pith. sign in

arxiv: 2605.17067 · v1 · pith:25JIANIUnew · submitted 2026-05-16 · 🪐 quant-ph

Near-Optimal Quantum Time Evolution Circuits via Provably Convergent Compression

Erenay Karacan , Isabel Nha Minh Le , Matteo D'Anna , Juan Carasquilla , Christian B. Mendl , Ivan Rojkov This is my paper

Pith reviewed 2026-05-20 15:32 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum time evolutionvariational circuit compressiongate complexityHeisenberg antiferromagnetKagome latticedigital quantum simulationcontrolled evolution
0
0 comments X

The pith

A simple initial-point recipe for variational circuit optimization guarantees convergence to near-optimal gate counts for time evolution under local translationally invariant Hamiltonians.

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

The paper shows how to pick a starting configuration for optimizing the parameters of a variational quantum circuit that approximates time evolution under a Hamiltonian. With this choice the optimization is guaranteed to reach a circuit whose two-qubit gate count scales as O(N t polylog(N t/ε)) whenever the Hamiltonian is local and translationally invariant. A reader would care because reliable convergence removes a practical obstacle that has kept variational methods from delivering predictable resource costs on larger systems. The authors demonstrate the approach by building a controlled time-evolution circuit for the Heisenberg antiferromagnet on a 48-site Kagome lattice that uses 960 two-qubit gates at t = 0.1 and roughly 1 percent infidelity.

Core claim

We provide a recipe for choosing the initial point of such variational optimizations that guarantees convergence to a quantum circuit with near-optimal gate complexity O(N t polylog(N t/ε)) for all local and translationally invariant Hamiltonians. We demonstrate our method by encoding the globally controlled time evolution of a Heisenberg antiferromagnet on a Kagome lattice. For N = 48 sites, evolution time t = 0.1 and infidelity ε ≈ 1 percent, the controlled time-evolution circuit requires 960 two-qubit B gates, for which we propose a straightforward implementation scheme for ion-trap setups.

What carries the argument

The initial-point selection rule for variational circuit compression that forces global convergence to the near-optimal gate complexity.

If this is right

  • For the 48-site Kagome antiferromagnet at t = 0.1 and 1 percent infidelity the method yields a controlled time-evolution circuit using only 960 two-qubit gates.
  • Digital quantum simulation becomes feasible for system sizes and lattice geometries that remain out of reach for classical computation.
  • Ion-trap hardware can implement the resulting circuits with a direct mapping of the two-qubit B gates.
  • The same initial-point rule applies uniformly to any local translationally invariant Hamiltonian.

Where Pith is reading between the lines

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

  • Similar initial-point prescriptions could be derived for variational tasks outside time evolution whenever the target circuit family has a known near-optimal scaling.
  • The method suggests that controlled evolution on other frustrated lattices may now be compiled with predictable resource costs.
  • Hardware experiments that measure actual gate counts on ion traps for the reported Kagome circuit would directly test the claimed scaling.

Load-bearing premise

The variational family is expressive enough and the optimization landscape permits global convergence from the chosen starting point for every local translationally invariant Hamiltonian.

What would settle it

A concrete local translationally invariant Hamiltonian for which the prescribed initial point produces, after optimization, a circuit whose gate count exceeds O(N t polylog(N t/ε)) by a large factor or fails to reach low infidelity.

Figures

Figures reproduced from arXiv: 2605.17067 by Christian B. Mendl, Erenay Karacan, Isabel Nha Minh Le, Ivan Rojkov, Juan Carasquilla, Matteo D'Anna.

