν2 is not Θ(n), the leakage bound of Theorem 1 does not imply confinement to one cluster, and the freezing conclusion collapses. Footnote [46] illustrates the same hazard for codes without linear soundness. Please provide a rigorous proof of the needed clustering for (16) or explicitly label the optimization applic
  2. [Sec. IV A, Prop. IV.1 and Eq. (17)] The freezing claim for the adiabatic algorithm is not rigorously derived from Theorem 1. Proposition IV.1 is stated for an initial Z-basis state inside a cluster, which is an eigenstate of H_L, but in the algorithmic evolution (17) the state at time s* is a general superposition, not a single Z-basis state. Theorem 1 controls leakage in the instantaneous eigenbasis of H(t), while the cluster decomposition is in the H_L eigenbasis. The paper's discussion after Eq. (17) asserts that states below E_B−2Λ are frozen and that the late-time evolution cannot improve or worsen the solution, but the change-of-basis argument and the treatment of superposed initial states are not given. Please supply the missing step or restrict the claim to evolutions with V(0)=0 and initial Z-basis states.
minor comments (4)
  1. [Sec. IV A after Eq. (17)] The expression for the total variation Λ of H(t)=s(t)H_L+(1−s(t))H_M from s to 1 appears incorrect: the total variation is (1−s)||H_L−H_M||_X (assuming monotone s), not (1−s)/s ||H_M||_X. Please correct or define the rescaled perturbation Hamiltonian explicitly.
  2. [Eq. (5)] The notation E_j(d) is used before it is defined; define the energy window in the text preceding Eq. (5).
  3. [Tables I and II; Sec. II] The notation "quasi-q⋆-local" and the star on q are used in tables and text before being formally defined in a way that distinguishes q from q⋆. Please add a definition near the first use.
  4. [Sec. II, Case 3] The condition [V(t), H'(t)]=0 in Case 3 is very restrictive in the dynamical setting (though automatically satisfied in the static reduction). The paper notes this, but it may be worth emphasizing that most natural time-dependent perturbations do not satisfy it, limiting the applicability of the improved bound (4).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the core theorem is derived from first principles, and the flagged p-spin OGP input is an unproven external assumption rather than a circular step.

full rationale

The central derivation chain is self-contained. Theorem 1 (Sec. II A, Eq. (3)) is obtained by bounding the growth of the 2k-th central moments of H(t) and applying Markov's inequality (Appendix B); no fitted parameter or assumed leakage bound enters. Theorem 2 is a T->0+ limit of the same theorem, not an assumed input. The LDPC applications use the clustering property (Definition 1) supplied by linear soundness, Eq. (9), via citations to independent code constructions; the paper then maps Theorem 1's energy-space bound to cluster confinement. The optimization application is explicitly conditional: Sec. IV A states "It was demonstrated, though not explicitly proved, in [62] that the model has the clustering property in a linear energy window [Eg, Eg+B] near the ground states." This is a genuine correctness/rigor caveat -- if that clustering/OGP input fails, the algorithmic-freezing conclusion does not follow -- but it is an external, clearly flagged hypothesis, not a re-labeled output of the paper's own derivation. The paper similarly flags the analogous hazard for codes without linear soundness in footnote [46] ("false vacuum"), reinforcing that the cluster input is separate from the theorem. There is a passing self-citation ([77]) but it is not load-bearing. No step of the derivation reduces by construction to its own input, and no fitted quantity is renamed as a prediction. Hence the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No parameters are fitted to data. The quantities M, q, λ, d are bounds/assumptions, not fitted. The detuning H_d uses random coefficients with fixed variance (a proof device, not a fit). No new physical entities are introduced; definitions (quantum distance, clustering property) are mathematical constructions.

