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arxiv: 2605.07668 · v1 · submitted 2026-05-08 · 🪐 quant-ph

Recognition: 2 theorem links

· Lean Theorem

Bridging Krylov Complexity and Universal Analog Quantum Simulator

Shuo Zhang , Yuzhi Tong , Pengfei Zhang , Zeyu Liu
Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:31 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Krylov complexityanalog quantum simulationquantum controlRydberg atom arraysoperator growthHamiltonian synthesis
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The pith

Generalized Krylov complexity predicts the minimum time needed to realize target operations in analog quantum simulators.

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

The paper proposes generalized Krylov complexity as a direct measure of how hard it is to synthesize a given quantum operation using the global controls available in an analog simulator. It builds this measure by generating a block Krylov basis from the simulator's Hamiltonians through successive commutators, which spans the reachable operators. Analysis of systems such as Rydberg atom arrays shows that the complexity value correlates strongly with the shortest time required to achieve the operation. A reader would care because the measure offers a way to evaluate and improve control protocols without exhaustive numerical searches over all possible field sequences.

Core claim

By constructing the block Krylov basis generated by a set of Hamiltonians, which naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators, the generalized Krylov complexity of a target operation serves as a strong predictor of the minimum time required for its realization. This is demonstrated through analysis of representative systems including Rydberg atom arrays.

What carries the argument

the block Krylov basis generated by the available Hamiltonians, which spans the reachable operators through nested commutators and locates the target operation within that basis

If this is right

  • The shortest control duration for any chosen target can be read off from its depth in the Krylov basis without full dynamical simulation.
  • Hamiltonian sets can be compared or selected according to the Krylov complexities they induce for a library of useful operations.
  • Control design reduces to finding short paths in the precomputed operator basis rather than optimizing continuous fields.
  • Universality of an analog simulator can be quantified by how small the complexities are for a fixed set of target gates or Hamiltonians.

Where Pith is reading between the lines

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

  • The same basis construction might be used to rank different physical platforms by their synthesis efficiency before any experiment is run.
  • If the correlation holds, it supplies a lower bound on control time that could be combined with existing speed-limit theorems.
  • The method could be tested on other global-control platforms such as trapped ions or superconducting circuits to check platform independence.

Load-bearing premise

That the position of a target operator in the block Krylov basis directly gives the minimum time needed to realize it with the simulator's controls, without extra assumptions on field shapes or system details.

What would settle it

An explicit computation for a Rydberg atom array in which an operation with low generalized Krylov complexity requires a longer minimum control time than one with higher complexity.

Figures

Figures reproduced from arXiv: 2605.07668 by Pengfei Zhang, Shuo Zhang, Yuzhi Tong, Zeyu Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of a quantum simulator under global con [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Structural properties of the block Krylov basis and [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Synthesis trajectories and residual Krylov weights. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
read the original abstract

Quantum simulation of complex many-body systems beyond classical computational capabilities provides a promising route toward understanding novel quantum phases and their transitions. In particular, analog quantum simulators with global control fields have attracted considerable attention due to their potential to simulate arbitrary Hamiltonians and perform quantum computing tasks. However, a clear, quantitative measure for the complexity of implementing specific quantum operations in such systems is still lacking. In this Letter, we address this challenge by introducing generalized Krylov complexity, a concept originating from operator growth dynamics, as a direct diagnosis for this synthesis complexity. We construct the block Krylov basis generated by a set of Hamiltonians, which naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators. By analyzing representative systems including Rydberg atom arrays, we demonstrate that the generalized Krylov complexity of a target operation serves as a strong predictor of the minimum time required for its realization. Our results establish Krylov complexity as an intuitive and predictive tool for designing efficient control protocols in analog quantum simulators.

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 introduces generalized Krylov complexity, defined via the dimension or growth of the block Krylov subspace generated by a simulator's native Hamiltonians and their nested commutators, and claims this quantity serves as a strong predictor of the minimal evolution time needed to realize a target operator in analog quantum simulators with global controls. The claim is supported by numerical analysis on representative systems, notably Rydberg atom arrays, where the complexity measure correlates with computed minimal realization times.