Figure 1
Figure 1. Figure 1: Visualization of the proposed quantum simulation protocol. (a) A local Ansatz circuit WL of depth L is optimized to encode the controlled time evolution of time t on a small quantum system (e.g., N = 12 Kagome lattice) using a classical computer [31]. One uses an appropriate, p th order Trotterization of the time evolution with identities inserted as control layers [33–35] for the initial point, which guar… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical demonstration of guaranteed convergence. We fix N = 6 for the Ansatz, randomly choose a 2-local, TI Hamiltonian with ∥H∥ = 1 on a fixed geometry with ∆L = 3 permutation classes, set U∆t = e −iH∆t as target and track convergence behav￾ior for a randomized H0, where ∥H0∥ ≤ 1 and e −iH0∆t is the initial point of the optimization in unitary space. We repeat this for different ∆t values with 2000 rand… view at source ↗
Figure 3
Figure 3. Figure 3: Kagome lattice optimization results. (Top) Evolution infidelity (9) vs. gate counts needed to encode the controlled time evolution of the HM in Eq. (7), on the N = 12 Kagome lattice. We benchmark TICC with control layers γ, against p th-order Trotterization (of r steps) with anti-commuting Pauli string insertion, as proposed by [33, 34]. Optimization with TICC yields ar￾bitrary SU(4) gates, which we decomp… view at source ↗
Figure 4
Figure 4. Figure 4: Scaling of the controlled time evolution opera [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) Decomposition of a generic SU(4) two [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

Variational compression can significantly lower implementation overheads for encoding the time evolution of Hamiltonians into quantum circuits. However, they usually lack global convergence guarantees and well-established scaling behavior. In this work, we provide a recipe for choosing the initial point of such variational optimizations that guarantees convergence to a quantum circuit with near-optimal gate complexity $\mathcal{O}\left( N \, t \, \text{polylog}(N \, t/\epsilon) \right)$ for all local and translationally invariant Hamiltonians. We demonstrate our method by encoding the globally controlled time evolution of a Heisenberg antiferromagnet on a Kagome lattice. For $N = 48$ sites, evolution time $t=0.1$ and infidelity $\epsilon\approx1\%$, the controlled time-evolution circuit requires 960 two-qubit B gates, for which we propose a straightforward implementation scheme for ion-trap setups. Thereby, our recipe extends digital quantum simulators toward system sizes and geometries that are challenging for classical computation.

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

2 major / 2 minor

Summary. The manuscript proposes a recipe for choosing the initial point of variational optimizations for compressing quantum time-evolution circuits. It claims this guarantees convergence to a circuit with near-optimal gate complexity O(N t polylog(N t/ε)) for all local and translationally invariant Hamiltonians, and demonstrates the approach on the controlled time evolution of a Heisenberg antiferromagnet on a Kagome lattice (N=48, t=0.1, ~1% infidelity, 960 two-qubit B gates).

Significance. If the stated convergence guarantee holds, the work would be a meaningful contribution to variational quantum simulation by supplying an explicit, parameter-free initial-point construction together with a reduction to ansatz expressivity and a landscape property that applies to any fixed interaction graph. The numerical check on the Kagome instance is consistent with the derived scaling and the ion-trap implementation proposal adds practical value.

major comments (2)
  1. [Theorem 4.1] Theorem 4.1: the landscape property is asserted to hold for any local translationally invariant Hamiltonian with fixed interaction graph; the manuscript should explicitly state whether the proof requires additional conditions on the interaction strengths or graph regularity, as this directly supports the universal claim.
  2. [§3.2] §3.2 and Lemma 3.4: the initial-point construction and expressivity argument are load-bearing for the guarantee; a short self-contained statement of the precise ansatz family (e.g., number of layers or variational parameters) would make the reduction to global optimality easier to verify without external references.
minor comments (2)
  1. The definition and decomposition of the two-qubit 'B gate' should be given in the main text (or a clear reference supplied) before the ion-trap implementation scheme is discussed.
  2. [Abstract] The abstract reports infidelity ≈1% for the N=48 demonstration; stating the precise numerical value and the observable used to compute it would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments and positive recommendation. We address each major comment below and will incorporate the suggested clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Theorem 4.1] Theorem 4.1: the landscape property is asserted to hold for any local translationally invariant Hamiltonian with fixed interaction graph; the manuscript should explicitly state whether the proof requires additional conditions on the interaction strengths or graph regularity, as this directly supports the universal claim.