assumptions (7)
  • domain assumption H(t) is q-local with bounded local norm ||H(t)||_loc ≤ M and total variation ∫||dH||_X ≤ λn.
    Assumptions of Theorem 1 (Sec. II A, Case 1). If violated, no exponential bound.
  • domain assumption Initial state is an eigenstate of H(0).
    Theorem 1; extended to general initial states by linearity in Eq. (5).
  • standard math Nested commutator growth ||ad^m_H(V)|| ≤ m!Δ^m||V||_X for q-local H,V (Δ=2qM).
    Proved in Appendix F; relies on norm subadditivity and local norm bound. This is the key estimate that makes the moment recursion work.
  • domain assumption Clustering property of LDPC codes: eigenstates below E_B are within ν1=2γn of a codeword and separated across clusters by ν2=d_C-2γn; guaranteed by linear soundness (Eq. (9)).
    Imported from [29,30,44,45] and [33] (Sec. III, Fig. 3). Load-bearing for configuration-space localization.
  • domain assumption Overlap gap property / clustering for random p-spin glasses and other optimization problems in a linear energy window.
    Sec. IV; for the q-spin glass example the paper states it was 'demonstrated, though not explicitly proved' in [62]. This is the weakest imported input.
  • domain assumption Quantum bottleneck theorem (Lemma E.1) for Gibbs sampler mixing time.
    Imported from [14]; used for Proposition III.4.
  • domain assumption Ground-state degeneracy of good qLDPC codes is stable against local perturbations [53,54].
    Used in Appendix D.3 to explain why eigenstate localization fails for qLDPC without logical detuning.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Universal energy-space localization and stable quantum phases against time-dependent perturbations." pith.science (2026). https://pith.science/paper/5KIXQNRK

@misc{pith2026251014160,
  author       = {Pith},
  title        = {Pith review of: Universal energy-space localization and stable quantum phases against time-dependent perturbations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KIXQNRK}},
  note         = {Machine review of arXiv:2510.14160}
}
abstract

Stability against perturbations is a highly nontrivial property of quantum systems and is often a requirement to define new phases. In most systems where stability can be rigorously established, only static perturbations are considered; whether any system can remain stable against generic time-dependent perturbations is largely elusive. In this work, we identify a universal phenomenon in driving $q$-local Hamiltonians called energy-space localization and prove that it can survive under generic time-dependent perturbations, where the evolving state is exponentially localized in an energy window of the instantaneous spectrum. For spin glass models where the configuration spaces are separated by large energy barriers, the localization in energy spaces can induce a true localization in configuration spaces and robustly break ergodicity. We then demonstrate its applications in several systems with such barriers. For certain LDPC codes, we show that the system remains localized near the original codeword for an exponentially long time even under generic time-dependent perturbations. For classical optimization problems with clustered solution space, the stability becomes an obstacle for quantum Hamiltonian-based algorithms to escape local minima. Our work provides a new lens for analyzing non-equilibrium dynamics of generic quantum systems, and versatile mathematical tools for establishing stability and for designing quantum algorithms.

Figures

Figures reproduced from arXiv: 2510.14160 by the authors.

Figure 1
Figure 1. FIG. 1. An illustrative example of the energy-space Localiza [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. An illustrative example of the clustering property [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of the clustering property in classi [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of the clustering property in hard opti [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

110 extracted references · 9 linked inside Pith

  1. [33]

    C. Yin, R. Nandkishore, and A. Lucas, Eigenstate local- ization in a many-body quantum system, Physical Re- view Letters133, 137101 (2024)

    2024

  2. [62]

    E. R. Anschuetz, D. Gamarnik, and B. Kiani, Combina- torial NLTS From the Overlap Gap Property, Quantum 8, 1527 (2024)

    2024

  3. [46]

    There are also codes with the clustered property but without linear soundness. Most of our results still ap- ply to these models, but the localized states may stay near a false vacuum—a local minimum only breaking a few checks but far away from all codewords—instead of true codewords

  4. [14]

    Rakovszky, B

    T. Rakovszky, B. Placke, N. P. Breuckmann, and V. Khemani, Bottlenecks in quantum channels and finite temperature phases of matter, arXiv preprint arXiv:2412.09598 (2024)

    arXiv 2024

  5. [1]

    P. W. Anderson, Absence of diffusion in certain random lattices, Physical review109, 1492 (1958)

    1958

  6. [2]

    Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev

    X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys.89, 041004 (2017)

    2017

  7. [3]

    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

  8. [4]

    R. M. Nandkishore and M. Hermele, Fractons, Annual Review of Condensed Matter Physics10, 295 (2019)

    2019

Show all 110 references
  1. [5]

    T. Mori, T. Kuwahara, and K. Saito, Rigorous bound on energy absorption and generic relaxation in periodically driven quantum systems, Phys. Rev. Lett.116, 120401 (2016)

    2016

  2. [6]

