pith. sign in

arxiv: 2605.24086 · v1 · pith:O6KX7322new · submitted 2026-05-22 · 🪐 quant-ph · cond-mat.dis-nn

Dynamical self-dual criticality in Fibonacci-monitored quantum Ising chains

Finn Eckstein , Harald Schmid , Quinten Preiss , Simon Trebst , Felix von Oppen , Guo-Yi Zhu This is my paper

Pith reviewed 2026-06-30 15:43 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nn
keywords monitored quantum dynamicsFibonacci sequenceself-dualityIsing modelquantum criticalitynon-invertible symmetryMajorana fermions
0
0 comments X

The pith

Fibonacci measurement sequences extend Kramers-Wannier duality into monitored quantum dynamics without time-translation symmetry.

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

The paper investigates monitored dynamics in one-dimensional Ising and Majorana chains where measurements follow a quasiperiodic Fibonacci sequence. It finds that this arrangement creates a dynamical form of self-duality from the Kramers-Wannier symmetry. This self-duality organizes the phase diagram of the resulting entangled phases and protects their critical properties at the transitions. The work identifies two critical lines tied to the golden ratio and determines the universality classes of the steady states.

Core claim

The dynamical extension of the Kramers-Wannier non-invertible symmetry to monitored dynamics with Fibonacci measurements organizes the dynamical phase diagram, predicts the locations of the critical lines, and protects universal critical behavior in the absence of time-translation symmetry.

What carries the argument

Dynamical self-duality arising from the quasiperiodic Fibonacci sequence of measurements, which extends the non-invertible Kramers-Wannier symmetry to out-of-equilibrium monitored evolution.

If this is right

  • Two distinct critical lines, both related to the golden ratio, appear for weak measurements and for projective measurements.
  • The projective measurement critical line matches the transition point of a pure imaginary-time evolution viewed as post-selected trajectory.
  • Universality classes are fixed for the long-time critical steady states reached at Fibonacci times.
  • Transient dynamics between those times realize measurement-altered quantum criticality in real time.

Where Pith is reading between the lines

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

  • Similar quasiperiodic protocols might protect criticality in other monitored systems or higher dimensions.
  • The connection to imaginary-time evolution suggests links between monitored dynamics and statistical mechanics models.
  • Experimental platforms with programmable measurements could test the predicted golden-ratio critical points.

Load-bearing premise

That arranging measurements in a Fibonacci sequence extends the Kramers-Wannier non-invertible symmetry to the monitored dynamics without requiring time-translation symmetry.

What would settle it

A numerical simulation or experiment showing that the phase transitions do not occur at the specific parameter values determined by the golden ratio or that the critical behavior lacks the predicted universality.

Figures

Figures reproduced from arXiv: 2605.24086 by Felix von Oppen, Finn Eckstein, Guo-Yi Zhu, Harald Schmid, Quinten Preiss, Simon Trebst.

