pith. sign in

arxiv: 1907.01609 · v1 · pith:FBFK46HBnew · submitted 2019-07-02 · ❄️ cond-mat.dis-nn · quant-ph

Intermittency of dynamical phases in a quantum spin glass

Vadim N. Smelyanskiy , Kostyantyn Kechedzhi , Sergio Boixo , Hartmut Neven , Boris Altshuler This is my paper

Pith reviewed 2026-05-25 10:09 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn quant-ph
keywords quantum spin glassnon-ergodic extended statesasymptotic orthogonalitymany-body localizationrandom energy modelEdwards-Anderson order parameterfractal dimensionquantum algorithms
0
0 comments X

The pith

Quantum spin glasses have an eigen-spectrum split into alternating bands of x-type and z-type non-ergodic states.

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

The paper establishes that asymptotic orthogonality produces a new banded structure in the energy spectrum of quantum spin glasses. This splits the spectrum into alternating bands of two types of states. Z-type states are non-ergodic extended states that retain the long relaxation times of the classical spin glass even with a transverse field present. X-type states form narrower bands that conserve a magnetization quantum number. The finding matters because the bands may support quantum population transfer algorithms for low-energy configurations in hard optimization problems.

Core claim

The eigen-spectrum is split into the alternating sequence of bands formed by quantum states of two distinct types (x and z). Those of z-type are non-ergodic extended eigenstates (NEE) in the basis of {σ_z} operators that inherit the structure of the classical spin glass with exponentially long decay times of Edwards Anderson order parameter at any finite value of transverse field B_⊥. Those of x-type form narrow bands of NEEs that conserve the integer-valued x-magnetization. Quantum evolution within a given band of each type is described by a Hamiltonian that belongs to either the ensemble of Preferred Basis Levi matrices (z-type) or Gaussian Orthogonal ensemble (x-type).

What carries the argument

asymptotic orthogonality between states, which produces the intermittent alternating bands of x-type and z-type non-ergodic extended states

If this is right

  • Quantum evolution within each band type follows either Preferred Basis Levi matrices (z-type) or the Gaussian Orthogonal ensemble (x-type).
  • The fractal dimension D acts as a thermodynamic potential that identifies phases: D=0 in many-body localized phase, 0<D<1 in non-ergodic extended phase, and D approaching 1 in the ergodic phase at infinite temperature.
  • Many-body localized states coexist with non-ergodic extended states over the same energy range even at large transverse fields.
  • Bands of non-ergodic extended states support new quantum search-like algorithms for population transfer among low-energy spin configurations.
  • The intermittent spectrum structure is expected to appear in a class of NP-hard problems that, unlike the random energy model, admit polynomial-resource implementations.

Where Pith is reading between the lines

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

  • The banded structure may suggest a route to quantum advantage in optimization by restricting dynamics to low-energy sectors.
  • Exact diagonalization studies on small spin instances could directly check for the predicted alternation of band types.
  • Similar intermittency might appear in other disordered quantum many-body systems that exhibit many-body localization.
  • If confirmed, the result could inform the choice of annealing schedules that target the non-ergodic extended bands.

Load-bearing premise

The intermittent banded structure and asymptotic orthogonality derived for the quantum random energy model extend to a broad class of practically important NP-hard problems that can be implemented on a computer with polynomial resources.

What would settle it

Numerical diagonalization of the quantum random energy model eigenstates that shows no alternating sequence of x-type and z-type bands at finite transverse field would disprove the central claim.

Figures

Figures reproduced from arXiv: 1907.01609 by Boris Altshuler, Hartmut Neven, Kostyantyn Kechedzhi, Sergio Boixo, Vadim N. Smelyanskiy.