    M. B. Hastings and X.-G. Wen, Quasiadiabatic contin- uation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance, Phys. Rev. B72, 045141 (2005)

    2005

  3. [7]

    Michalakis and J

    S. Michalakis and J. P. Zwolak, Stability of frustration- free hamiltonians, Communications in Mathematical Physics322, 277 (2013)

    2013

  4. [8]

    J. Z. Imbrie, On many-body localization for quantum spin chains, Journal of Statistical Physics163, 998 (2016)

    2016

  5. [9]

    Abanin, W

    D. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems, Com- munications in Mathematical Physics354, 809 (2017)

    2017

  6. [10]

    D. V. Else, B. Bauer, and C. Nayak, Prethermal phases of matter protected by time-translation symmetry, Physical Review X7, 011026 (2017)

    2017

  7. [11]

    Machado, D

    F. Machado, D. V. Else, G. D. Kahanamoku-Meyer, C. Nayak, and N. Y. Yao, Long-range prethermal phases of nonequilibrium matter, Phys. Rev. X10, 011043 (2020)

    2020

  8. [12]

    Z. Gong, R. Hamazaki, and M. Ueda, Discrete time- crystalline order in cavity and circuit qed systems, Phys. Rev. Lett.120, 040404 (2018)

    2018

  9. [13]

    Y. Hong, J. Guo, and A. Lucas, Quantum memory at nonzero temperature in a thermodynamically trivial sys- tem, Nature Communications16, 316 (2025)

    2025

  10. [15]

    L. A. Wray, S.-Y. Xu, Y. Xia, Y. S. Hor, D. Qian, A. V. Fedorov, H. Lin, A. Bansil, R. J. Cava, and M. Z. Hasan, Observation of topological order in a supercon- ducting doped topological insulator, Nature Physics6, 855 (2010)

    2010

  11. [16]

    Gring, M

    M. Gring, M. Kuhnert, T. Langen, T. Kitagawa, B. Rauer, M. Schreitl, I. Mazets, D. A. Smith, E. Demler, and J. Schmiedmayer, Relaxation and prethermalization in an isolated quantum system, Science337, 1318 (2012)

    2012

  12. [17]

    S. Choi, J. Choi, R. Landig, G. Kucsko, H. Zhou, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, V. Khemani,et al., Ob- servation of discrete time-crystalline order in a disordered dipolar many-body system, Nature543, 221 (2017)

    2017

  13. [18]

    P. Peng, C. Yin, X. Huang, C. Ramanathan, and P. Cap- pellaro, Floquet prethermalization in dipolar spin chains, Nature Physics17, 444 (2021)

    2021

  14. [19]

    Kongkhambut, J

    P. Kongkhambut, J. Skulte, L. Mathey, J. G. Cosme, A. Hemmerich, and H. Keßler, Observation of a continu- ous time crystal, Science377, 670 (2022)

    2022

  15. [20]

    Y. Guo, S. Dhar, A. Yang, Z. Chen, H. Yao, M. Hor- vath, L. Ying, M. Landini, and H.-C. N¨ agerl, Observa- tion of many-body dynamical localization, Science389, 716 (2025)

    2025

  16. [21]

    F. J. Dyson, The s matrix in quantum electrodynamics, Physical Review75, 1736 (1949)

    1949

  17. [22]

    E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Communications in mathemat- ical physics28, 251 (1972)

    1972

  18. [23]

    Kuwahara, T

    T. Kuwahara, T. Mori, and K. Saito, Floquet–magnus theory and generic transient dynamics in periodically driven many-body quantum systems, Annals of Physics 367, 96 (2016)

    2016

  19. [24]

    J. M. Deutsch, Quantum statistical mechanics in a closed system, Physical review a43, 2046 (1991)

  20. [25]

    Srednicki, Chaos and quantum thermalization, Phys- ical review e50, 888 (1994)

    M. Srednicki, Chaos and quantum thermalization, Phys- ical review e50, 888 (1994). 12

    1994

  21. [26]

    Rigol, V

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

    2008

  22. [27]

    Nandkishore and D

    R. Nandkishore and D. A. Huse, Many-body localiza- tion and thermalization in quantum statistical mechan- ics, Annu. Rev. Condens. Matter Phys.6, 15 (2015)

    2015

  23. [28]