Significance. If the correlation can be placed on a rigorous footing, the work would supply a computationally inexpensive diagnostic for control complexity in analog simulators, potentially guiding protocol design without exhaustive numerical optimization. The connection between operator-growth concepts and quantum control is novel and could be useful for assessing simulability of many-body operations.

major comments (3)
  1. [Introduction / Theoretical construction] The abstract and introduction assert that the block Krylov basis 'naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators,' yet no theorem, proposition, or speed-limit inequality is provided that converts subspace dimension (or growth rate) into a lower bound on evolution time under bounded control amplitudes. Without such a relation, the numerical correlation on Rydberg arrays remains post-hoc rather than predictive.
  2. [Numerical results on Rydberg arrays] In the Rydberg-array demonstrations (presumably §4 or the numerical-results section), it is not stated how the minimal realization times are obtained (e.g., via GRAPE-type optimal control, variational ansatz, or analytic bounds) nor whether data points were selected or filtered before reporting the correlation strength. This leaves open whether the 'strong predictor' claim survives without post-selection.
  3. [Definition of generalized Krylov complexity] The precise definition of the generalized Krylov complexity (dimension of the block subspace, maximal growth rate, or a normalized quantity) is not accompanied by an explicit statement of how it is computed from the nested commutators; Eq. (X) or the corresponding algorithmic description should make clear whether the measure is parameter-free or involves any auxiliary cutoffs.
minor comments (2)
  1. [Discussion / Outlook] A short discussion of the method's limitations (e.g., applicability to time-dependent vs. time-independent controls, or to systems with only local rather than global fields) would strengthen the conclusions.
  2. [Theoretical framework] Notation for the block Krylov basis and the target operator should be introduced with a small example (e.g., a two-qubit gate) to improve readability for readers outside the Krylov-complexity community.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each of the major comments below and will revise the manuscript accordingly to strengthen the presentation and clarify the theoretical and numerical aspects.

read point-by-point responses

  1. Referee: The abstract and introduction assert that the block Krylov basis 'naturally organizes the operator space achievable through the simulator's native interactions and their nested commutators,' yet no theorem, proposition, or speed-limit inequality is provided that converts subspace dimension (or growth rate) into a lower bound on evolution time under bounded control amplitudes. Without such a relation, the numerical correlation on Rydberg arrays remains post-hoc rather than predictive.

    Authors: We agree that a rigorous theorem establishing a lower bound on the evolution time in terms of the Krylov subspace dimension would provide a stronger theoretical foundation. In the current manuscript, we present the generalized Krylov complexity as a diagnostic tool supported by extensive numerical evidence across representative systems, including Rydberg atom arrays, where it correlates strongly with the minimal realization times. While deriving a general speed-limit inequality is an important open question that we plan to pursue in future work, we will add a heuristic argument in the revised introduction explaining why the dimension of the block Krylov subspace provides a natural measure of the 'effort' required to reach the target operator, based on the number of independent nested commutators needed. This will clarify that the correlation is not merely post-hoc but grounded in the structure of the reachable operator space. revision: partial

  2. Referee: In the Rydberg-array demonstrations (presumably §4 or the numerical-results section), it is not stated how the minimal realization times are obtained (e.g., via GRAPE-type optimal control, variational ansatz, or analytic bounds) nor whether data points were selected or filtered before reporting the correlation strength. This leaves open whether the 'strong predictor' claim survives without post-selection.

    Authors: We apologize for the lack of clarity in the manuscript. We will explicitly state in the revised manuscript that the minimal realization times were obtained via numerical optimization of the control fields to achieve the target operator in minimal time, and confirm that the reported correlation includes all sampled points without post-selection. revision: yes

  3. Referee: The precise definition of the generalized Krylov complexity (dimension of the block subspace, maximal growth rate, or a normalized quantity) is not accompanied by an explicit statement of how it is computed from the nested commutators; Eq. (X) or the corresponding algorithmic description should make clear whether the measure is parameter-free or involves any auxiliary cutoffs.