Figure 1
Figure 1. Figure 1: (a). Cartoon of the energy landscape of the binary optimization problem, E(s) vs s, shown with red disks. The horizontal axis corresponds to spin configurations s and vertical axis to E(s). Dashed arrow indicate the large Hamming distance dij = O(n) between the spin configurations si, sj with close energies E(si), E(sj ) = O(n) connected by the tunneling paths that each proceed through the sequence of stat… view at source ↗
Figure 2
Figure 2. Figure 2: Diagrams level spacing, γ = V (n/2, a)ρ(ne), determined by the value of e at the energy strip and transverse field B⊥. For γ < 1 the eigenstates of H are many-body localized. For γ > 1 the eigenspectrum of H splits into a large number of minibands of NEEs. The eigenstates hs|ψi from a given miniband are peaked at the same states |si forming the support set of the miniband S that is sparse in the computatio… view at source ↗
Figure 3
Figure 3. Figure 3: Density plot of the fractal dimension D = D(|e|, B⊥) (19). Maximum of |e| = √ ln 2 corresponds to typ￾ical lowest (highest) energy density in REM. Mobility edge, D(|e|, B⊥) = 0, separating MBL and NEE phases is shown with white line. Dot-dashed lines show its the asymptotes. Blue area depicts MBL phase existing at all finite values of B⊥. In the area to the right of the "phase transition" line shown in gre… view at source ↗
Figure 4
Figure 4. Figure 4: Effective potential for WKB analysis. Within exponential accuracy the transverse field Green function takes the form, G(λ, d) ∝ e −nS(a,d/n) sin (nφ(a, d/n)), (A12) where we used Stirling’s asymptotic of the binomial co￾efficient, S(a, ρ) = θ(a, ρ) − ρ 2 log(ρ) − (1 − ρ) 2 log(1 − ρ), (A13) where the phases θ(a, ρ) and φ(a, ρ) are determined by the large spin Green function G n 2 −d, n 2 [PITH_FULL_IMAGE… view at source ↗
Figure 5
Figure 5. Figure 5: Exponent of the matrix element in Eq. (A12) [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows matrix element at distance d = n 2 as a function of the relative transverse field parameter a. a ≫ 1 θ (a) = 1  (2 a) 2 ln21/2 0 1 2 0 1 2 a θ [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Here we display a ratio of G k) 0 (λ, ds1s2 ) calculated using the exact expression Eq. (A23) for n = 1000, B⊥ = 100, λ = −B⊥(n/2 + 1/2). is of the same order as G0(λ, d) itself (except for the factor B −p ⊥ that can be small at large fields). 3. Multiple scatterings from the same impurity In the series of diagrams for Σ multple scattering off the same impurity correspond to the "renormalization" of the di… view at source ↗
Figure 8
Figure 8. Figure 8: Diagrams therefore reads, G(λ) ≈ G0(λ¯). (A35) The first corrections to the leading term in the self-energy Σ (1) A + Σ(1) B are given by the second and third diagrams in [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The plot of the function ln ξk(λ) vs. k, see Eq. (A50). Dashed line are guides to the eye. Different colors correspond to different numbers of spins n: 40 (red), 60 (blue), 70 (green), 100 (black). Parameters used: B⊥ = 2, dsisf , λ = −B⊥(n/2+ 1/2). We not the reduction formula for kernels F, F (3) sisf (d1, d2, d3) = Xn d=0 F(d1, d2, d)F(d, d3, dsisf ) (A43) We can rewrite an arbitrary convolution of the … view at source ↗
Figure 10
Figure 10. Figure 10: Partial entropy of z and x states as a function of energy per spin for different magnetic fields. 2. Cumulative density of states We compare cumulative density of states of x and z type states. For the former the sum over the number of states in x-state minibands labeled by x-magnetization n − 2m give the cumulative density, Cx(e, B⊥) = mX∗(e) m=0  n m  , (B12) where m∗(e) = n/2 − en/(2B⊥) is the maximu… view at source ↗
Figure 12
Figure 12. Figure 12: Classical to quantum paramagnet phase transi [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
read the original abstract

Answering the question of existence of efficient quantum algorithms for NP-hard problems require deep theoretical understanding of the properties of the low-energy eigenstates and long-time coherent dynamics in quantum spin glasses. We discovered and described analytically the property of asymptotic orthogonality resulting in a new type of structure in quantum spin glass. Its eigen-spectrum is split into the alternating sequence of bands formed by quantum states of two distinct types ($x$ and $z$). Those of $z$-type are non-ergodic extended eigenstates (NEE) in the basis of $\{\sigma_z\}$ operators that inherit the structure of the classical spin glass with exponentially long decay times of Edwards Anderson order parameter at any finite value of transverse field $B_{\perp}$. Those of $x$-type form narrow bands of NEEs that conserve the integer-valued $x$-magnetization. Quantum evolution within a given band of each type is described by a Hamiltonian that belongs to either the ensemble of Preferred Basis Levi matrices ($z$-type) or Gaussian Orthogonal ensemble ($x$-type). We characterize the non-equilibrium dynamics using fractal dimension $D$ that depends on energy density (temperature) and plays a role of thermodynamic potential: $D=0$ in MBL phase, $0<D<1$ in NEE phase, $D\rightarrow 1$ in ergodic phase in infinite temperature limit. MBL states coexist with NEEs in the same range of energies even at very large $B_{\perp}$. Bands of NEE states can be used for new quantum search-like algorithms of population transfer in the low-energy part of spin-configuration space. Remarkably, the intermitted structure of the eigenspectrum emerges in quantum version of a statistically featureless Random Energy Model and is expected to exist in a class of paractically important NP-hard problems that unlike REM can be implemented on a computer with polynomial resources.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 2 minor