    J. M. Deutsch, Eigenstate thermalization hypothesis, Re- ports on Progress in Physics81, 082001 (2018)

    2018

  24. [29]

    Dinur, M.-H

    I. Dinur, M.-H. Hsieh, T.-C. Lin, and T. Vidick, Good quantum ldpc codes with linear time decoders, inPro- ceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023 (Association for Computing Machinery, New York, NY, USA, 2023) pp. 905–918

    2023

  25. [30]

    Panteleev and G

    P. Panteleev and G. Kalachev, Asymptotically good quantum and locally testable classical ldpc codes, inPro- ceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 (Association for Computing Machinery, New York, NY, USA, 2022) pp. 375–388

    2022

  26. [31]

    G. Zhao, A. C. Doherty, and I. H. Kim, Energy barrier of hypergraph product codes, Phys. Rev. Lett.134, 180601 (2025)

    2025

  27. [32]

    Gamarnik, The overlap gap property: A topologi- cal barrier to optimizing over random structures, Pro- ceedings of the National Academy of Sciences118, e2108492118 (2021)

    D. Gamarnik, The overlap gap property: A topologi- cal barrier to optimizing over random structures, Pro- ceedings of the National Academy of Sciences118, e2108492118 (2021)

    2021

  28. [34]

    D. A. Abanin, W. De Roeck, W. W. Ho, and F. Hu- veneers, Effective hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems, Physical Review B95, 014112 (2017)

    2017

  29. [35]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A universal operator growth hypothesis, Phys. Rev. X9, 041017 (2019)

    2019

  30. [36]

    Avdoshkin and A

    A. Avdoshkin and A. Dymarsky, Euclidean operator growth and quantum chaos, Phys. Rev. Res.2, 043234 (2020)

    2020

  31. [37]

    Weisse, R

    A. Weisse, R. Gerstner, and J. Sirker, Operator growth in disordered spin chains: Indications for the absence of many-body localization, Phys. Rev. Res.7, 033018 (2025)

    2025

  32. [38]

    Mezard and A

    M. Mezard and A. Montanari,Information, physics, and computation(Oxford University Press, 2009)

    2009

  33. [39]

    Gamarnik and M

    D. Gamarnik and M. Sudan, Limits of local algorithms over sparse random graphs, THE ANNALS of PROBA- BILITY , 2353 (2017)

    2017

  34. [40]

    Gamarnik and A

    D. Gamarnik and A. Jagannath, The overlap gap prop- erty and approximate message passing algorithms for p- spin models, The Annals of Probability49, 180 (2021)

    2021

  35. [41]

    Placke, T

    B. Placke, T. Rakovszky, N. P. Breuckmann, and V. Khe- mani, Topological quantum spin glass order and its real- ization in qldpc codes, arXiv preprint arXiv:2412.13248 (2024)

    2024 arXiv

  36. [42]

    For quasi-q-local perturbations, we require ν2 ∼Θ(n)

    When consideringq-local perturbations,ν 2 > qis usu- ally sufficient. For quasi-q-local perturbations, we require ν2 ∼Θ(n). Although we mainly focus on models with ν2 ∼Θ(n) in this work, the results can easily extend to other scalings ofν 2

  37. [43]

    Gamarnik, B

    D. Gamarnik, B. T. Kiani, and A. Zlokapa, Slow mixing of quantum gibbs samplers, arXiv preprint arXiv:2411.04300 (2024)

    2024 arXiv

  38. [44]

    Sipser and D

    M. Sipser and D. Spielman, Expander codes, IEEE Trans- actions on Information Theory42, 1710 (1996)

    1996

  39. [45]

    Dinur, S

    I. Dinur, S. Evra, R. Livne, A. Lubotzky, and S. Mozes, Locally testable codes with constant rate, distance, and locality, inProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022 (Asso- ciation for Computing Machinery, New York, NY, USA,

    2022

  40. [47]

    Rademaker and D

    L. Rademaker and D. A. Abanin, Slow nonthermalizing dynamics in a quantum spin glass, Physical review letters 125, 260405 (2020)

    2020

  41. [48]

    C. Yin, F. M. Surace, and A. Lucas, Theory of metastable states in many-body quantum systems, Phys. Rev. X15, 011064 (2025)

    2025

  42. [49]

    Dennis, A

    E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological quantum memory, Journal of Mathematical Physics43, 4452 (2002)