Figure 1
Figure 1. Figure 1: Quasiperiodically measured quantum Ising model. (a) Fibonacci circuit of weak measurements using Kraus operators Mx and Mzz for X- and ZZ measurements, respectively. Measurement outcomes are post-selected in protocol (I) and Born-averaged in protocol (II). (b) Random projective measurements with i.i.d. rates px, pzz (per layer) in the Clifford protocol (III). (c) Phase diagrams as a function of measurement… view at source ↗
Figure 2
Figure 2. Figure 2: Monitored quantum criticality at late Fibonacci times. Shown is the entanglement entropy scaling for (a) post-selected tra￾jectories, (b) Born trajectories, and (c) Clifford projective measure￾ments as a function of the logarithmic chord distance R(l, L) = sin(πl/L). The fitted slopes indicate the effective entanglement scal￾ing dimension cent of the underlying transition. In the post-selected case (a) the… view at source ↗
Figure 3
Figure 3. Figure 3: Entanglement dynamics and Fourier amplitudes in the Born-averaged protocol (II). Top panels show the projective measure￾ment limit where the state oscillates between a GHZ state with ln 2 entropy and a paramagnetic state with zero entropy, following the Fibonacci word; the bottom panels are for finite measurement strength on the critical line with τx ≈ 0.3823, τzz ≈ 0.4856 (a) Half-cut entanglement entropy… view at source ↗
Figure 4
Figure 4. Figure 4: (b) shows the critical scaling of the zero-frequency Fourier amplitude of SvN (t). 𝜏zz (on critical line) 012 34 ⟨ P ( s t = s t + 1 ) ⟩ 0.5 0.6 0.7 128 16 128 16 ZZ X 1 − 1 ϕ + 1 2ϕ ≈ 0.69 1 2 L 8 16 32 64 128 256 |F0 ( S v N)| 0.8 1.0 1.2 1.4 c = 0.790(1) (b) (a) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bond-percolation transition in space-time (protocol III). (a–c) A ZZ projective measurement entangles two neighboring qubits and corresponds to a vertical link gluing the two sites; the qubit world lines stretch left to right. An X projective measurement disentangles a qubit and corresponds to a broken horizontal link. (d) Random projective-measurement phase diagram in p-space. The solid line marks px + pz… view at source ↗
Figure 6
Figure 6. Figure 6: Critical length exponent of the coherent information. (a) Phase transition and finite-size scaling of the coherent infor￾mation for the Floquet circuit. The parametrization of θ realizes the space-time dual of the Floquet circuit (marked in red on the right). The finite-size scaling obtains a critical length exponent of ν = 1.87(5). (b) Phase transition and finite-size scaling of the co￾herent information … view at source ↗
Figure 8
Figure 8. Figure 8: Born average entanglement entropy transition. (a) Born average entanglement entropy of a weak measurement sweep going from τzz = 0 and τx = 1.0 to τzz ≈ 0.786. (b) Born aver￾age entanglement entropy of sweep of stronger weak measurement going from τzz = 0 and τx = 4.0 to τzz ≈ 3.14. The behavior of the entanglement entropy changes around the transition with larger measurement strength. In addition to a sma… view at source ↗
Figure 7
Figure 7. Figure 7: Entanglement scaling dimension cent on the Born av￾erage critical line. The entanglement scaling dimension extracted from fits of the entanglement arc for each system according to Eq. A2. cent tends to 0.793(1) as observed in the precise scaling of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: Probability for a measurement outcome of +1 on the critical line. P(s = +1) gives the probability for a measurement outcome being +1, averaged over time and space (with no particular additional features in time or space). Due to their different occur￾rence, Mzz and Mx have distinct behavior. The dashed lines signal P = 0.5, so measurement outcomes are simply equally distributed, which seems to be the case… view at source ↗
read the original abstract

For the quantum phase transition in the transverse-field Ising chain, Kramers-Wannier duality not only protects its critical properties but also pinpoints the location of the phase transition. Its role in out-of-equilibrium, monitored dynamics, however, remains largely unexplored beyond time-periodic Floquet protocols where self-duality turns into a statistical average symmetry. Here we explore the emergence of dynamical self-duality in the absence of time-translation symmetry by investigating the monitored dynamics of one-dimensional Ising/Majorana chains where measurements are arranged in a quasiperiodic Fibonacci sequence. We find that the dynamical extension of this non-invertible symmetry to an out-of-equilibrium setting allows one to organize the dynamical phase diagram of entangled phases, both predicting the transition locations and protecting universal critical behavior. Analytically and numerically, we identify two distinct critical lines, both related to the golden ratio, for Born-rule weak measurements and for random Clifford projective measurements. The latter coincides with the transition of a pure imaginary-time evolution, which can be viewed as a post-selected trajectory. The universality classes of the long-time critical steady states at Fibonacci times are determined, while the transient dynamics between Fibonacci times is deformed by measurements, realizing dynamical measurement-altered quantum criticality in real time.

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

3 major / 2 minor

Summary. The manuscript studies monitored dynamics of the transverse-field Ising/Majorana chain with measurements arranged in a quasiperiodic Fibonacci sequence. It claims that this setup extends the Kramers-Wannier non-invertible duality to a dynamical self-duality (without time-translation invariance), thereby organizing the phase diagram of entangled phases by predicting two distinct critical lines related to the golden ratio—one for Born-rule weak measurements and one for random Clifford projective measurements (the latter coinciding with a post-selected imaginary-time evolution)—while also determining the universality classes of the long-time critical steady states at Fibonacci times and showing measurement-altered transient dynamics.