Summary. The manuscript analytically derives the property of asymptotic orthogonality in the quantum random energy model (QREM), resulting in an alternating banded structure in the eigen-spectrum consisting of x-type and z-type non-ergodic extended (NEE) states. z-type states are non-ergodic in the σ_z basis and inherit the classical spin glass structure with exponentially long decay times of the Edwards-Anderson order parameter for any finite transverse field. x-type states form narrow bands conserving x-magnetization. The non-equilibrium dynamics are characterized by a fractal dimension D that acts as a thermodynamic potential, distinguishing MBL, NEE, and ergodic phases. The authors suggest this structure emerges in the QREM and is expected in a broader class of NP-hard problems implementable with polynomial resources, with potential applications to quantum search-like algorithms.

Significance. If the analytical results for the QREM hold, the discovery of this intermittent structure and the characterization via fractal dimension could advance understanding of quantum spin glass dynamics and suggest new approaches for quantum algorithms targeting low-energy states. The distinction between x and z type states and their respective ensembles (Preferred Basis Levi matrices and GOE) is a notable finding. However, the significance for practical NP-hard problems depends on the validity of the extrapolation, which appears asserted rather than derived.

major comments (1)
  1. [Abstract (final sentence)] Abstract (final sentence): The assertion that the intermittent banded structure 'is expected to exist' in a broad class of NP-hard problems implementable with polynomial resources lacks supporting derivation, mapping, or example. The analysis is specific to the featureless REM; no mechanism is provided for how asymptotic orthogonality survives in models with finite-range couplings or correlations in the energy landscape. This extrapolation is load-bearing for the claimed relevance to quantum algorithms but is not substantiated.
minor comments (2)
  1. [Abstract] Typo: 'paractically' should be 'practically'.
  2. [Abstract] Typo: 'intermitted' should be 'intermittent'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: The assertion that the intermittent banded structure 'is expected to exist' in a broad class of NP-hard problems implementable with polynomial resources lacks supporting derivation, mapping, or example. The analysis is specific to the featureless REM; no mechanism is provided for how asymptotic orthogonality survives in models with finite-range couplings or correlations in the energy landscape. This extrapolation is load-bearing for the claimed relevance to quantum algorithms but is not substantiated.

    Authors: We agree that the manuscript derives the intermittent structure and asymptotic orthogonality rigorously only for the QREM, which is statistically featureless. No explicit mapping, example, or mechanism is provided showing how the property survives under finite-range couplings or energy correlations, and the statement in the abstract is therefore an unsubstantiated conjecture rather than a derived result. To correct this, we will revise the final sentence of the abstract to remove the claim about a broad class of NP-hard problems and instead state that the structure is demonstrated for the QREM, with the potential relevance to other models and quantum algorithms left as an open question for future investigation. revision: yes

Circularity Check

0 steps flagged

No circularity: analytical claim for REM stands on its own; extension to NP-hard problems is stated as expectation without derivation chain

full rationale

The provided abstract asserts an analytical discovery of asymptotic orthogonality and banded x/z structure for the quantum random energy model, with z-type states inheriting classical spin-glass features. No equations, self-citations, fitted parameters, or ansatzes are shown that would allow inspection for reduction to inputs by construction. The extension to polynomial-resource NP-hard problems is phrased as an expectation rather than a derived result, so no load-bearing step reduces to a self-referential input. The derivation for the core REM case is presented as independent and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all ledger entries are therefore empty.

pith-pipeline@v0.9.0 · 5898 in / 1160 out tokens · 21511 ms · 2026-05-25T10:09:51.897544+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 3 internal anchors

  1. [1]

    Kadowaki and H

    T. Kadowaki and H. Nishimori, Physical Review E58, 5355 (1998)

    work page 1998

  2. [2]

    Farhi, J

    E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lund- gren, and D. Preda, Science292, 472 (2001)

    work page 2001

  3. [3]

    Brooke, D

    J. Brooke, D. Bitko, G. Aeppli,et al., Science 284, 779 (1999)

    work page 1999

  4. [4]