    2002

  43. [50]

    McEwen, L

    M. McEwen, L. Faoro, K. Arya, A. Dunsworth, T. Huang, S. Kim, B. Burkett, A. Fowler, F. Arute, J. C. Bardin,et al., Resolving catastrophic error bursts from cosmic rays in large arrays of superconducting qubits, Nature Physics18, 107 (2022)

    2022

  44. [51]

    S. J. S. Tan, C. A. Pattison, M. McEwen, and J. Preskill, Resilience of the surface code to error bursts, arXiv preprint arXiv:2406.18897 (2024)

    2024 arXiv

  45. [52]

    Sriram, N

    A. Sriram, N. O’Dea, Y. Li, T. Rakovszky, and V. Khe- mani, Rare events and griffiths phases in topologi- cal quantum error correction, Physical Review B111, 224208 (2025)

    2025

  46. [53]

    De Roeck, V

    W. De Roeck, V. Khemani, Y. Li, N. O’Dea, and T. Rakovszky, Low-density parity-check stabilizer codes as gapped quantum phases: Stability under graph-local perturbations, PRX Quantum6, 030330 (2025)

    2025

  47. [54]

    Yin and A

    C. Yin and A. Lucas, Low-density parity-check codes as stable phases of quantum matter, PRX Quantum6, 030329 (2025)

    2025

  48. [55]

    C.-F. Chen, M. J. Kastoryano, and A. Gily´ en, An effi- cient and exact noncommutative quantum gibbs sampler, arXiv preprint arXiv:2311.09207 (2023)

    2023

  49. [56]

    D. Fang, J. Lu, and Y. Tong, Mixing time of open quan- tum systems via hypocoercivity, Phys. Rev. Lett.134, 140405 (2025)

    2025

  50. [57]

    Z. Ding, B. Li, and L. Lin, Efficient quantum gibbs sam- plers with kubo–martin–schwinger detailed balance con- dition, Communications in Mathematical Physics406, 67 (2025)

    2025

  51. [58]

    M. X. Goemans and D. P. Williamson, Improved approx- imation algorithms for maximum cut and satisfiability problems using semidefinite programming, Journal of the ACM (JACM)42, 1115 (1995)

    1995

  52. [59]

    Sherrington and S

    D. Sherrington and S. Kirkpatrick, Solvable model of a spin-glass, Phys. Rev. Lett.35, 1792 (1975)

    1975

  53. [60]

    Montanari, Optimization of the sherrington– kirkpatrick hamiltonian, SIAM Journal on Computing 54, FOCS19 (2025)

    A. Montanari, Optimization of the sherrington– kirkpatrick hamiltonian, SIAM Journal on Computing 54, FOCS19 (2025)

    2025

  54. [61]

    W.-K. Chen, D. Gamarnik, D. Panchenko, and M. Rah- man, Suboptimality of local algorithms for a class of max- cut problems, The Annals of Probability47, 1587 (2019). 13

    2019

  55. [63]

    M´ ezard, G

    M. M´ ezard, G. Parisi, and R. Zecchina, Analytic and al- gorithmic solution of random satisfiability problems, Sci- ence297, 812 (2002)

    2002

  56. [64]

    Achlioptas, A

    D. Achlioptas, A. Coja-Oghlan, and F. Ricci-Tersenghi, On the solution-space geometry of random constraint sat- isfaction problems, Random Structures & Algorithms38, 251 (2011)

    2011

  57. [65]

    J. Ding, A. Sly, and N. Sun, Proof of the satisfiability con- jecture for largek, Annals of Mathematics196, 1 (2022)

    2022

  58. [66]

    Bresler and B

    G. Bresler and B. Huang, The algorithmic phase transi- tion of random k-sat for low degree polynomials, in2021 IEEE 62nd annual symposium on foundations of com- puter science (FOCS)(IEEE, 2022) pp. 298–309

    2022

  59. [67]

    Coja-Oghlan and C

    A. Coja-Oghlan and C. Efthymiou, On independent sets in random graphs, Random Structures & Algorithms47, 436 (2015)

    2015

  60. [68]

    M´ ezard, T

    M. M´ ezard, T. Mora, and R. Zecchina, Clustering of so- lutions in the random satisfiability problem, Phys. Rev. Lett.94, 197205 (2005)

    2005

  61. [69]