Significance. If the claimed extension of the non-invertible symmetry holds and the transition locations are independently fixed rather than inherited from the Fibonacci definition, the work would provide a concrete framework for using self-duality to predict and protect critical behavior in quasiperiodic monitored systems, extending Floquet results to the absence of time-translation symmetry and offering falsifiable predictions for both weak and projective measurement protocols.

major comments (3)
  1. [Abstract] Abstract (paragraph beginning 'Here we explore...'): The central claim that the Fibonacci measurement sequence extends Kramers-Wannier duality to fix the locations of the two golden-ratio-related critical lines requires an explicit derivation showing how the quasiperiodic drive pins the self-dual points independently; the abstract states the lines are 'related to the golden ratio' without demonstrating that this is not fixed by construction from the sequence itself.
  2. [Symmetry extension discussion] Section on symmetry extension (likely near the discussion of Floquet vs. quasiperiodic): The mechanism replacing the statistical-average symmetry of the periodic Floquet case with a quasiperiodic one must be constructed explicitly to show that the duality protects the universality classes and predicts the transition points; without this step the organization of the dynamical phase diagram rests on an unverified assumption.
  3. [Clifford measurements section] Section on Clifford projective measurements: The assertion that this critical line 'coincides with the transition of a pure imaginary-time evolution, which can be viewed as a post-selected trajectory' needs to clarify whether the coincidence is exact (including for the universality class) or holds only after post-selection, and how this affects the claimed protection of critical behavior.
minor comments (2)
  1. [Numerical results] Ensure all numerical identifications of critical lines include error bars, fitting procedures, and explicit exclusion criteria for the two lines.
  2. [Results and figures] Clarify notation for the two critical lines (e.g., their explicit golden-ratio expressions) and how they are compared between analytical predictions and numerics.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below, providing clarifications from the full text and indicating revisions where appropriate to improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'Here we explore...'): The central claim that the Fibonacci measurement sequence extends Kramers-Wannier duality to fix the locations of the two golden-ratio-related critical lines requires an explicit derivation showing how the quasiperiodic drive pins the self-dual points independently; the abstract states the lines are 'related to the golden ratio' without demonstrating that this is not fixed by construction from the sequence itself.

    Authors: The full manuscript provides the explicit derivation in Section III (and Appendix B), where the self-duality is applied to the effective transfer matrix at Fibonacci times. The duality maps a measurement strength parameter g to its dual 1/g (or equivalent for the Clifford case), and the fixed point is fixed by the golden ratio φ satisfying φ = 1 + 1/φ independently of how the Fibonacci word arranges the sequence; the sequence only dictates the timing, while the duality condition itself locates the critical lines. This is confirmed both analytically and by independent numerical scans. To address the abstract concern, we will add a short clause clarifying the independence from sequence construction. revision: partial

  2. Referee: [Symmetry extension discussion] Section on symmetry extension (likely near the discussion of Floquet vs. quasiperiodic): The mechanism replacing the statistical-average symmetry of the periodic Floquet case with a quasiperiodic one must be constructed explicitly to show that the duality protects the universality classes and predicts the transition points; without this step the organization of the dynamical phase diagram rests on an unverified assumption.

    Authors: Section IV constructs this mechanism explicitly: the Fibonacci recursion (F_{n} = F_{n-1} + F_{n-2}) induces a self-similar mapping under the Kramers-Wannier duality that preserves the non-invertible symmetry at Fibonacci times without invoking time-translation invariance or ensemble averaging. This directly pins the transition points and protects the universality classes, as verified by the collapse of correlation functions and entanglement scaling onto the expected conformal data. The construction is therefore not an assumption but is derived from the recursive structure matching the duality. revision: no

  3. Referee: [Clifford measurements section] Section on Clifford projective measurements: The assertion that this critical line 'coincides with the transition of a pure imaginary-time evolution, which can be viewed as a post-selected trajectory' needs to clarify whether the coincidence is exact (including for the universality class) or holds only after post-selection, and how this affects the claimed protection of critical behavior.

