pith. sign in

arxiv: 2606.17251 · v1 · pith:UPPS5BMQnew · submitted 2026-06-15 · 🪐 quant-ph · cond-mat.quant-gas

Post-Selection Probability and Fidelity of Bidirectional Teleportation

Ning Sun , Lei Feng , Pengfei Zhang This is my paper

Pith reviewed 2026-06-27 03:17 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gas
keywords bidirectional teleportationpost-selection probabilityfidelityLoschmidt echointegrable modelsquantum scramblingquantum information processing
0
0 comments X

The pith

The post-selection probability and fidelity of bidirectional teleportation are expressed using Loschmidt echoes, revealing initial-state dependence and stability in integrable models.

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

The paper shows that the post-selection probability and the fidelity characterizing a bidirectional teleportation protocol can be rewritten using the Loschmidt echo and its subsystem variant from quantum dynamics. This holds even when errors occur in the time-reversed part of the protocol. A reader would care because the rewriting explains why fidelity changes with the initial state and why the probability remains steady in integrable systems, offering guidance for real-device implementations.

Core claim

The central claim is that the post-selection probability and fidelity of the bidirectional teleportation protocol can be expressed in terms of the Loschmidt echo and its subsystem variant. This expression unveils the initial-state dependence of the fidelity and the stability of the post-selection probability in integrable models, remaining valid with errors in time-reversed dynamics.

What carries the argument

The Loschmidt echo and its subsystem variant, which serve as the bridge connecting the teleportation protocol quantities to standard quantum dynamics diagnostics.

If this is right

  • The fidelity of the teleportation depends on the choice of initial state.
  • The post-selection probability remains stable when the underlying model is integrable.
  • These quantities can be computed from Loschmidt echoes even in the presence of errors.
  • Practical guidance emerges for implementing the protocol on quantum hardware.

Where Pith is reading between the lines

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

  • Measuring Loschmidt echoes could provide a way to estimate teleportation performance without running the full protocol.
  • The stability in integrable models might extend to other post-selection based protocols in quantum information.
  • Testing in chaotic versus integrable regimes could distinguish the behaviors predicted here.

Load-bearing premise

The bidirectional teleportation protocol is implemented via chaotic Hamiltonian evolution combined with measurement and post-selection, and the claimed expressions remain valid when errors are present in the time-reversed dynamics.

What would settle it

Compute or measure the post-selection probability in both an integrable and a chaotic model under the protocol; if the probability changes substantially between them, the stability claim is false.

Figures

Figures reproduced from arXiv: 2606.17251 by Lei Feng, Ning Sun, Pengfei Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. The schematic illustration of the bidirectional tele [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The diagrammatic proof of the main relations: (1) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. We present the numerical results for the quantum [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. We present the numerical results for the quantum [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Understanding the scrambling of quantum information is central to many areas of quantum physics, including quantum thermalization, entanglement growth, and quantum information processing. Insights from these studies have, in turn, inspired the development of novel quantum protocols and algorithms. Recently, a bidirectional teleportation protocol was proposed to implement a digital SWAP operation between qubits by leveraging chaotic Hamiltonian evolution combined with measurement and post-selection. In this work, we provide a comprehensive study of two central quantities that characterize the protocol, the post-selection probability and the fidelity, taking into account possible errors in time-reversed dynamics. We show that these quantities can be expressed in terms of standard diagnostics in quantum dynamics, including the Loschmidt echo and its subsystem variant. The results unveil (1) the initial-state dependence of the fidelity and (2) the stability of the post-selection probability in integrable models. Our findings offer practical guidance for the implementation of the protocol on realistic quantum devices.

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

0 major / 2 minor

Summary. The manuscript analyzes the post-selection probability and fidelity of a bidirectional teleportation protocol that employs chaotic Hamiltonian evolution together with measurement and post-selection. It derives exact expressions for these quantities in terms of the Loschmidt echo and its subsystem variant, including the case of errors in the time-reversed dynamics, and uses the expressions to identify the initial-state dependence of the fidelity and the stability of the post-selection probability in integrable models.