    Farhi, D

    E. Farhi, D. Gamarnik, and S. Gutmann, The quan- tum approximate optimization algorithm needs to see the whole graph: A typical case, arXiv preprint arXiv:2004.09002 (2020)

    2004 arXiv

  62. [70]

    Basso, D

    J. Basso, D. Gamarnik, S. Mei, and L. Zhou, Perfor- mance and limitations of the qaoa at constant levels on large sparse hypergraphs and spin glass models, in2022 IEEE 63rd Annual Symposium on Foundations of Com- puter Science (FOCS)(IEEE, 2022) pp. 335–343

    2022

  63. [71]

    Farhi, J

    E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lund- gren, and D. Preda, A quantum adiabatic evolution al- gorithm applied to random instances of an np-complete problem, Science292, 472 (2001)

    2001

  64. [72]

    Farhi, J

    E. Farhi, J. Goldstone, and S. Gutmann, A quan- tum approximate optimization algorithm, arXiv preprint arXiv:1411.4028 (2014)

    2014 arXiv

  65. [73]

    Knysh, Zero-temperature quantum annealing bottle- necks in the spin-glass phase, Nature communications7, 12370 (2016)

    S. Knysh, Zero-temperature quantum annealing bottle- necks in the spin-glass phase, Nature communications7, 12370 (2016)

    2016

  66. [74]

    S. C. Benjamin, L. Zhao, and J. F. Fitzsimons, Measurement-driven quantum computing: Performance of a 3-sat solver, arXiv preprint arXiv:1711.02687 (2017)

    2017 arXiv

  67. [75]

    Venturelli, S

    D. Venturelli, S. Mandr` a, S. Knysh, B. O’Gorman, R. Biswas, and V. Smelyanskiy, Quantum optimization of fully connected spin glasses, Phys. Rev. X5, 031040 (2015)

    2015

  68. [76]

    Boulebnane and A

    S. Boulebnane and A. Montanaro, Solving boolean sat- isfiability problems with the quantum approximate opti- mization algorithm, PRX Quantum5, 030348 (2024)

    2024

  69. [77]

    H. Yu, F. Wilczek, and B. Wu, Quantum algorithm for approximating maximum independent sets, Chinese Physics Letters38, 030304 (2021)

    2021

  70. [78]

    Ebadi, A

    S. Ebadi, A. Keesling, M. Cain, T. T. Wang, H. Levine, D. Bluvstein, G. Semeghini, A. Omran, J.-G. Liu, R. Samajdar,et al., Quantum optimization of maximum independent set using rydberg atom arrays, Science376, 1209 (2022)

    2022

  71. [79]

    Munoz-Bauza and D

    H. Munoz-Bauza and D. Lidar, Scaling advantage in ap- proximate optimization with quantum annealing, Phys. Rev. Lett.134, 160601 (2025)

    2025

  72. [80]

    A. D. King, A. Nocera, M. M. Rams, J. Dziarmaga, R. Wiersema, W. Bernoudy, J. Raymond, N. Kaushal, N. Heinsdorf, R. Harris,et al., Beyond-classical compu- tation in quantum simulation, Science388, 199 (2025)

    2025

  73. [81]

    Kadowaki and H

    T. Kadowaki and H. Nishimori, Quantum annealing in the transverse ising model, Phys. Rev. E58, 5355 (1998)

    1998

  74. [82]

    Crosson and D

    E. Crosson and D. Lidar, Prospects for quantum en- hancement with diabatic quantum annealing, Nature Re- views Physics3, 466 (2021)

    2021

  75. [83]

    Farhi, J

    E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution, arXiv preprint quant-ph/0001106 (2000)

    2000 arXiv

  76. [84]

    Albash and D

    T. Albash and D. A. Lidar, Adiabatic quantum compu- tation, Rev. Mod. Phys.90, 015002 (2018)

    2018

  77. [85]

    J. Leng, E. Hickman, J. Li, and X. Wu, Quantum hamil- tonian descent, arXiv preprint arXiv:2303.01471 (2023)

    2023 arXiv

  78. [86]

    Bravyi, M

    S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb- robinson bounds and the generation of correlations and topological quantum order, Physical review letters97, 050401 (2006)

    2006

  79. [87]

    A. H. Moosavian, S. S. Kahani, and S. Beigi, Limits of Short-Time Evolution of Local Hamiltonians, Quantum 6, 744 (2022)

    2022

  80. [88]