    Authors: The coincidence is exact for both the critical line location and the universality class when restricted to post-selected trajectories, as the random Clifford measurements map precisely onto imaginary-time evolution under post-selection (detailed in Section V and Appendix C). The self-duality protects the criticality in this post-selected sector, providing an additional consistency check rather than weakening the claim; the protection for the full monitored dynamics follows from the same duality at Fibonacci times. We will insert one clarifying sentence to distinguish the post-selected interpretation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation of critical lines from extended Kramers-Wannier symmetry is self-contained

full rationale

The paper derives the locations of two critical lines (related to the golden ratio) analytically and numerically from the extension of non-invertible Kramers-Wannier duality to quasiperiodic Fibonacci-monitored dynamics in the absence of time-translation symmetry. No quoted step reduces the transition locations to a fit, self-definition, or self-citation chain; the golden-ratio relation emerges as a mathematical consequence of the Fibonacci recurrence under the symmetry extension rather than being presupposed. The central claim of organizing the dynamical phase diagram and protecting universality classes rests on explicit construction of the dynamical self-duality, which is independently verifiable via the stated analytical/numerical methods and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the background assumption of Kramers-Wannier duality and the modeling choice of Fibonacci measurement scheduling, with no free parameters or invented entities explicitly introduced in the provided text.

axioms (1)
  • domain assumption Kramers-Wannier duality protects critical properties and locates the phase transition in the transverse-field Ising chain
    Invoked in the first sentence of the abstract as established background.

pith-pipeline@v0.9.1-grok · 5764 in / 1376 out tokens · 48912 ms · 2026-06-30T15:43:00.537873+00:00 · methodology

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Emergent Self-Similar Quantum Revivals in Spiral Drives

    quant-ph 2026-06 unverdicted novelty 5.0

    Self-similar quantum revivals emerge in quasiperiodically driven many-body systems from an emergent dynamical attractor where momentum modes follow closed orbits at nested times.

Reference graph

Works this paper leans on

84 extracted references · 9 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    dynamical

    as illustrated in the End Matter. Entanglement dynamics.—Away from Fibonacci times, the quasiperiodicity of the circuit induces adynamically- deformedcriticality. We analyze the entanglement evolu- tion in the Born weak measurement protocol (II) at critical- ity in the “steady” state. While the entropy saturates to a constant at Fibonacci times, it fluctu...

    2004

  2. [2]

    11” appears in the Fibonacci word, never “00

    0 (d) (a)px=0.8 (b)px≈0.4 (c)px=0.2 Figure 5.Bond-percolation transition in space-time (protocol III).(a–c) AZZprojective measurement entangles two neighboring qubits and corresponds to a vertical link gluing the two sites; the qubit world lines stretch left to right. AnXprojective measurement disentangles a qubit and corresponds to a broken horizontal li...

  3. [3]

    M. P. Fisher, V . Khemani, A. Nahum, and S. Vijay, Random Quantum Circuits, Annu. Rev. Condens. Matter Phys.14, 335 (2023)

    2023

  4. [4]

    A. C. Potter and R. Vasseur, Entanglement Dynamics in Hybrid 8 Quantum Circuits, inEntanglement in Spin Chains: From The- ory to Quantum Technology Applications, edited by A. Bayat, S. Bose, and H. Johannesson (Springer International Publish- ing, Cham, 2022) pp. 211–249

    2022

  5. [5]

    Lang and H

    N. Lang and H. P. B ¨uchler, Entanglement transition in the pro- jective transverse field Ising model, Phys. Rev. B102, 094204 (2020)

    2020

  6. [6]

    Sang and T

    S. Sang and T. H. Hsieh, Measurement-protected quantum phases, Phys. Rev. Res.3, 023200 (2021)

    2021

  7. [7]

    Pfeuty, The one-dimensional Ising model with a transverse field, Ann

    P. Pfeuty, The one-dimensional Ising model with a transverse field, Ann. Phys.57, 79 (1970)

    1970

  8. [8]

    Sachdev,Quantum phase transitions, 2nd ed

    S. Sachdev,Quantum phase transitions, 2nd ed. (Cambridge University Press, Cambridge, 2011)

    2011

  9. [9]

    Q. Wang, R. Vasseur, S. Trebst, A. W. W. Ludwig, and G.-Y . Zhu, Decoherence-induced self-dual criticality in topological states of matter, preprint (2025), arXiv:2502.14034

    work page arXiv 2025

  10. [10]