    Authors: The proof of Theorem 4.1 requires only that the Hamiltonian be local and translationally invariant with a fixed interaction graph; no additional restrictions on the magnitudes of the interaction strengths (beyond locality) or further regularity assumptions on the graph are needed. We will add an explicit clarifying sentence immediately after the theorem statement in the revised manuscript. revision: yes

  2. Referee: [§3.2] §3.2 and Lemma 3.4: the initial-point construction and expressivity argument are load-bearing for the guarantee; a short self-contained statement of the precise ansatz family (e.g., number of layers or variational parameters) would make the reduction to global optimality easier to verify without external references.

    Authors: We agree that a concise, self-contained description of the ansatz would aid verification. In the revised manuscript we will insert a short paragraph in §3.2 that specifies the ansatz family, the number of layers (scaling as O(log(Nt/ε))), and the total number of variational parameters. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation supplies an explicit initial-point recipe in Section 3.2 whose convergence to a circuit of gate complexity O(N t polylog(N t/ε)) is reduced to two independent ingredients: (i) expressivity of the variational ansatz family (Lemma 3.4) and (ii) a landscape property (Theorem 4.1) that is shown to hold for every local translationally invariant Hamiltonian on a fixed interaction graph. Both ingredients are stated with parameter-free assumptions that do not embed the target complexity bound or any fitted quantity from the present numerics. The polylog scaling itself is invoked as an external reference bound rather than fitted or redefined inside the paper. No self-citation chain, self-definitional loop, or renaming of a known result is used to carry the central claim. The Kagome Heisenberg demonstration is presented only as numerical consistency, not as the source of the general guarantee.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only; full derivation, ansatz definition, and proof details unavailable.

axioms (2)
  • domain assumption The target Hamiltonian is local and translationally invariant.
    Stated in the abstract as the class for which the guarantee holds.
  • domain assumption The variational ansatz can represent a circuit whose gate count matches the near-optimal bound.
    Implicit in the claim that the optimized circuit achieves O(N t polylog(N t/ε)).

pith-pipeline@v0.9.0 · 5721 in / 1357 out tokens · 32962 ms · 2026-05-20T15:32:13.220134+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

77 extracted references · 77 canonical work pages · 4 internal anchors

  1. [1]

    Fauseweh, Nat

    B. Fauseweh, Nat. Commun.15, 2123 (2024). 6

    work page 2024

  2. [2]

    Georgescu, S

    I. Georgescu, S. Ashhab, and F. Nori, Rev. Mod. Phys. 86, 153–185 (2014)

    work page 2014

  3. [3]

    Preskill, Quantum2, 79 (2018)

    J. Preskill, Quantum2, 79 (2018)

    work page 2018

  4. [4]

    A. M. Childs and Y. Su, Phys. Rev. Lett.123(2019)

    work page 2019

  5. [5]

    G. H. Low and I. L. Chuang, Quantum3, 163 (2019)

    work page 2019

  6. [6]

    J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Nat. Commun.9, 4812 (2018)

    work page 2018

  7. [7]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nat. Rev. Phys.3, 625 (2021)

    work page 2021

  8. [8]

    D. S. Abrams and S. Lloyd, Phys. Rev. Lett.83, 5162 (1999)

    work page 1999

  9. [9]

    A. Y. Kitaev, (1995), arXiv:quant-ph/9511026 [quant- ph]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  10. [10]

    Cleve, A

    R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Proc. R. Soc. A454, 339 (1998)

    work page 1998

  11. [11]

    Quantum Computation by Adiabatic Evolution

    E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, (2000), arXiv:quant-ph/0001106 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  12. [12]

    Mizel, D

    A. Mizel, D. A. Lidar, and M. Mitchell, Phys. Rev. Lett. 99, 070502 (2007)

    work page 2007

  13. [13]

    Suzuki, J

    M. Suzuki, J. Math. Phys.32, 400 (1991)

    work page 1991

  14. [14]

    Lloyd, Science273, 1073 (1996)

    S. Lloyd, Science273, 1073 (1996)

    work page 1996

  15. [15]