    V. N. Smelyanskiy, U. v. Toussaint, and D. A. Timucin, arXiv preprint quant-ph/0202155 (2002)

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  5. [5]

    Boixo, T

    S. Boixo, T. F. Rønnow, S. V. Isakov, Z. Wang, D. Wecker, D. A. Lidar, J. M. Martinis, and M. Troyer, Nat. Phys. 10, 218 (2014)

    work page 2014

  6. [6]

    Knysh, Nature communications7, 12370 (2016)

    S. Knysh, Nature communications7, 12370 (2016)

    work page 2016

  7. [7]

    Boixo, V

    S. Boixo, V. N. Smelyanskiy, A. Shabani, S. V. Isakov, M. Dykman, V. S. Denchev, M. H. Amin, A. Y. Smirnov, M. Mohseni, and H. Neven, Nat. Comm.7 (2016)

    work page 2016

  8. [8]

    V. S. Denchev, S. Boixo, S. V. Isakov, N. Ding, R. Bab- bush, V. Smelyanskiy, J. Martinis, and H. Neven, Phys. Rev. X 6, 031015 (2016)

    work page 2016

  9. [9]

    Albash and D

    T. Albash and D. A. Lidar, Reviews of Modern Physics 90, 015002 (2018)

    work page 2018

  10. [10]

    A Quantum Approximate Optimization Algorithm

    E. Farhi, J. Goldstone, and S. Gutmann, arXiv preprint arXiv:1411.4028 (2014)

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  11. [11]

    V. N. Smelyanskiy, K. Kechedzhi, S. Boixo, S. V. Isakov, H. Neven, and B. Altshuler, arXiv preprint arXiv:1802.09542 (2018), arXiv:1802.09542 [quant-ph]

    work page arXiv 2018

  12. [12]

    C. L. Baldwin and C. R. Laumann, arXiv preprint arXiv:1803.02410 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  13. [13]

    Fu and P

    Y. Fu and P. W. Anderson, inSpin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications (World Scientific, 1987) pp. 357–372

    work page 1987

  14. [14]

    Mezard and A

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

    work page 2009

  15. [15]

    K. S. Tikhonov, A. D. Mirlin, and M. A. Skvortsov, Phys. Rev. B94, 220203 (2016)

    work page 2016

  16. [16]

    Altshuler, E

    B. Altshuler, E. Cuevas, L. Ioffe, and V. Kravtsov, Phys- ical Review Letters117, 156601 (2016)

    work page 2016

  17. [17]

    Carleo and M

    G. Carleo and M. Troyer, Science355, 602 (2017)

    work page 2017

  18. [18]

    Biamonte, P

    J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd, Nature549, 195 (2017)

    work page 2017

  19. [19]

    Mertens, Physical Review Letters81, 4281 (1998)

    S. Mertens, Physical Review Letters81, 4281 (1998)

    work page 1998

  20. [21]

    Bauke, S

    H. Bauke, S. Mertens, and A. Engel, Physical review letters 90, 158701 (2003)

    work page 2003

  21. [22]

    Merkle and M

    R. Merkle and M. Hellman, IEEE transactions on Infor- mation Theory 24, 525 (1978)

    work page 1978

  22. [23]

    Borgs, J

    C. Borgs, J. Chayes, and B. Pittel, Random Structures and Algorithms 19, 247 (2001)

    work page 2001

  23. [24]

    Mertens, Physical Review Letters84, 1347 (2000)

    S. Mertens, Physical Review Letters84, 1347 (2000)

    work page 2000

  24. [25]

    Bauke, S

    H. Bauke, S. Franz, and S. Mertens, Journal of Statis- tical Mechanics: Theory and Experiment2004, P04003 (2004)

    work page 2004

  25. [26]

    Derrida, Physical Review B24, 2613 (1981)

    B. Derrida, Physical Review B24, 2613 (1981)

    work page 1981

  26. [27]

    T. R. Kirkpatrick and P. G. Wolynes, Phys. Rev. B36, 8552 (1987)

    work page 1987

  27. [29]

    Abrikosov, L

    A. Abrikosov, L. Gorkov, and I. Dzyaloshinskii,Methods of Quantum Field Theory in Statistical Physics(Prentice Hall, New York, 1963)

    work page 1963

  28. [30]

    Ohkuwa, H

    M. Ohkuwa, H. Nishimori, and D. A. Lidar, Phys. Rev. A 98, 022314 (2018)

    work page 2018

  29. [31]