    Authors: We will revise the manuscript to include a detailed algorithmic description of how the generalized Krylov complexity is computed. Specifically, it is defined as the dimension of the block Krylov subspace generated by iteratively applying nested commutators with the native Hamiltonians until the subspace stabilizes, and this procedure is parameter-free with no auxiliary cutoffs. We will add pseudocode or a step-by-step explanation following the definition to make the computation explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: Krylov complexity defined independently via commutators; numerical correlation with control time is empirical, not definitional.

full rationale

The paper defines the block Krylov basis and generalized Krylov complexity directly from the native Hamiltonians and their nested commutators, which organizes the operator space by algebraic construction. The minimum realization time is a separate dynamical quantity obtained from control protocols or optimization. The abstract and described analysis present a numerical demonstration of correlation on Rydberg arrays as evidence that complexity predicts time, without any equation or step that substitutes the time metric back into the complexity definition or fits parameters to force the relation. No self-citation chain or ansatz smuggling is indicated in the provided text that would reduce the central claim to its inputs by construction. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the construction of a block Krylov basis from Hamiltonians and the assumption that its complexity metric correlates with physical implementation time; no explicit free parameters are mentioned, but the correlation itself may implicitly involve fitting or selection.

axioms (1)
  • domain assumption The block Krylov basis generated by a set of Hamiltonians organizes the operator space achievable through native interactions and nested commutators
    Invoked in the construction of generalized Krylov complexity as the foundation for the complexity measure.
invented entities (1)
  • generalized Krylov complexity no independent evidence
    purpose: To serve as a direct diagnosis and predictor of synthesis complexity and minimum realization time in analog quantum simulators
    New concept introduced to quantify the complexity of implementing operations via global control fields.

pith-pipeline@v0.9.0 · 5473 in / 1346 out tokens · 62846 ms · 2026-05-11T02:31:41.216181+00:00 · methodology

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Reference graph

Works this paper leans on

125 extracted references · 125 canonical work pages · 1 internal anchor

  1. [1]

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

    work page arXiv

  2. [2]

    Jefferson and R

    R. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, JHEP10, 107, arXiv:1707.08570 [hep-th]

    work page arXiv

  3. [3]

    Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys

    R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield double states, Phys. Rev. D97, 066004 (2018), arXiv:1709.00921 [hep-th]

    work page arXiv 2018

  4. [4]

    Chapman, M

    S. Chapman, M. P. Heller, H. Marrochio, and F. Pastawski, Toward a Definition of Complexity for Quantum Field Theory States, Phys. Rev. Lett.120, 121602 (2018), arXiv:1707.08582 [hep-th]

    work page arXiv 2018

  5. [5]

    R. Khan, C. Krishnan, and S. Sharma, Circuit Com- plexity in Fermionic Field Theory, Phys. Rev. D98, 126001 (2018), arXiv:1801.07620 [hep-th]

    work page arXiv 2018

  6. [6]

    Yang, Y.-S

    R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang, and K.- Y. Kim, Principles and symmetries of complexity in quantum field theory, Eur. Phys. J. C79, 109 (2019), arXiv:1803.01797 [hep-th]

    work page arXiv 2019

  7. [7]

    Lucas, Operator size at finite temperature and Planckian bounds on quantum dynamics, Phys

    A. Lucas, Operator size at finite temperature and Planckian bounds on quantum dynamics, Phys. Rev. Lett.122, 216601 (2019), arXiv:1809.07769 [cond- mat.str-el]

    work page arXiv 2019

  8. [8]

    Quantum Complexity of Time Evolution with Chaotic Hamiltonians,

    V. Balasubramanian, M. Decross, A. Kar, and O. Par- rikar, Quantum Complexity of Time Evolution with Chaotic Hamiltonians, JHEP01, 134, arXiv:1905.05765 [hep-th]

    work page arXiv 1905

  9. [9]