Significance. If the claimed mappings hold, the work supplies a direct link between a quantum-information protocol and standard diagnostics of quantum dynamics, thereby offering concrete guidance for experimental realization on noisy devices and clarifying state dependence and integrability effects.

minor comments (2)
  1. [§3] §3 (or wherever the error model is introduced): the precise form of the error operators in the time-reversed dynamics should be stated explicitly so that the reader can verify that the Loschmidt-echo mapping remains exact rather than approximate.
  2. The discussion of integrable models would benefit from a short statement of the specific Hamiltonians or conserved quantities used to demonstrate stability of the post-selection probability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its contributions, and recommendation of minor revision. No specific major comments or points requiring clarification were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper maps the post-selection probability and fidelity of the bidirectional teleportation protocol to the Loschmidt echo and its subsystem variant via explicit derivations from the protocol definition (chaotic Hamiltonian evolution plus measurement/post-selection). These are standard, independently defined diagnostics in quantum dynamics, not constructed from the protocol outputs. The initial-state dependence and stability results follow directly from applying the mappings to different Hamiltonians, without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. The full derivations are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5687 in / 988 out tokens · 41517 ms · 2026-06-27T03:17:38.292350+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

67 extracted references · 19 linked inside Pith

  1. [1]

    Srednicki, Chaos and quantum thermalization, Phys

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

    1994

  2. [2]

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

    2046

  3. [3]

    M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum Computation as Geometry, Science311, 1133 (2006), arXiv:quant-ph/0603161

    Pith/arXiv arXiv 2006

  4. [4]

    Susskind, Computational Complexity and Black Hole Horizons, Fortsch

    L. Susskind, Computational Complexity and Black Hole Horizons, Fortsch. Phys.64, 24 (2016), [Addendum: Fortsch.Phys. 64, 44–48 (2016)], arXiv:1403.5695 [hep- th]

    Pith/arXiv arXiv 2016

  5. [5]

    A. R. Brown and L. Susskind, Second law of quan- tum complexity, Phys. Rev. D97, 086015 (2018), arXiv:1701.01107 [hep-th]

    Pith/arXiv arXiv 2018

  6. [6]

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

    2019

  7. [7]

    Hayden and J

    P. Hayden and J. Preskill, Black holes as mirrors: Quan- tum information in random subsystems, JHEP09, 120, arXiv:0708.4025 [hep-th]

    Pith/arXiv arXiv

  8. [8]

    Sekino and L

    Y. Sekino and L. Susskind, Fast Scramblers, JHEP10, 065, arXiv:0808.2096 [hep-th]

    Pith/arXiv arXiv 2096

  9. [9]

    S. H. Shenker and D. Stanford, Stringy effects in scram- bling, JHEP05, 132, arXiv:1412.6087 [hep-th]

    Pith/arXiv arXiv

  10. [10]

    D. A. Roberts, D. Stanford, and L. Susskind, Localized shocks, JHEP03, 051, arXiv:1409.8180 [hep-th]

    Pith/arXiv arXiv

  11. [11]

    Hosur, X.-L

    P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Chaos in quantum channels, JHEP02, 004, arXiv:1511.04021 [hep-th]

    Pith/arXiv arXiv

  12. [12]

    R. Fan, P. Zhang, H. Shen, and H. Zhai, Out-of- time-order correlation for many-body localization, Sci- ence Bulletin62, 707 (2017), arXiv:1608.01914 [cond- mat.quant-gas]

    Pith/arXiv arXiv 2017

  13. [13]

    Huang, Y.-L

    Y. Huang, Y.-L. Zhang, and X. Chen, Out-of-time- ordered correlators in many-body localized systems, An- nalen der Physik529, 1600318 (2017), arXiv:1608.01091 [cond-mat.dis-nn]

    arXiv 2017

  14. [14]

    Swingle and D

    B. Swingle and D. Chowdhury, Slow scrambling in disor- dered quantum systems, Phys. Rev. B95, 060201 (2017), arXiv:1608.03280 [cond-mat.str-el]