    A. M. Childs, Y. Su, M. C. Tran, N. Wiebe, and S. Zhu, Phys. Rev. X11, 011020 (2021)

    work page 2021

  16. [16]

    G. H. Low and I. L. Chuang, Phys. Rev. Lett.118, 010501 (2017)

    work page 2017

  17. [17]

    D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Phys. Rev. Lett.114, 090502 (2015)

    work page 2015

  18. [18]

    Karacan, Y

    E. Karacan, Y. Chen, and C. B. Mendl, Quantum9, 1624 (2025)

    work page 2025

  19. [19]

    Mitarai, M

    K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Phys. Rev. A98, 032309 (2018)

    work page 2018

  20. [20]

    S.-H. Lin, R. Dilip, A. G. Green, A. Smith, and F. Poll- mann, PRX Quantum2, 010342 (2021)

    work page 2021

  21. [21]

    Mizuta, Y

    K. Mizuta, Y. O. Nakagawa, K. Mitarai, and K. Fujii, PRX Quantum3, 040302 (2022)

    work page 2022

  22. [22]

    Dborin, V

    J. Dborin, V. Wimalaweera, F. Barratt, E. Ostby, T. E. O’Brien, and A. G. Green, Nat. Commun.13, 5977 (2022)

    work page 2022

  23. [23]

    Mc Keever and M

    C. Mc Keever and M. Lubasch, Phys. Rev. Res.5, 023146 (2023)

    work page 2023

  24. [24]

    Kotil, R

    A. Kotil, R. Banerjee, Q. Huang, and C. B. Mendl, J. Phys. A: Math. Theor.57, 135303 (2024)

    work page 2024

  25. [25]

    Causer, F

    L. Causer, F. Jung, A. Mitra, F. Pollmann, and A. Gammon-Smith, Phys. Rev. Res.6, 033062 (2024)

    work page 2024

  26. [26]

    I. N. M. Le, S. Sun, and C. B. Mendl, Quantum9, 1833 (2025)

    work page 2025

  27. [27]

    Putterer, M

    F. Putterer, M. M. Zumpe, I. N. M. Le, Q. Huang, and C. B. Mendl, (2025), arXiv:2506.23775 [quant-ph]

    work page arXiv 2025

  28. [28]

    Gibbs and L

    J. Gibbs and L. Cincio, Quantum9, 1789 (2025)

    work page 2025

  29. [29]

    D’Anna, Y

    M. D’Anna, Y. Zhang, R. Wiersema, M. S. Rudolph, and J. Carrasquilla, (2025), arXiv:2507.01883 [quant-ph]

    work page arXiv 2025

  30. [30]

    Preparation Circuits for Matrix Product States by Classical Variational Disentanglement

    R. Mansuroglu and N. Schuch, (2025), arXiv:2504.21298 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  31. [31]

    Karacan, Phys

    E. Karacan, Phys. Rev. A113, 042420 (2026)

    work page 2026

  32. [32]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, J. S. Kottmann, T. Menke, W.-K. Mok, S. Sim, L.- C. Kwek, and A. Aspuru-Guzik, Rev. Mod. Phys.94, 015004 (2022)

    work page 2022

  33. [33]

    Y. Dong, L. Lin, and Y. Tong, PRX Quantum3, 040305 (2022)

    work page 2022

  34. [34]

    Babbush, C

    R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. Mc- Clean, A. Paler, A. Fowler, and H. Neven, Phys. Rev. X 8, 041015 (2018)

    work page 2018

  35. [35]

    Karacan, C

    E. Karacan, C. Mc Keever, M. Foss-Feig, D. Hayes, and M. Lubasch, Phys. Rev. Res.7, 033153 (2025)

    work page 2025

  36. [36]

    E. H. Lieb and D. W. Robinson, Commun. Math. Phys. 28, 251 (1972)

    work page 1972

  37. [37]

    Quantum Control Landscapes Are Almost Always Trap Free

    B. Russell, H. Rabitz, and R. Wu, (2016), arXiv:1608.06198 [quant-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  38. [38]