    Balasubramanian, M

    V. Balasubramanian, M. DeCross, A. Kar, Y. C. Li, and O. Parrikar, Complexity growth in integrable and chaotic models, JHEP07, 011, arXiv:2101.02209 [hep- th]

    work page arXiv

  10. [10]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information(Cambridge University Press, 2012)

    work page 2012

  11. [11]

    Elementary gates for quantum computation

    A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin, and H. We- infurter, Elementary gates for quantum computation, Phys. Rev. A52, 3457 (1995), arXiv:quant-ph/9503016

    work page arXiv 1995

  12. [12]

    Deutsch, A

    D. Deutsch, A. Barenco, and A. Ekert, Universality in quantum computation, Proc. Roy. Soc. Lond. A449, 669 (1995), arXiv:quant-ph/9505018

    work page arXiv 1995

  13. [13]

    Lloyd, Universal Quantum Simulators, Science273, 1073 (1996)

    S. Lloyd, Universal Quantum Simulators, Science273, 1073 (1996)

    work page 1996

  14. [14]

    M. J. Bremner, C. M. Dawson, J. L. Dodd, A. Gilchrist, A. W. Harrow, D. Mortimer, M. A. Nielsen, and T. J. Osborne, Practical Scheme for Quantum Computation with Any Two-Qubit Entangling Gate, Phys. Rev. Lett. 89, 247902 (2002), arXiv:quant-ph/0207072

    work page arXiv 2002

  15. [15]

    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]

    work page arXiv 2016

  16. [16]

    Stanford and L

    D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D90, 126007 (2014), arXiv:1406.2678 [hep-th]

    work page arXiv 2014

  17. [17]

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Holographic Complexity Equals Bulk Action?, Phys. Rev. Lett.116, 191301 (2016), 6 arXiv:1509.07876 [hep-th]

    work page arXiv 2016

  18. [18]

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Complexity, action, and black holes, Phys. Rev. D93, 086006 (2016), arXiv:1512.04993 [hep-th]

    work page arXiv 2016

  19. [19]

    Caputa, N

    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, and K. Watanabe, Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories, Phys. Rev. Lett.119, 071602 (2017), arXiv:1703.00456 [hep- th]

    work page arXiv 2017

  20. [20]

    Caputa, N

    P. Caputa, N. Kundu, M. Miyaji, T. Takayanagi, and K. Watanabe, Liouville Action as Path-Integral Complexity: From Continuous Tensor Networks to AdS/CFT, JHEP11, 097, arXiv:1706.07056 [hep-th]

    work page arXiv

  21. [21]

    A. R. Brown, H. Gharibyan, G. Penington, and L. Susskind, The Python’s Lunch: geometric obstruc- tions to decoding Hawking radiation, JHEP08, 121, arXiv:1912.00228 [hep-th]

    work page arXiv 1912

  22. [22]

    Belin, R.C

    A. Belin, R. C. Myers, S.-M. Ruan, G. S´ arosi, and A. J. Speranza, Does Complexity Equal Anything?, Phys. Rev. Lett.128, 081602 (2022), arXiv:2111.02429 [hep- th]

    work page arXiv 2022

  23. [23]

    Belin, R

    A. Belin, R. C. Myers, S.-M. Ruan, G. S´ arosi, and A. J. Speranza, Complexity equals anything II, JHEP01, 154, arXiv:2210.09647 [hep-th]

    work page arXiv

  24. [24]

    Miyaji, T

    M. Miyaji, T. Numasawa, N. Shiba, T. Takayanagi, and K. Watanabe, Distance between Quantum States and Gauge-Gravity Duality, Phys. Rev. Lett.115, 261602 (2015), arXiv:1507.07555 [hep-th]

    work page arXiv 2015

  25. [25]

    Miyaji, Butterflies from Information Metric, JHEP 09, 002, arXiv:1607.01467 [hep-th]

    M. Miyaji, Butterflies from Information Metric, JHEP 09, 002, arXiv:1607.01467 [hep-th]

    work page arXiv

  26. [26]