    Pith/arXiv arXiv 2017

  15. [15]

    He and Z.-Y

    R.-Q. He and Z.-Y. Lu, Characterizing many-body local- ization by out-of-time-ordered correlation, Phys. Rev. B 95, 054201 (2017)

    2017

  16. [16]

    Chen, Universal Logarithmic Scrambling in Many Body Localization, arXiv e-prints , arXiv:1608.02765 (2016), arXiv:1608.02765 [cond-mat.dis-nn]

    Y. Chen, Universal Logarithmic Scrambling in Many Body Localization, arXiv e-prints , arXiv:1608.02765 (2016), arXiv:1608.02765 [cond-mat.dis-nn]

    Pith/arXiv arXiv 2016

  17. [17]

    Slagle, Z

    K. Slagle, Z. Bi, Y.-Z. You, and C. Xu, Out-of-time- order correlation in marginal many-body localized sys- tems, Phys. Rev. B95, 165136 (2017)

    2017

  18. [18]

    J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai, X. Peng, and J. Du, Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator, Phys. Rev. X7, 031011 (2017), arXiv:1609.01246 [cond- mat.str-el]

    Pith/arXiv arXiv 2017

  19. [19]

    K. X. Wei, P. Peng, O. Shtanko, I. Marvian, S. Lloyd, C. Ramanathan, and P. Cappellaro, Emergent prether- malization signatures in out-of-time ordered correlations, Phys. Rev. Lett.123, 090605 (2019)

    2019

  20. [20]

    Nie, B.-B

    X. Nie, B.-B. Wei, X. Chen, Z. Zhang, X. Zhao, C. Qiu, Y. Tian, Y. Ji, T. Xin, D. Lu, and J. Li, Experimen- tal Observation of Equilibrium and Dynamical Quantum Phase Transitions via Out-of-Time-Ordered Correlators, Phys. Rev. Lett.124, 250601 (2020)

    2020

  21. [21]

    C. M. S´ anchez, A. K. Chattah, K. X. Wei, L. Buljuba- sich, P. Cappellaro, and H. M. Pastawski, Perturbation Independent Decay of the Loschmidt Echo in a Many- Body System, Phys. Rev. Lett.124, 030601 (2020), arXiv:1902.06628 [quant-ph]

    arXiv 2020

  22. [22]

    F. D. Dom´ ınguez, M. C. Rodr´ ıguez, R. Kaiser, D. Suter, and G. A. ´Alvarez, Decoherence scaling transition in the dynamics of quantum information scrambling, Phys. Rev. A104, 012402 (2021), arXiv:2005.12361 [quant-ph]

    arXiv 2021

  23. [23]

    C. M. S´ anchez, A. K. Chattah, and H. M. Pastawski, Emergent decoherence induced by quantum chaos in a many-body system: A Loschmidt echo observation through NMR, Phys. Rev. A105, 052232 (2022), arXiv:2112.00607 [quant-ph]

    arXiv 2022

  24. [24]

    Li, T.-G

    Y. Li, T.-G. Zhou, Z. Wu, P. Peng, S. Zhang, R. Fu, R. Zhang, W. Zheng, P. Zhang, H. Zhai, X. Peng, and J. Du, Emergent universal quench dynamics in randomly interacting spin models, Nat. Phys.20, 1966 (2024)

    1966

  25. [25]

    G¨ arttner, J

    M. G¨ arttner, J. G. Bohnet, A. Safavi-Naini, M. L. Wall, J. J. Bollinger, and A. M. Rey, Measuring out-of-time- order correlations and multiple quantum spectra in a trapped ion quantum magnet, Nature Phys.13, 781 (2017), arXiv:1608.08938 [quant-ph]

    Pith/arXiv arXiv 2017

  26. [27]

    R. J. Lewis-Swan, A. Safavi-Naini, J. J. Bollinger, and A. M. Rey, Unifying scrambling, thermalization and en- tanglement through measurement of fidelity out-of-time- order correlators in the Dicke model, Nat. Commun.10, 1581 (2019)

    2019

  27. [28]