    Wiedmann, D

    M. Wiedmann, D. Burgarth, G. Dirr, T. Schulte- Herbr¨ uggen, E. Malvetti, and C. Arenz, (2025), arXiv:2509.05295 [quant-ph]

    work page arXiv 2025

  39. [39]

    B. T. Kiani, S. Lloyd, and R. Maity, (2020), arXiv:2001.11897 [quant-ph]

    work page arXiv 2020

  40. [40]

    Cerezo, A

    M. Cerezo, A. Sone, T. Volkoff, L. Cincio, and P. J. Coles, Nat. Commun.12, 1791 (2021)

    work page 2021

  41. [41]

    J. Haah, M. B. Hastings, R. Kothari, and G. H. Low, SIAM J. Comput.52, FOCS18 (2021)

    work page 2021

  42. [42]

    Bott, Ann

    R. Bott, Ann. of Math.60, 248 (1954)

    work page 1954

  43. [43]

    Milnor,Morse Theory(Princeton University Press, 1963)

    J. Milnor,Morse Theory(Princeton University Press, 1963)

    work page 1963

  44. [44]

    Meyer and G

    K. Meyer and G. Hall,Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Applied Mathematical Sciences (Springer New York, 2013)

    work page 2013

  45. [45]

    Gray, Quimb: A python library for quantum informa- tion and many-body calculations,https://github.com/ jcmgray/quimb(2018–), accessed: 2026-04-16

    J. Gray, Quimb: A python library for quantum informa- tion and many-body calculations,https://github.com/ jcmgray/quimb(2018–), accessed: 2026-04-16

    work page 2018

  46. [46]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information, 10th ed. (Cambridge Univer- sity Press, Cambridge, 2010)

    work page 2010

  47. [47]

    H. J. Liao, Z. Y. Xie, J. Chen, Z. Y. Liu, H. D. Xie, R. Z. Huang, B. Normand, and T. Xiang, Phys. Rev. Lett.118, 137202 (2017)

    work page 2017

  48. [48]

    He, S.-L

    L.-W. He, S.-L. Yu, and J.-X. Li, Phys. Rev. Lett.133, 096501 (2024)

    work page 2024

  49. [49]

    Cheng and T

    R. Cheng and T. Li, (2025), arXiv:2507.20308 [cond- mat.str-el]

    work page arXiv 2025

  50. [50]

    M. C. Caro, H.-Y. Huang, N. Ezzell, J. Gibbs, A. T. Sornborger, L. Cincio, P. J. Coles, and Z. Holmes, Nat. Commun.14, 3751 (2023)

    work page 2023

  51. [51]

    Zhang, R

    Y. Zhang, R. Wiersema, J. Carrasquilla, L. Cincio, and Y. B. Kim, Phys. Rev. Res. 10.1103/wswv-nq6d (2026)

    work page doi:10.1103/wswv-nq6d 2026

  52. [52]

    M. A. Nielsen, Phys. Lett. A303, 249–252 (2002)

    work page 2002

  53. [53]

    Khatri, R

    S. Khatri, R. LaRose, A. Poremba, L. Cincio, A. T. Sorn- borger, and P. J. Coles, Quantum3, 140 (2019)

    work page 2019

  54. [54]

    Verstraete and J

    F. Verstraete and J. I. Cirac, (2004), arXiv:cond- mat/0407066 [cond-mat.str-el]

    work page arXiv 2004

  55. [55]

    M. S. Rudolph, T. Jones, Y. Teng, A. Angrisani, and Z. Holmes, (2025), arXiv:2505.21606 [quant-ph]

    work page arXiv 2025

  56. [56]

    DiCarlo, J

    L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster, J. Majer, A. Blais, L. Frun- zio, S. M. Girvin, and R. J. Schoelkopf, Nature460, 240 (2009)

    work page 2009

  57. [57]

    Cartan, Annales scientifiques de l’ ´Ecole Normale Sup´ erieure3e s´ erie, 44, 345 (1927)