    Belin, A

    A. Belin, A. Lewkowycz, and G. S´ arosi, Complexity and the bulk volume, a new York time story, JHEP03, 044, arXiv:1811.03097 [hep-th]

    work page arXiv

  27. [27]

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

    work page arXiv 2018

  28. [28]

    Haferkamp, P

    J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eis- ert, and N. Y. Halpern, Linear growth of quan- tum circuit complexity, Nature Phys.18, 528 (2022), arXiv:2106.05305 [quant-ph]

    work page arXiv 2022

  29. [29]

    Chiuet al., Continuous operation of a co- herent 3,000-qubit system, Nature646, 1075 (2025), arXiv:2506.20660 [quant-ph]

    N.-C. Chiuet al., Continuous operation of a co- herent 3,000-qubit system, Nature646, 1075 (2025), arXiv:2506.20660 [quant-ph]

    work page arXiv 2025

  30. [30]

    H. J. Manetsch, G. Nomura, E. Bataille, X. Lv, K. H. Leung, and M. Endres, A tweezer array with 6,100 highly coherent atomic qubits, Nature647, 60 (2025), arXiv:2403.12021 [quant-ph]

    work page arXiv 2025

  31. [31]

    S. A. Guoet al., A site-resolved two-dimensional quan- tum simulator with hundreds of trapped ions, Nature 630, 613 (2024), arXiv:2311.17163 [quant-ph]

    work page arXiv 2024

  32. [32]

    Semeghini, H

    G. Semeghiniet al., Probing topological spin liquids on a programmable quantum simulator, Science374, abi8794 (2021), arXiv:2104.04119 [quant-ph]

    work page arXiv 2021

  33. [33]

    J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall, A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Quan- tum spin dynamics and entanglement generation with hundreds of trapped ions, Science352, 1297 (2016), arXiv:1512.03756 [quant-ph]

    work page arXiv 2016

  34. [34]

    Nature595, 233–238 (2021).2012.12268

    P. Schollet al., Quantum simulation of 2D antiferro- magnets with hundreds of Rydberg atoms, Nature595, 233 (2021), arXiv:2012.12268 [quant-ph]

    work page arXiv 2021

  35. [35]

    Large-scale quantum reservoir learning with an analog quantum computer.arXiv:2407.02553, 2024

    M. Kornjaˇ caet al., Large-scale quantum reservoir learning with an analog quantum computer, (2024), arXiv:2407.02553 [quant-ph]

    work page arXiv 2024

  36. [36]

    Manovitzet al., Quantum coarsening and collective dynamics on a programmable simulator, Nature638, 86 (2025), arXiv:2407.03249 [quant-ph]

    T. Manovitzet al., Quantum coarsening and collective dynamics on a programmable simulator, Nature638, 86 (2025), arXiv:2407.03249 [quant-ph]

    work page arXiv 2025

  37. [37]

    Hartke, B

    T. Hartke, B. Oreg, C. Turnbaugh, N. Jia, and M. Zwierlein, Direct observation of nonlocal fermion pairing in an attractive Fermi-Hubbard gas, Science 381, 82 (2023), arXiv:2208.05948 [cond-mat.quant-gas]

    work page arXiv 2023

  38. [38]

    T. I. Andersenet al., Thermalization and criticality on an analogue–digital quantum simulator, Nature638, 79 (2025), arXiv:2405.17385 [quant-ph]

    work page arXiv 2025

  39. [39]

    Kohlert, S

    T. Kohlert, S. Scherg, P. Sala, F. Pollmann, B. Hebbe Madhusudhana, I. Bloch, and M. Aidels- burger, Exploring the regime of fragmentation in strongly tilted fermi-hubbard chains, Phys. Rev. Lett. 130, 010201 (2023)

    work page 2023

  40. [40]

    Liang, Z

    X. Liang, Z. Yue, Y.-X. Chao, Z.-X. Hua, Y. Lin, M. K. Tey, and L. You, Observation of Anomalous Information Scrambling in a Rydberg Atom Array, Phys. Rev. Lett. 135, 050201 (2025), arXiv:2410.16174 [quant-ph]

    work page arXiv 2025

  41. [41]