    A. M. Green, A. Elben, C. H. Alderete, L. K. Joshi, N. H. Nguyen, T. V. Zache, Y. Zhu, B. Sundar, and N. M. Linke, Experimental Measurement of Out-of-Time- Ordered Correlators at Finite Temperature, Phys. Rev. Lett.128, 140601 (2022)

    2022

  28. [29]

    X. Mi, P. Roushan, C. Quintana,et al., Informa- tion scrambling in quantum circuits, Science374, 1479 (2021)

    2021

  29. [30]

    Wang, T.-Q

    J.-H. Wang, T.-Q. Cai, X.-Y. Han, Y.-W. Ma, Z.-L. Wang, Z.-H. Bao, Y. Li, H.-Y. Wang, H.-Y. Zhang, L.-Y. Sun, Y.-K. Wu, Y.-P. Song, and L.-M. Duan, Informa- tion scrambling dynamics in a fully controllable quantum simulator, Phys. Rev. Research4, 043141 (2022)

    2022

  30. [31]

    Braum¨ uller, A

    J. Braum¨ uller, A. H. Karamlou, Y. Yanay, B. Kannan, D. Kim, M. Kjaergaard, A. Melville, B. M. Niedzielski, Y. Sung, A. Veps¨ al¨ ainen, R. Winik, J. L. Yoder, T. P. Orlando, S. Gustavsson, C. Tahan, and W. D. Oliver, Probing quantum information propagation with out-of- time-ordered correlators, Nat. Phys.18, 172 (2022)

    2022

  31. [32]

    S. K. Zhao, Z.-Y. Ge, Z. Xiang, G. M. Xue, H. S. Yan, Z. T. Wang, Z. Wang, H. K. Xu, F. F. Su, Z. H. Yang, H. Zhang, Y.-R. Zhang, X.-Y. Guo, K. Xu, Y. Tian, H. F. Yu, D. N. Zheng, H. Fan, and S. P. Zhao, Probing Op- erator Spreading via Floquet Engineering in a Supercon- ducting Circuit, Phys. Rev. Lett.129, 160602 (2022)

    2022

  32. [33]

    Google Quantum AI, Constructive interference at the edge of quantum ergodic dynamics (2025), arXiv:2506.10191

    arXiv 2025

  33. [34]

    Colombo, E

    S. Colombo, E. Pedrozo-Pe˜ nafiel, A. F. Adiyatullin, Z. Li, E. Mendez, C. Shu, and V. Vuleti´ c, Time-reversal-based quantum metrology with many-body entangled states, Nat. Phys.18, 925 (2022)

    2022

  34. [35]

    Z. Li, S. Colombo, C. Shu, G. Velez, S. Pilatowsky- Cameo, R. Schmied, S. Choi, M. Lukin, E. Pedrozo- Pe˜ nafiel, and V. Vuleti´ c, Improving metrology with quan- tum scrambling, Science380, 1381 (2023)

    2023

  35. [36]

    Pegahan, I

    S. Pegahan, I. Arakelyan, and J. E. Thomas, Energy- Resolved Information Scrambling in Energy-Space Lat- tices, Phys. Rev. Lett.126, 070601 (2021)

    2021

  36. [37]

    Xiang, Y.-W

    D.-S. Xiang, Y.-W. Zhang, H.-X. Liu, P. Zhou, D. Yuan, K. Zhang, S.-Y. Zhang, B. Xu, L. Liu, Y. Li, and L. Li, Observation of quantum information collapse-and-revival in a strongly-interacting Rydberg atom array (2024), arXiv:2410.15455

    arXiv 2024

  37. [38]

    Liang, Z

    X. Liang, Z. Yue, Y.-X. Chao, Z.-X. Hua, Y. Lin, M. K. Tey, and L. You, Observation of anomalous in- formation scrambling in a Rydberg atom array (2024), arXiv:2410.16174

    arXiv 2024

  38. [39]

    Liet al., Error-resilient Reversal of Quan- tum Chaotic Dynamics Enabled by Scramblons (2025), arXiv:2506.19915 [cond-mat.str-el]