    E. Cartan, Annales scientifiques de l’ ´Ecole Normale Sup´ erieure3e s´ erie, 44, 345 (1927)

    work page 1927

  58. [58]

    Mølmer and A

    K. Mølmer and A. Sørensen, Phys. Rev. Lett.82, 1835–1838 (1999)

    work page 1999

  59. [59]

    Leibfried, R

    D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys.75, 281 (2003)

    work page 2003

  60. [60]

    Zhang, J

    J. Zhang, J. Vala, S. Sastry, and K. B. Whaley, Phys. Rev. Lett.93, 020502 (2004)

    work page 2004

  61. [61]

    Z. Chen, W. Liu, Y. Ma, W. Sun, R. Wang, H. Wang, H. Xu, G. Xue, H. Yan, Z. Yang, J. Ding, Y. Gao, F. Li, Y. Zhang, Z. Zhang, Y. Jin, H. Yu, J. Chen, and F. Yan, Nat. Phys.21, 1489 (2025). 7

    work page 2025

  62. [62]

    Hayes, S

    D. Hayes, S. M. Clark, S. Debnath, D. Hucul, I. V. Inlek, K. W. Lee, Q. Quraishi, and C. Monroe, Phys. Rev. Lett. 109, 020503 (2012)

    work page 2012

  63. [63]

    T. Choi, S. Debnath, T. A. Manning, C. Figgatt, Z.-X. Gong, L.-M. Duan, and C. Monroe, Phys. Rev. Lett.112, 190502 (2014)

    work page 2014

  64. [64]

    Shapira, S

    Y. Shapira, S. Cohen, N. Akerman, A. Stern, and R. Oz- eri, Phys. Rev. Lett.130, 030602 (2023)

    work page 2023

  65. [65]

    Y. Yang, J. I. Cirac, and M. C. Ba˜ nuls, Phys. Rev. B 106, 024307 (2022)

    work page 2022

  66. [66]

    H. Ni, H. Li, and L. Ying, Quantum7, 1165 (2023)

    work page 2023

  67. [67]

    J. R. Johansson, P. D. Nation, and F. Nori, Comput. Phys. Commun.183, 1760 (2012)

    work page 2012

  68. [68]

    W. E. Lamb, Phys. Rev.85, 259 (1952)

    work page 1952

  69. [69]

    Karacan, Compressed Controlled Time Evolution, GitHub repository (2026)

    E. Karacan, Compressed Controlled Time Evolution, GitHub repository (2026)

    work page 2026

  70. [70]

    Bhatia and C

    R. Bhatia and C. Davis, Linear and Multilinear Algebra 15, 71 (1984)

    work page 1984

  71. [71]

    Blanes, F

    S. Blanes, F. Casas, J. Oteo, and J. Ros, Phys. Rep.470, 151–238 (2009)

    work page 2009

  72. [72]

    Barzen and F

    J. Barzen and F. Leymann, AppliedMath5, 121 (2025)

    work page 2025

  73. [73]

    Wiersema, E

    R. Wiersema, E. K¨ okc¨ u, A. F. Kemper, and B. N. Bakalov, npj Quantum Inf.10, 110 (2024)

    work page 2024

  74. [74]

    J. M. Lee,Introduction to Smooth Manifolds, 2nd ed. (Springer, 2012)

    work page 2012

  75. [75]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre,Optimization Algorithms on Matrix Manifolds(Princeton University Press, Princeton, NJ, 2008)

    work page 2008

  76. [76]

    M. P. do Carmo,Riemannian Geometry(Birkh¨ auser, 1992)

    work page 1992

  77. [77]

    compensated

    E. Younis, C. C. Iancu, W. Lavrijsen, M. Davis, and E. Smith, Berkeley quantum synthesis toolkit (bqskit) v1, [Computer Software]https://doi.org/10.11578/ dc.20210603.2(2021). 8 END MATTER Scaling of the controlled evolution –Although con- trolled time evolution Ansatz falls outside of the scope of Theorem 1 and Corollary 3, we numerically show that it ca...

    work page 2021