    M. Xu, L. H. Kendrick, A. Kale, Y. Gang, C. Feng, S. Zhang, A. W. Young, M. Lebrat, and M. Greiner, A neutral-atom Hubbard quantum simulator in the cryo- genic regime, Nature642, 909 (2025), arXiv:2502.00095 [cond-mat.quant-gas]

    work page arXiv 2025

  42. [42]

    Mazurenko, C

    A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kan´ asz-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom fermi–hubbard antiferromagnet, Nature545, 462 (2017)

    work page 2017

  43. [43]

    X. Wang, E. Khatami, F. Fei, J. Wyrick, P. Nam- boodiri, R. Kashid, A. F. Rigosi, G. Bryant, and R. Silver, Experimental realization of an extended Fermi-Hubbard model using a 2D lattice of dopant- based quantum dots, Nature Communications13, 6824 (2022), arXiv:2110.08982 [quant-ph]

    work page arXiv 2022

  44. [44]

    Hofrichter, L

    C. Hofrichter, L. Riegger, F. Scazza, M. H¨ ofer, D. R. Fernandes, I. Bloch, and S. F¨ olling, Direct Prob- ing of the Mott Crossover in the SU (N ) Fermi- Hubbard Model, Physical Review X6, 021030 (2016), arXiv:1511.07287 [cond-mat.quant-gas]

    work page arXiv 2016

  45. [45]

    Shaoet al., Antiferromagnetic phase transition in a 3D fermionic Hubbard model, Nature632, 267 (2024), arXiv:2402.14605 [cond-mat.quant-gas]

    H.-J. Shaoet al., Antiferromagnetic phase transition in a 3D fermionic Hubbard model, Nature632, 267 (2024), arXiv:2402.14605 [cond-mat.quant-gas]

    work page arXiv 2024

  46. [46]

    Gross and I

    C. Gross and I. Bloch, Quantum simulations with ultra- cold atoms in optical lattices, Science357, 995 (2017)

    work page 2017

  47. [47]

    Sompetet al., Realising the Symmetry-Protected Haldane Phase in Fermi-Hubbard Ladders, Nature606, 484 (2022), arXiv:2103.10421 [cond-mat.quant-gas]

    P. Sompetet al., Realising the Symmetry-Protected Haldane Phase in Fermi-Hubbard Ladders, Nature606, 484 (2022), arXiv:2103.10421 [cond-mat.quant-gas]

    work page arXiv 2022

  48. [48]

    Observation of average topological phase in disordered Rydberg atom array

    Z. Yueet al., Observation of average topological phase in disordered Rydberg atom array, (2025), arXiv:2505.06286 [cond-mat.quant-gas]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  49. [49]

    Y. Lu, C. Wang, S. K. Kanungo, F. B. Dunning, and T. C. Killian, Probing the topological phase transition in the Su-Schrieffer-Heeger Hamiltonian using Rydberg- atom synthetic dimensions, Phys. Rev. A110, 023318 (2024), arXiv:2404.18420 [quant-ph]

    work page arXiv 2024

  50. [50]

    de L´ es´ eleuc, V

    S. de L´ es´ eleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. B¨ uchler, T. Lahaye, and A. Browaeys, Observation of a symmetry-protected topological phase of interacting bosons with Rydberg atoms, Science365, aav9105 (2019), arXiv:1810.13286 [quant-ph]

    work page arXiv 2019

  51. [51]

    Gonzalez-Cuadraet al., Nature642, 321 (2025), arXiv:2410.16558 [quant-ph]

    D. Gonzalez-Cuadraet al., Observation of string break- ing on a (2 + 1)D Rydberg quantum simulator, Nature 642, 321 (2025), arXiv:2410.16558 [quant-ph]. 7

    work page arXiv 2025

  52. [52]

    Mildenberger, W

    J. Mildenberger, W. Mruczkiewicz, J. C. Halimeh, Z. Jiang, and P. Hauke, Confinement in aZ 2 lattice gauge theory on a quantum computer, Nature Phys.21, 312 (2025), arXiv:2203.08905 [quant-ph]

    work page arXiv 2025

  53. [53]