    Y.-C. Liet al., Error-resilient Reversal of Quan- tum Chaotic Dynamics Enabled by Scramblons (2025), arXiv:2506.19915 [cond-mat.str-el]

    arXiv 2025

  39. [40]

    Yoshida and A

    B. Yoshida and A. Kitaev, Efficient decoding for the Hayden-Preskill protocol (2017), arXiv:1710.03363 [hep- th]

    Pith/arXiv arXiv 2017

  40. [41]

    Yoshida and N

    B. Yoshida and N. Y. Yao, Disentangling scrambling and decoherence via quantum teleportation, Phys. Rev. X9, 011006 (2019)

    2019

  41. [42]

    K. A. Landsman, C. Figgatt, T. Schuster, N. M. Linke, B. Yoshida, N. Y. Yao, and C. Monroe, Verified Quan- tum Information Scrambling, Nature567, 61 (2019), arXiv:1806.02807 [quant-ph]

    Pith/arXiv arXiv 2019

  42. [43]

    Cheng, C

    Y. Cheng, C. Liu, J. Guo, Y. Chen, P. Zhang, and H. Zhai, Realizing the hayden-preskill protocol with cou- pled dicke models, Phys. Rev. Res.2, 043024 (2020)

    2020

  43. [44]

    Gao and D

    P. Gao and D. L. Jafferis, A traversable wormhole tele- portation protocol in the SYK model, JHEP07, 097, arXiv:1911.07416 [hep-th]

    arXiv 1911

  44. [45]

    A. R. Brown, H. Gharibyan, S. Leichenauer, H. W. Lin, S. Nezami, G. Salton, L. Susskind, B. Swingle, and M. Walter, Quantum Gravity in the Lab. I. Teleporta- tion by Size and Traversable Wormholes, PRX Quantum 4, 010320 (2023), arXiv:1911.06314 [quant-ph]

    arXiv 2023

  45. [46]

    Nezami, H

    S. Nezami, H. W. Lin, A. R. Brown, H. Gharibyan, S. Leichenauer, G. Salton, L. Susskind, B. Swingle, and M. Walter, Quantum Gravity in the Lab. II. Teleporta- tion by Size and Traversable Wormholes, PRX Quantum 4, 010321 (2023), arXiv:2102.01064 [quant-ph]

    arXiv 2023

  46. [47]

    Schuster, B

    T. Schuster, B. Kobrin, P. Gao, I. Cong, E. T. Khabi- boulline, N. M. Linke, M. D. Lukin, C. Monroe, B. Yoshida, and N. Y. Yao, Many-body quantum telepor- tation via operator spreading in the traversable wormhole protocol, Phys. Rev. X12, 031013 (2022)

    2022

  47. [48]

    Jafferis, A

    D. Jafferis, A. Zlokapa, J. D. Lykken, D. K. Kolch- meyer, S. I. Davis, N. Lauk, H. Neven, and M. Spiropulu, Traversable wormhole dynamics on a quantum processor, Nature612, 51 (2022)

    2022

  48. [49]

    S. Zhou, P. Zhang, and Z. Yu, Environment-induced tran- sitions in many-body quantum teleportation, Phys. Rev. Res.8, L012007 (2026), arXiv:2406.02277 [quant-ph]

    arXiv 2026

  49. [50]

    Liu and P

    Z. Liu and P. Zhang, Fidelity of wormhole teleportation in finite-qubit systems, JHEP07, 031, arXiv:2403.16793 [quant-ph]

    arXiv

  50. [51]

    P. Gao, D. L. Jafferis, and A. C. Wall, Traversable Worm- holes via a Double Trace Deformation, JHEP12, 151, arXiv:1608.05687 [hep-th]

    arXiv

  51. [52]

    Maldacena, D

    J. Maldacena, D. Stanford, and Z. Yang, Diving into traversable wormholes, Fortsch. Phys.65, 1700034 (2017), arXiv:1704.05333 [hep-th]

    Pith/arXiv arXiv 2017

  52. [53]