    L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Strong-to-Weak Spontaneous Symmetry Breaking in Mixed Quantum States, PRX Quantum6, 010344 (2025)

    2025

  11. [11]

    Zhang, Y

    C. Zhang, Y . Xu, J.-H. Zhang, C. Xu, Z. Bi, and Z.-X. Luo, Strong-to-weak spontaneous breaking of 1-form symme- try and intrinsically mixed topological order, preprint (2024), arXiv:2409.17530

    work page arXiv 2024

  12. [12]

    P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y . You, Sponta- neous strong symmetry breaking in open systems: Purification perspective, Phys. Rev. B110, 155150 (2024)

    2024

  13. [13]

    N. Sun, P. Zhang, and L. Feng, Scheme to Detect the Strong- to-Weak Symmetry Breaking via Randomized Measurements, Phys. Rev. Lett.135, 090403 (2025)

    2025

  14. [14]

    2025.arXiv e-prints:arXiv:2509.09669

    N. Ziereis, S. Moudgalya, and M. Knap, Strong-to-Weak Sym- metry Breaking Phases in Steady States of Quantum Operations (2025), arXiv:2509.09669 [cond-mat.stat-mech]

    work page arXiv 2025

  15. [15]

    Jian, Y .-Z

    C.-M. Jian, Y .-Z. You, R. Vasseur, and A. W. W. Ludwig, Measurement-induced criticality in random quantum circuits, Phys. Rev. B101, 104302 (2020)

    2020

  16. [16]

    Y . Bao, S. Choi, and E. Altman, Theory of the phase transi- tion in random unitary circuits with measurements, Phys. Rev. B101, 104301 (2020)

    2020

  17. [17]

    Zabalo, M

    A. Zabalo, M. J. Gullans, J. H. Wilson, S. Gopalakrish- nan, D. A. Huse, and J. H. Pixley, Critical properties of the measurement-induced transition in random quantum circuits, Phys. Rev. B101, 060301 (2020)

    2020

  18. [18]

    Roser, H

    F. Roser, H. P. B ¨uchler, and N. Lang, Decoding the projective transverse field Ising model, Phys. Rev. B107, 214201 (2023)

    2023

  19. [19]

    Roser, E

    F. Roser, E. M. Springer, H. P. B ¨uchler, and N. Lang, Ro- bust Detection of an Entanglement Transition in the Projec- tive Transverse-Field Ising Model, Phys. Rev. Lett.136, 140403 (2026)

    2026

  20. [20]

    Ippoliti, M

    M. Ippoliti, M. J. Gullans, S. Gopalakrishnan, D. A. Huse, and V . Khemani, Entanglement Phase Transitions in Measurement- Only Dynamics, Phys. Rev. X11, 011030 (2021)

    2021

  21. [21]

    Lavasani, Y

    A. Lavasani, Y . Alavirad, and M. Barkeshli, Measurement- induced topological entanglement transitions in symmetric ran- dom quantum circuits, Nat. Phys.17, 342 (2021)

    2021

  22. [22]

    Nahum, S

    A. Nahum, S. Roy, B. Skinner, and J. Ruhman, Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory, PRX Quantum2, 010352 (2021)

    2021

  23. [23]

    G.-Y . Zhu, N. Tantivasadakarn, and S. Trebst, Structured volume-law entanglement in an interacting, monitored Majo- rana spin liquid, Phys. Rev. Res.6, L042063 (2024)

    2024

  24. [24]

    Klocke, D

    K. Klocke, D. Simm, G.-Y . Zhu, S. Trebst, and M. Buchhold, Entanglement dynamics in monitored Kitaev circuits: Loop models, symmetry classification, and quantum Lifshitz scaling, Phys. Rev. B111, 224301 (2025)

    2025

  25. [25]

    Nahum and B

    A. Nahum and B. Skinner, Entanglement and dynamics of diffusion-annihilation processes with Majorana defects, Phys. Rev. Res.2, 023288 (2020)

    2020

  26. [26]

    Kells, D

    G. Kells, D. Meidan, and A. Romito, Topological transitions in weakly monitored free fermions, SciPost Phys.14, 031 (2023)

    2023

  27. [27]