    D. K. Mark, H.-Y. Hu, J. Kwan, C. Kokail, S. Choi, and S. F. Yelin, Efficiently Measuring d-Wave Pairing and Beyond in Quantum Gas Microscopes, Phys. Rev. Lett. 135, 123402 (2025), arXiv:2412.13186 [cond-mat.quant- gas]

    work page arXiv 2025

  54. [54]

    S. C. Benjamin, Schemes for Parallel Quantum Com- putation Without Local Control of Qubits, Phys. Rev. A 10.1103/PhysRevA.61.020301 (1999), arXiv:quant- ph/9909007

    work page doi:10.1103/physreva.61.020301 1999

  55. [55]

    S. C. Benjamin, Quantum Computing Without Local Control of Qubit-Qubit Interactions, Phys. Rev. Lett. 88, 017904 (2001), arXiv:quant-ph/0104117 [quant-ph]

    work page arXiv 2001

  56. [56]

    Lloyd, Quantum approximate optimization is compu- tationally universal, (2018), arXiv:1812.11075 [quant- ph]

    S. Lloyd, Quantum approximate optimization is compu- tationally universal, (2018), arXiv:1812.11075 [quant- ph]

    work page arXiv 2018

  57. [57]

    Cesa and H

    F. Cesa and H. Pichler, Universal Quantum Com- putation in Globally Driven Rydberg Atom Arrays, Phys. Rev. Lett.131, 170601 (2023), arXiv:2305.19220 [quant-ph]

    work page arXiv 2023

  58. [58]

    Oszmaniec and Z

    M. Oszmaniec and Z. Zimbor´ as, Universal extensions of restricted classes of quantum operations, Phys. Rev. Lett.119, 220502 (2017), arXiv:1705.11188 [quant-ph]

    work page arXiv 2017

  59. [59]

    H.-Y. Hu, A. M. Gomez, L. Chen, A. Trowbridge, A. J. Goldschmidt, Z. Manchester, F. T. Chong, A. Jaffe, and S. F. Yelin, Universal Dynamics with Glob- ally Controlled Analog Quantum Simulators, (2025), arXiv:2508.19075 [quant-ph]

    work page arXiv 2025

  60. [60]

    D’Alessandro, Lie Algebraic Analysis and Control of Quantum Dynamics, (2009), arXiv:0803.1193 [quant- ph]

    D. D’Alessandro, Lie Algebraic Analysis and Control of Quantum Dynamics, (2009), arXiv:0803.1193 [quant- ph]

    work page arXiv 2009

  61. [61]

    d’Alessandro,Introduction to quantum control and dynamics(Chapman and hall/CRC, 2021)

    D. d’Alessandro,Introduction to quantum control and dynamics(Chapman and hall/CRC, 2021)

    work page 2021

  62. [62]

    Khaneja and S

    N. Khaneja and S. J. Glaser, Cartan decomposition of SU(2 n ) and control of spin systems , Chem. Phys.267, 11 (2001)

    work page 2001

  63. [63]

    Ragone, B

    M. Ragone, B. N. Bakalov, F. Sauvage, A. F. Kemper, C. O. Marrero, M. Larocca, and M. Cerezo, A Lie alge- braic theory of barren plateaus for deep parameterized quantum circuits, Nature Commun.15, 7172 (2024), arXiv:2309.09342 [quant-ph]

    work page arXiv 2024

  64. [64]

    Wiersema, E

    R. Wiersema, E. K¨ okc¨ u, A. F. Kemper, and B. N. Bakalov, Classification of dynamical Lie algebras of 2- local spin systems on linear, circular and fully con- nected topologies, npj Quantum Inf.10, 110 (2024), arXiv:2309.05690 [quant-ph]

    work page arXiv 2024

  65. [65]

    Allcock, M

    J. Allcock, M. Santha, P. Yuan, and S. Zhang, On the dynamical Lie algebras of quantum approximate opti- mization algorithms, (2024), arXiv:2407.12587 [quant- ph]

    work page arXiv 2024

  66. [66]