    Born and V

    M. Born and V. Fock, Beweis des adiabatensatzes, Zeitschrift f¨ ur Physik51, 165 (1928)

    1928

  81. [89]

    Altshuler, H

    B. Altshuler, H. Krovi, and J. Roland, Anderson local- ization makes adiabatic quantum optimization fail, Pro- ceedings of the National Academy of Sciences107, 12446 (2010)

    2010

  82. [90]

    One can also choose chooseH L =H E(G) and setVto re- spect the symmetry that conserves the total vertex num- ber (i.e., [V, HV] = 0). Such a symmetry-preserving par- ticle number is more common, and one can verify that its energy landscape exhibits clustering property in a lin...

  83. [91]

    Bernien, S

    H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om- ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner,et al., Probing many-body dynamics on a 51-atom quantum simulator, Nature551, 579 (2017)

    2017

  84. [92]

    Serbyn, D

    M. Serbyn, D. A. Abanin, and Z. Papi´ c, Quantum many- body scars and weak breaking of ergodicity, Nature Physics17, 675 (2021)

    2021

  85. [93]

    Sachdev and J

    S. Sachdev and J. Ye, Gapless spin-fluid ground state in a random quantum heisenberg magnet, Phys. Rev. Lett. 70, 3339 (1993)

    1993

  86. [94]

    M. C. Tran, C.-F. Chen, A. Ehrenberg, A. Y. Guo, A. Deshpande, Y. Hong, Z.-X. Gong, A. V. Gorshkov, and A. Lucas, Hierarchy of linear light cones with long- range interactions, Phys. Rev. X10, 031009 (2020)

    2020

  87. [95]

    Nandy, A

    P. Nandy, A. S. Matsoukas-Roubeas, P. Mart ´ ınez- Azcona, A. Dymarsky, and A. del Campo, Quantum dynamics in krylov space: Methods and applications, Physics Reports1125, 1 (2025)

    2025

  88. [96]

    T. K. Philips and R. Nelson, The moment bound is tighter than chernoff’s bound for positive tail probabili- ties, The American Statistician49, 175 (1995). 14 Appendix A: Definition of different norms and total variations In this work, we define various norms and total varia- t...

    1995

  89. [97]

    Main proof for Cases 1, 2, 4 For Cases. 1 and 2, we define the total variation at timetwith 0≤t≤Tas λ(t)≡ 1 n Z t 0 ∂H(τ) ∂τ X dτ.(B9) Then it is obvious thatλ(t)≥0 increases monotonically withtand, for brevity, we defineλ t ≡λ(t)≤λ(T)≡ λT =λ. Additionally, according to the de...

  90. [98]

    In this case, we can use the same defini- tion forλ(t) andλ ′(t) as (B9) and (B10)

    Improved bounds for Case 3 In Case 3, we assume that theH(t) admits a partition into a mutually commuting coreHC(t) and the remainder V(t) H(t) =H C(t) +V(t),(B39) such that [V(t), H ′(t)] = 0 for alltand all terms in theH C(t) = P X hX (t) are mutually commuting with each oth...

  91. [99]

    Here, we roughly estimate the leakage bounds in general settings where all parameters can scale differently withn

    Bounds for perturbations of general scalings In previous sections we discussed cases with linear- scaling perturbations and energy windows, with bounded local norms. Here, we roughly estimate the leakage bounds in general settings where all parameters can scale differently wit...

  92. [100]

    The h n j i also has a combinatorial interpretation:h n j i counts the number of permutations when dividing thenelements intojdisjoint cycles

    Properties off (1) k In this section, we prove that thef (1) k (x)’s we found f (1) n (x) = n−1Y m=0 (x+m) = nX j=0 n j xj (B57) indeed satisfies the recurrence relation (B33), where h n j i is the unsigned Stirling number of the first kind by defi- nition. The h n j i also ha...

  93. [101]

    The n n j o has a combinatorial interpretation: n n j o counts the number of ways to partitionnelements intojunlabeled non-empty sets

    Properties off (2) k In this section, we want to confirm that f (2) n (x) = nX j=0 n j xj (B61) satisfies the recurrence relation (B42), where n n j o is the Stirling number of the second kind. The n n j o has a combinatorial interpretation: n n j o counts the number of ways t...