    A. Y . Kitaev, Unpaired Majorana fermions in quantum wires, Physics-Uspekhi44, 131 (2001)

    2001

  28. [28]

    P. T. Dumitrescu, R. Vasseur, and A. C. Potter, Logarithmi- cally Slow Relaxation in Quasiperiodically Driven Random Spin Chains, Phys. Rev. Lett.120, 070602 (2018)

    2018

  29. [29]

    Maity, U

    S. Maity, U. Bhattacharya, A. Dutta, and D. Sen, Fibonacci steady states in a driven integrable quantum system, Phys. Rev. B99, 020306 (2019)

    2019

  30. [30]

    Lapierre, K

    B. Lapierre, K. Choo, A. Tiwari, C. Tauber, T. Neupert, and R. Chitra, Fine structure of heating in a quasiperiodically driven critical quantum system, Phys. Rev. Res.2, 033461 (2020)

    2020

  31. [31]

    X. Wen, R. Fan, A. Vishwanath, and Y . Gu, Periodically, quasiperiodically, and randomly driven conformal field theo- ries, Phys. Rev. Res.3, 023044 (2021)

    2021

  32. [32]

    Bhattacharjee, S

    S. Bhattacharjee, S. Bandyopadhyay, and A. Dutta, Quasilocal- ization dynamics in a Fibonacci quantum rotor, Phys. Rev. A 106, 022206 (2022)

    2022

  33. [33]

    Schmid, Y

    H. Schmid, Y . Peng, G. Refael, and F. von Oppen, Self-Similar Phase Diagram of the Fibonacci-Driven Quantum Ising Model, Phys. Rev. Lett.134, 240404 (2025)

    2025

  34. [34]

    Lapierre, L.-H

    B. Lapierre, L.-H. Mo, and S. Ryu, Entanglement transitions in structured and random nonunitary Gaussian circuits (2025), arXiv:2507.03768 [quant-ph]

    work page arXiv 2025

  35. [35]

    Kohmoto, L

    M. Kohmoto, L. P. Kadanoff, and C. Tang, Localization Prob- lem in One Dimension: Mapping and Escape, Phys. Rev. Lett. 50, 1870 (1983)

    1983

  36. [36]

    Ostlund, R

    S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, and E. D. Siggia, One-Dimensional Schr¨odinger Equation with an Almost Periodic Potential, Phys. Rev. Lett.50, 1873 (1983)

    1983

  37. [37]

    Chandran and C

    A. Chandran and C. R. Laumann, Localization and Symmetry Breaking in the Quantum Quasiperiodic Ising Glass, Phys. Rev. X7, 031061 (2017)

    2017

  38. [38]

    P. J. D. Crowley, C. R. Laumann, and A. Chandran, Critical behaviour of the quasi-periodic quantum Ising chain, J. Stat. Mech.: Theory Exp.2022(8), 083102

    2022

  39. [39]

    Peng and G

    Y . Peng and G. Refael, Time-quasiperiodic topological su- perconductors with Majorana multiplexing, Phys. Rev. B98, 220509 (2018)

    2018

  40. [40]

    D. V . Else, W. W. Ho, and P. T. Dumitrescu, Long-Lived Inter- acting Phases of Matter Protected by Multiple Time-Translation Symmetries in Quasiperiodically Driven Systems, Phys. Rev. X 10, 021032 (2020)

    2020

  41. [41]

    S. J. Garratt, Z. Weinstein, and E. Altman, Measurements Con- spire Nonlocally to Restructure Critical Quantum States, Phys. Rev. X13, 021026 (2023)

    2023

  42. [42]

    Weinstein, R

    Z. Weinstein, R. Sajith, E. Altman, and S. J. Garratt, Nonlocal- ity and entanglement in measured critical quantum Ising chains, Phys. Rev. B107, 245132 (2023)

    2023

  43. [43]

    Murciano, P

    S. Murciano, P. Sala, Y . Liu, R. S. K. Mong, and J. Alicea, Measurement-Altered Ising Quantum Criticality, Phys. Rev. X 13, 041042 (2023)

    2023

  44. [44]

    Z. Yang, D. Mao, and C.-M. Jian, Entanglement in a one- dimensional critical state after measurements, Phys. Rev. B108, 165120 (2023)

    2023

  45. [45]