    Araiza Bravo, J

    R. Araiza Bravo, J. G. Ponce, H.-y. Hu, and S. F. Yelin, Circumventing traps in analog quantum machine learning algorithms through co-design, APL Quantum 1, 046121 (2024), arXiv:2408.14697 [quant-ph]

    work page arXiv 2024

  67. [67]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi, and E. Altman, A Universal Operator Growth Hypothesis, Phys. Rev. X9, 041017 (2019), arXiv:1812.08657 [cond- mat.stat-mech]

    work page arXiv 2019

  68. [68]

    Avdoshkin, A

    A. Avdoshkin, A. Dymarsky, and M. Smolkin, Krylov complexity in quantum field theory, and beyond, JHEP 06, 066, arXiv:2212.14429 [hep-th]

    work page arXiv

  69. [69]

    Balasubramanian, P

    V. Balasubramanian, P. Caputa, J. M. Magan, and Q. Wu, Quantum chaos and the complexity of spread of states, Phys. Rev. D106, 046007 (2022), arXiv:2202.06957 [hep-th]

    work page arXiv 2022

  70. [70]

    C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems, Phys. Rev. Res.5, 033085 (2023), arXiv:2207.13603 [cond-mat.str-el]

    work page arXiv 2023

  71. [71]

    J. L. F. Barb´ on, E. Rabinovici, R. Shir, and R. Sinha, On The Evolution Of Operator Complexity Beyond Scrambling, JHEP10, 264, arXiv:1907.05393 [hep-th]

    work page arXiv 1907

  72. [72]

    Dymarsky and A

    A. Dymarsky and A. Gorsky, Quantum chaos as delo- calization in Krylov space, Phys. Rev. B102, 085137 (2020), arXiv:1912.12227 [cond-mat.stat-mech]

    work page arXiv 2020

  73. [73]

    J. L. F. Barb´ on, J. Mart´ ın-Garc´ ıa, and M. Sasieta, Mo- mentum/Complexity Duality and the Black Hole Inte- rior, JHEP07, 169, arXiv:1912.05996 [hep-th]

    work page arXiv 1912

  74. [74]

    J. M. Mag´ an and J. Sim´ on, On operator growth and emergent Poincar´ e symmetries, JHEP05, 071, arXiv:2002.03865 [hep-th]

    work page arXiv 2002

  75. [75]

    S.-K. Jian, B. Swingle, and Z.-Y. Xian, Complexity growth of operators in the SYK model and in JT grav- ity, JHEP03, 014, arXiv:2008.12274 [hep-th]

    work page arXiv 2008

  76. [76]

    Rabinovici, A

    E. Rabinovici, A. S´ anchez-Garrido, R. Shir, and J. Son- ner, Operator complexity: a journey to the edge of Krylov space, JHEP06, 062, arXiv:2009.01862 [hep-th]

    work page arXiv 2009

  77. [77]

    Chen and A

    C.-F. Chen and A. Lucas, Operator Growth Bounds from Graph Theory, Commun. Math. Phys.385, 1273 (2021), arXiv:1905.03682 [math-ph]

    work page arXiv 2021

  78. [78]

    J. D. Noh, Operator growth in the transverse-field Ising spin chain with integrability-breaking longitudinal field, Phys. Rev. E104, 034112 (2021), arXiv:2107.08287 [quant-ph]

    work page arXiv 2021

  79. [79]

    Caputa and S

    P. Caputa and S. Datta, Operator growth in 2d CFT, Journal of High Energy Physics2021, 188 (2021), arXiv:2110.10519 [hep-th]

    work page arXiv 2021

  80. [80]

    Patramanis, Probing the entanglement of operator growth, PTEP2022, 063A01 (2022), arXiv:2111.03424 [hep-th]

    D. Patramanis, Probing the entanglement of operator growth, PTEP2022, 063A01 (2022), arXiv:2111.03424 [hep-th]

    work page arXiv 2022

Showing first 80 references.