  94. [102]

    We require thatV(t) satisfies R T 0− ∥H ′ ext(t)∥X ≤λnfor strictlyq-local case

    Exponentially long time dynamical localization In this section, we consider the evolution under the following time-dependent Hamiltonian: H(t) =H C +V(t),(C9) whereV(t) is a time-dependent perturbation. We require thatV(t) satisfies R T 0− ∥H ′ ext(t)∥X ≤λnfor strictlyq-local ...

  95. [103]

    Infinitely long time dynamical localization In this section, we are going to show that the dynamical localization can be infinitely long in some circumstances. We consider the following static Hamiltonian (similarly defined in [33]) H=H C +V 0 +H d,(C28) whereH C is a good cLD...

  96. [104]

    Thus, we can choose the energy window to be [E ′ − dn, E′ +dn], withd=b−ε ′

    Eigenstate localization in cLDPC codes with strictlyq-local perturbations For the perturbed eigenstate|ψ⟩with energyE ′ ≡ ε′n+E g, its energy difference from the top ofWis (b− ε′)n. Thus, we can choose the energy window to be [E ′ − dn, E′ +dn], withd=b−ε ′. Note that here the...

  97. [105]

    We consider an eigenstate|ψ⟩of H=H C +V 0 +H d,(D18) with energyE ′ =ε ′n+E g

    Eigenstate localization in cLDPC codes with quasi-q-local perturbations For a quasi-q⋆-local perturbationV 0, we can follow sim- ilar procedures. We consider an eigenstate|ψ⟩of H=H C +V 0 +H d,(D18) with energyE ′ =ε ′n+E g. From Theorem 2, we know that the leakage outside all...

  98. [106]

    However, an issue arises when we want to apply Lemma D.1 to bound the∥(1−P w)|ψ⟩ ∥, where we need to lower boundδ W and ensure that it is much larger than the∥P > |ψ⟩ ∥

    Eigenstate localization in qLDPC codes To investigate the eigenstate localization in qLDPC codes, we can still use Theorem 2 to bound the∥P > |ψ⟩ ∥ exponentially. However, an issue arises when we want to apply Lemma D.1 to bound the∥(1−P w)|ψ⟩ ∥, where we need to lower boundδ ...

  99. [107]

    Suppose that for any|E i⟩ ∈Vand |Ej⟩/∈V, we defineδ V ≡min i,j |Ej −E i|

    Proof of Lemma D.1 Lemma D.1For any HamiltonianH, if one can find an approximate eigensolution (H−E ′)|Ψ⟩=|ϵ⟩, where|Ψ⟩ is normalized and∥ |ϵ⟩ ∥=ϵ <1, then there is at least one exact energyE ⋆ ofHsatisfying |E⋆ −E ′| ≤ϵ.(D31) LetVbe some subspace constructed from eigenstates ...

  100. [108]

    Proof of Lemma D.2 Lemma D.2. If the small diagonal detuningH d is added Hd ≡ 1 n2 X j hz j Zj,(D41) for any 0< ξ2 < ξ1 −ln 4, we have P δW < e−ξ1n < e−ξ2n.(D42) Here,δ W is defined in the eigenbasis of HW ≡P W (HC +V 0 +H d)PW ,(D43) whose eigenstates are all exactly localize...

  101. [109]

    This gives PA =P w, PB =P >, ρ0 = 1 Z e−βH .(E5) whereZ ≡tre −βH is the partition function

    Strictlyq-local perturbations We can then chooseA=w,Cbeing all the other clus- ters, andBbeing the high energy contribution withB 1 contain all eigenstates ofH C that can be connected by local channelMwith eigenstates inw. This gives PA =P w, PB =P >, ρ0 = 1 Z e−βH .(E5) where...

  102. [110]

    This can be bounded by tr P>e−βH ≤2 n max P>|ϕ⟩̸=0 ϕ e−βH ϕ .(E16) From Theorem 2 and Eq

    Quasi-q-local perturbations One can verify that for Quasi-q⋆-local perturbationV 0, most of the steps are similar to strictlyq-local cases, ex- cept upper bounding tr(P >e−βH ). This can be bounded by tr P>e−βH ≤2 n max P>|ϕ⟩̸=0 ϕ e−βH ϕ .(E16) From Theorem 2 and Eq. (E10) we ...

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.