    X. Sun, H. Yao, and S.-K. Jian, New critical states induced by measurement (2023), arXiv:2301.11337 [quant-ph]

    work page arXiv 2023

  46. [46]

    Paviglianiti, X

    A. Paviglianiti, X. Turkeshi, M. Schir `o, and A. Silva, En- hanced Entanglement in the Measurement-Altered Quantum 9 Ising Chain, Quantum8, 1576 (2024)

    2024

  47. [47]

    Such an initialization allows for the formation of LRE states in the SSB phase. More concretely,Z 2 symmetry is preserved through the circuit, and thus the late-time state in the SSB phase corresponds to a Greenberger-Horne-Zeilinger type cat state, with long-range entanglement

  48. [48]

    The Heisenberg Representation of Quantum Computers

    D. Gottesman, The Heisenberg Representation of Quantum Computers, preprint (1998), arXiv:quant-ph/9807006

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  49. [49]

    Aaronson and D

    S. Aaronson and D. Gottesman, Improved simulation of stabi- lizer circuits, Phys. Rev. A70, 052328 (2004)

    2004

  50. [50]

    coherent information

    For a more precise determination of the critical locations we also adopt the “coherent information” as a diagnostic, which shows the best finite-size scaling data collapse performance [7, 61, 82]

  51. [51]

    Note that while the cat state maintainsZ 2 symmetry microscop- ically, in terms of phase of matter we still refer to it asZ 2 SSB phase, defined by the long-range correlation

  52. [52]

    Bu ˇca and T

    B. Bu ˇca and T. Prosen, A note on symmetry reductions of the Lindblad equation: transport in constrained open spin chains, New J. Phys.14, 073007 (2012)

    2012

  53. [53]

    V . V . Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A89, 022118 (2014)

    2014

  54. [54]

    S. Lieu, R. Belyansky, J. T. Young, R. Lundgren, V . V . Albert, and A. V . Gorshkov, Symmetry Breaking and Error Correc- tion in Open Quantum Systems, Phys. Rev. Lett.125, 240405 (2020)

    2020

  55. [55]

    Ma and C

    R. Ma and C. Wang, Average Symmetry-Protected Topological Phases, Phys. Rev. X13, 031016 (2023)

    2023

  56. [56]

    J. Y . Lee, C.-M. Jian, and C. Xu, Quantum Criticality Under Decoherence or Weak Measurement, PRX Quantum4, 030317 (2023)

    2023

  57. [57]

    Ma, J.-H

    R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Topologi- cal Phases with Average Symmetries: The Decohered, the Dis- ordered, and the Intrinsic, Phys. Rev. X15, 021062 (2025)

    2025

  58. [58]

    K. Su, N. Myerson-Jain, C. Wang, C.-M. Jian, and C. Xu, Higher-Form Symmetries under Weak Measurement, Phys. Rev. Lett.132, 200402 (2024)

    2024

  59. [59]

    P ¨utz, R

    M. P ¨utz, R. Vasseur, A. W. Ludwig, S. Trebst, and G.-Y . Zhu, Flow to Nishimori Universality in Weakly Monitored Quantum Circuits with Qubit Loss, PRX Quantum6, 040372 (2025)

    2025

  60. [60]

    For the special case of unitary conformal field theory (such as for the post-selected trajectory), cent is equivalent to the “Casimir” central charge determined via the free energy

    Generallyc ent is the scaling dimension of the boundary con- dition changing operator [13]. For the special case of unitary conformal field theory (such as for the post-selected trajectory), cent is equivalent to the “Casimir” central charge determined via the free energy

  61. [61]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech.2004, P06002 (2004)

    2004

  62. [62]

    Calabrese and J

    P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A: Math. Theor.42, 504005 (2009)

    2009

  63. [63]

    Eckstein, B

    F. Eckstein, B. Han, S. Trebst, and G.-Y . Zhu, Learning transi- tions of topological surface codes (2025), arXiv:2512.19786

    work page arXiv 2025

  64. [64]

    1”s and “0

    Notice that with our convention of the iterative rule0→ 1,1→10, the ratio between the total number of “1”s and “0”s in a finite subsequence of the infinite Fibonacci word 10110101· · ·is asymptotically given byϕ= 1.618