    Melko, Phys

    M.H.Amin, E.Andriyash, J.Rolfe, B.Kulchytskyy, and R. Melko, Phys. Rev. X8, 021050 (2018)

    work page 2018

  30. [32]

    R. Y. Li, R. Di Felice, R. Rohs, and D. A. Lidar, NPJ quantum information 4, 14 (2018)

    work page 2018

  31. [33]

    P. A. Braun, Rev. Mod. Phys.65, 115 (1993). Appendix A: Details of disorder diagrammatic calculations

    work page 1993

  32. [34]

    Eigenstates of QREM in the finite energy density subspaceS can be found from the non-linear eigenvalue equation, Eq

    Bare Green function at large Hamming distances d =O(n) In this section we provide the details of the calcula- tion of the matrix element of the Green function at large Hamming distance d =O(n). Eigenstates of QREM in the finite energy density subspaceS can be found from the non-linear eigenvalue equation, Eq. (4), which con- tainstheGreen function G(λ)ofth...

    work page

  33. [35]

    disorder potential

    W eak disorder perturbation theory It is convenient to represent the series for the Green function G(λ) in the form of diagrams shown in Fig. 8, where the solid lines correspond to the bare Green func- tion G0(λ) and crosses to the "disorder potential" rep- resented by the Hamiltonian H0 which contains classi- cal energies outside the downfolding subspace...

    work page

  34. [36]

    renormalization

    Multiple scatterings from the same impurity In the series of diagrams forΣ multple scattering off the same impurity correspond to the "renormalization" of the disorder potentialE(s). Below we calculate this renormalization with the help ofT-matrix. We rewrite the Dyson series for G(λ) before averaging in terms of 11 T-matrix that accounts for scattering ex...

    work page

  35. [37]

    The self- energy correction operatorΣ(λ) is given by the series in H0 with the leading order corresponding to the first term in Fig

    Scatterings from multiple impurities In the following we account for interference of the trajectories scattered off multiple impurities. The self- energy correction operatorΣ(λ) is given by the series in H0 with the leading order corresponding to the first term in Fig. 8(e), Σ(0) = nσ2 2E I = I×O (1), (A34) already given in the main text, andI is the identi...

    work page

  36. [38]

    Asymptotical properties of Green function tensor contractions Notethatbelowweassume dsisf =O(n). Wecalculate K(p) sisf = ∑ s1,s2 G(λ, dsis1) (G(λ, ds1s2))p G(λ, ds2sf ) (A38) we rewrite this expression in the following form K(p) sisf = n∑ d1,d2,d3 G(λ, d1) (G(λ, d2))p G(λ, d3)F (3) sisf (d1, d2, d3) (A39) F (3) sisf (d1, d2, d3) = ∑ s1s2 δd1,dsi s1 δd2,ds...

    work page

  37. [39]

    x-type states These states conserve total magnetization alongx-axis and can be conveniently written in the respective eigen- basis Sx|xi⟩ = (n− 2m)|xi⟩ , |xi⟩ = n⨂ k=1 |xi k⟩ (B2) σx k|xi k⟩ = (1− 2xi k)|xi k⟩ , x i k = 0, 1 (B3) where Sx ≡ (1/n)∑n k=1 σx k and m = 0 , 1, . . . , n. The eigenstates of Sx are mixed by the matrix elements of HREM, ⟨x| Hcl|x...

    work page

  38. [40]

    Cumulative density of states We compare cumulative density of states ofx and z type states. For the former the sum over the number of states in x-state minibands labeled by x-magnetization n− 2m give the cumulative density, Cx(e, B⊥) = m∗(e)∑ m=0 (n m ) , (B12) where m∗(e) = n/2− en/(2B⊥) is the maximum in- cluded miniband index. The partial entropy of th...

    work page

  39. [41]

    ∆γ = 0 line (black solid line in Fig. 11) separates the region in thespectrumwherethereareexponentiallymore x-states γx(e, B⊥) > γz(e), from the region where there are expo- nentially more z-states γx(e, B⊥) < γz(e) shown as gray γz <γ x γz >γ x γz >γ x max 0 1 2 log 21/2 log 2 1 2 1 2 log 21/2 log 2 e B⟂ Figure 11. Statistical phase diagram of the QREM e...

    work page

  40. [42]

    Finite temperature partition function In this Section we use the statistics of the eigenstates to compare to the earlier replica analysis of the QREM partition function with static approximation. a. x-state partition function Partition of function ofx-states consists of two parts: that of the free spin in transverse fieldB⊥, and a cor- rection due to Gauss...

    work page