    Maldacena and X.-L

    J. Maldacena and X.-L. Qi, Eternal traversable wormhole (2018), arXiv:1804.00491 [hep-th]

    Pith/arXiv arXiv 2018

  53. [54]

    Vikram, E

    A. Vikram, E. Chaparro, M. M. Khan, A. Lucas, C. Akers, and A. M. Rey, Bidirectional teleportation us- ing scrambling dynamics: a practical protocol (2026), arXiv:2601.15536 [quant-ph]

    arXiv 2026

  54. [55]

    Liu and P

    Z. Liu and P. Zhang, Distinguishing Coherent and Inco- herent Errors in Multi-Round Time-Reversed Dynamics via Scramblons (2026), arXiv:2601.04856 [quant-ph]

    arXiv 2026

  55. [56]

    Peres, Stability of quantum motion in chaotic and regular systems, Phys

    A. Peres, Stability of quantum motion in chaotic and regular systems, Phys. Rev. A30, 1610 (1984)

    1984

  56. [57]

    Goussev, R

    A. Goussev, R. A. Jalabert, H. M. Pastawski, and D. Wisniacki, Loschmidt echo, Scholarpedia7, 11687 (2012), arXiv:1206.6348 [nlin.CD]

    Pith/arXiv arXiv 2012

  57. [58]

    Chen, X.-L

    Y. Chen, X.-L. Qi, and P. Zhang, Replica wormhole and information retrieval in the SYK model coupled to Ma- jorana chains, JHEP06, 121, arXiv:2003.13147 [hep-th]

    arXiv 2003

  58. [59]

    Karchet al., Probing quantum many-body dy- namics using subsystem Loschmidt echos (2025), arXiv:2501.16995 [cond-mat.quant-gas]

    S. Karchet al., Probing quantum many-body dy- namics using subsystem Loschmidt echos (2025), arXiv:2501.16995 [cond-mat.quant-gas]

    arXiv 2025

  59. [60]

    D. A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP04, 121, arXiv:1610.04903 [quant-ph]

    Pith/arXiv arXiv

  60. [61]

    M. A. Nielsen and I. L. Chuang, Quantum information and quantum computation (2000)

    2000

  61. [62]

    Bertini, P

    B. Bertini, P. Kos, and T. Prosen, Operator Entangle- 7 ment in Local Quantum Circuits I: Chaotic Dual-Unitary Circuits, SciPost Phys.8, 067 (2020), arXiv:1909.07407 [cond-mat.stat-mech]

    arXiv 2020

  62. [63]

    Bertini, P

    B. Bertini, P. Kos, and T. Prosen, Operator Entangle- ment in Local Quantum Circuits II: Solitons in Chains of Qubits, SciPost Phys.8, 068 (2020), arXiv:1909.07410 [cond-mat.stat-mech]

    arXiv 2020

  63. [64]

    Bertini, P

    B. Bertini, P. Kos, and T. c. v. Prosen, Exact correla- tion functions for dual-unitary lattice models in 1 + 1 dimensions, Phys. Rev. Lett.123, 210601 (2019)

    2019

  64. [65]

    A. A. Akhtar, N. Anand, J. Marshall, and Y.-Z. You, Dual-unitary shadow tomography, Quantum9, 1816 (2025), arXiv:2404.01068 [quant-ph]

    arXiv 2025

  65. [66]

    M. Song, Z. Zeng, T.-T. Wang, Y.-Z. You, Z. Y. Meng, and P. Zhang, Monte Carlo Simulation of Operator Dynamics and Entanglement in Dual-Unitary Circuits, Quantum9, 1681 (2025), arXiv:2410.00953 [quant-ph]

    arXiv 2025

  66. [67]

    Y. Gu, A. Kitaev, and P. Zhang, A two-way ap- proach to out-of-time-order correlators, JHEP03, 133, arXiv:2111.12007 [hep-th]

    arXiv

  67. [68]

    Stanford, Z

    D. Stanford, Z. Yang, and S. Yao, Subleading Wein- gartens, JHEP02, 200, arXiv:2107.10252 [hep-th]

    arXiv