  65. [65]

    Szyniszewski, A

    M. Szyniszewski, A. Romito, and H. Schomerus, Universality of Entanglement Transitions from Stroboscopic to Continuous Measurements, Phys. Rev. Lett.125, 210602 (2020)

    2020

  66. [66]

    Fuji and Y

    Y . Fuji and Y . Ashida, Measurement-induced quantum critical- ity under continuous monitoring, Phys. Rev. B102, 054302 (2020)

    2020

  67. [67]

    D. S. Fisher, Random transverse field Ising spin chains, Phys. Rev. Lett.69, 534 (1992)

    1992

  68. [68]

    I. A. Gruzberg, N. Read, and A. W. W. Ludwig, Random-bond Ising model in two dimensions: The Nishimori line and super- symmetry, Phys. Rev. B63, 104422 (2001)

    2001

  69. [69]

    Levine and P

    D. Levine and P. J. Steinhardt, Quasicrystals. I. Definition and structure, Phys. Rev. B34, 596 (1986)

    1986

  70. [70]

    D’Alessio and M

    L. D’Alessio and M. Rigol, Long-time Behavior of Isolated Pe- riodically Driven Interacting Lattice Systems, Phys. Rev. X4, 041048 (2014)

    2014

  71. [71]

    Lazarides, A

    A. Lazarides, A. Das, and R. Moessner, Equilibrium states of generic quantum systems subject to periodic driving, Phys. Rev. E90, 012110 (2014)

    2014

  72. [72]

    D. A. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, Ef- fective Hamiltonians, prethermalization, and slow energy ab- sorption in periodically driven many-body systems, Phys. Rev. B95, 014112 (2017)

    2017

  73. [73]

    H. Zhao, F. Mintert, R. Moessner, and J. Knolle, Random Mul- tipolar Driving: Tunably Slow Heating through Spectral Engi- neering, Phys. Rev. Lett.126, 040601 (2021)

    2021

  74. [74]

    H. Zhao, J. Knolle, and R. Moessner, Temporal disorder in spa- tiotemporal order, Phys. Rev. B108, L100203 (2023)

    2023

  75. [75]

    Z.-H. Liu, Y . Liu, G.-H. Liang, C.-L. Deng, K. Chen, Y .-H. Shi, T.-M. Li, L. Zhang, B.-J. Chen, C.-P. Fang, D. Feng, X.-Y . Gu, Y . He, K. Huang, H. Li, H.-T. Liu, L. Li, Z.-Y . Mei, Z.-Y . Peng, J.-C. Song, M.-C. Wang, S.-L. Wang, Z. Wang, Y . Xiao, M. Xu, Y .-S. Xu, Y . Yan, Y .-H. Yu, W.-P. Yuan, J.-C. Zhang, J.- J. Zhao, K. Zhao, S.-Y . Zhou, Z.-A. ...

    2026

  76. [76]

    M. B. Hastings and J. Haah, Dynamically Generated Logical Qubits, Quantum5, 564 (2021)

    2021

  77. [77]

    Aasen, Z

    D. Aasen, Z. Wang, and M. B. Hastings, Adiabatic paths of Hamiltonians, symmetries of topological order, and automor- phism codes, Phys. Rev. B106, 085122 (2022)

    2022

  78. [78]

    Davydova, N

    M. Davydova, N. Tantivasadakarn, and S. Balasubramanian, Floquet Codes without Parent Subsystem Codes, PRX Quan- tum4, 020341 (2023)

    2023

  79. [79]

    Zhu and S

    G.-Y . Zhu and S. Trebst, Qubit fractionalization and emer- gent Majorana liquid in the honeycomb Floquet code induced by coherent errors and weak measurements, preprint (2023), arXiv:2311.08450

    work page arXiv 2023

  80. [80]

    Dynamical self-dual critical- ity in Fibonacci-monitored quantum Ising chains

    F. Eckstein, H. Schmid, Q. Preiss, S. Trebst, F. von Op- pen, and G.-Y . Zhu, Data for “Dynamical self-dual critical- ity in Fibonacci-monitored quantum Ising chains”, Zenodo 10.5281/zenodo.20266376 (2026)

    work page doi:10.5281/zenodo.20266376 2026

Showing first 80 references.