pith. sign in

arxiv: 2512.13549 · v2 · pith:ID3SNPU4new · submitted 2025-12-15 · 🪐 quant-ph · math.OC

Pontryagin Maximum Principle for Rydberg-blockaded state-to-state transfers: A semi-analytic approach

Federico Alberto Astolfi , Sven Jandura , Guido Pupillo This is my paper

Pith reviewed 2026-05-21 16:49 UTC · model grok-4.3

classification 🪐 quant-ph math.OC
keywords Rydberg blockadePontryagin Maximum Principleoptimal quantum controlneutral-atom qubitsstate-to-state transfersemi-analytic methodquartic potentialtime-optimal gates
0
0 comments X

The pith

Block-diagonalization of the Rydberg Hamiltonian lets the Pontryagin Maximum Principle map optimal laser detuning to classical motion in a quartic potential.

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

The paper develops a semi-analytic method to find time-optimal laser controls for state-to-state transfers in neutral-atom qubits under Rydberg blockade. It begins by block-diagonalizing the Hamiltonian to simplify the multi-qubit dynamics, then applies the Pontryagin Maximum Principle to classify extremals. For normal extremals in the two-qubit case, the laser detuning is shown to correspond to the position of a classical particle moving in a quartic potential, turning the control problem into a reduced classical mechanics task. This hybrid analytic-numerical approach aims to deliver high-fidelity gates faster than pure numerical searches alone.

Core claim

Block-diagonalization of the Hamiltonian in the Rydberg blockade regime simplifies the dynamics enough to permit a semi-analytic Pontryagin Maximum Principle treatment. For N qubits a general formalism is given; for two qubits normal extremals are mapped to the motion of a classical particle in a quartic potential, with laser detuning playing the role of the dynamical variable. This correspondence reduces the search for time-optimal controls to solving classical equations of motion, which can then be combined with numerical refinement.

What carries the argument

The correspondence that equates laser detuning from atomic transitions to the position of a classical particle in a quartic potential, which carries the argument by converting the optimal-control problem into a lower-dimensional classical mechanics problem for normal extremals.

If this is right

  • Time-optimal two-qubit gates can be obtained by solving the classical equations for a particle in a quartic potential and then refining with numerics.
  • Abnormal extremals are either absent or suboptimal for the cases examined, so the normal-extremal solutions dominate the search.
  • The same block-diagonalization plus Pontryagin procedure extends to N-qubit systems, giving a general formalism for larger neutral-atom registers.
  • High-fidelity controls emerge from the combination of the analytic reduction and numerical polishing rather than from either method alone.

Where Pith is reading between the lines

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

  • The quartic-potential picture may supply analytic approximations for systems too large for exhaustive numerical search.
  • Similar effective-potential reductions could appear in other blockade-based quantum control settings outside neutral atoms.
  • Hardware tests of the derived pulses would show how robust the semi-analytic solutions remain under realistic noise and decoherence.
  • The method suggests that classical-mechanics analogies can systematically guide gate design in strongly interacting quantum platforms.

Load-bearing premise

Block-diagonalization of the Hamiltonian sufficiently simplifies the dynamics to allow direct application of a semi-analytic Pontryagin Maximum Principle approach in the Rydberg blockade regime.

What would settle it

A full numerical optimal-control calculation or hardware experiment that produces a shorter-duration pulse sequence with equal or higher fidelity than the one obtained from the quartic-potential reduction.

Figures

Figures reproduced from arXiv: 2512.13549 by Federico Alberto Astolfi, Guido Pupillo, Sven Jandura.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Schematic approach to OC problems. A [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Trajectories on the Bloch sphere corresponding to different normal extremals obtained with the PMP. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Depicted are [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
read the original abstract

We study time-optimal state-to-state control for two- and multi-qubit operations motivated by neutral-atom quantum processors within the Rydberg blockade regime. Block-diagonalization of the Hamiltonian simplifies the dynamics and enables the application of a semi-analytic approach to the Pontryagin Maximum Principle to derive optimal laser controls. We provide a general formalism for $N$ qubits. For $N=2$ qubits, we classify normal and abnormal extremals, showcasing examples where abnormal solutions are either absent or suboptimal. For normal extremals, we establish a correspondence between the laser detuning from atomic transitions and the motion of a classical particle in a quartic potential, yielding a reduced, semi-analytic formulation of the control problem. Combining PMP-based insights with numerical optimization, our approach bridges analytic and computational methods for high-fidelity, time-optimal control.

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 develops a semi-analytic framework for time-optimal state-to-state control of Rydberg-blockaded neutral-atom qubits by block-diagonalizing the Hamiltonian and applying the Pontryagin Maximum Principle (PMP). For general N it provides a formalism; for N=2 it classifies normal and abnormal extremals, shows abnormal solutions are often absent or suboptimal, and maps laser detuning for normal extremals to classical motion in a quartic potential, thereby reducing the control problem. The approach is combined with numerical optimization to target high-fidelity gates.

Significance. If the reduction is shown to preserve optimality and boundary conditions in the full space, the quartic-potential correspondence supplies a concrete analytic handle on optimal detuning schedules that could accelerate gate design in Rydberg processors. The explicit classification of extremals and the hybrid analytic-numerical strategy are useful contributions to quantum control literature.

major comments (2)
  1. [General formalism for N qubits and the N=2 normal-extremal analysis] The central reduction step (block-diagonalization followed by PMP) is load-bearing for the claim of semi-analytic optimality. The manuscript does not explicitly demonstrate that the reduced PMP equations and the quartic-potential mapping preserve the original state-to-state boundary conditions (initial |00…⟩ to target state) once the neglected off-block terms are restored; without this verification the derived controls may satisfy the reduced dynamics yet fail to be feasible or optimal in the full Hilbert space.
  2. [Classification of normal and abnormal extremals for N=2] The classification that abnormal extremals are absent or suboptimal for the N=2 case rests on the reduced Hamiltonian; it is not shown whether reintroducing the blockade-approximation error terms can promote an abnormal extremal to optimality or alter the normal/abnormal distinction.
minor comments (2)
  1. [Normal extremals and quartic-potential correspondence] Notation for the reduced control variable (laser detuning) and the quartic potential parameters should be introduced with explicit reference to the original Hamiltonian matrix elements to avoid ambiguity when readers reconstruct the mapping.
  2. [Numerical optimization section] The abstract states that the method yields 'high-fidelity' controls, yet no quantitative fidelity or gate-error metrics are referenced in the summary of numerical results; adding a brief table or statement of achieved infidelities would strengthen the bridge between analytic and computational parts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below, clarifying the scope of the blockade approximation and agreeing to strengthen the presentation where needed.

read point-by-point responses

  1. Referee: [General formalism for N qubits and the N=2 normal-extremal analysis] The central reduction step (block-diagonalization followed by PMP) is load-bearing for the claim of semi-analytic optimality. The manuscript does not explicitly demonstrate that the reduced PMP equations and the quartic-potential mapping preserve the original state-to-state boundary conditions (initial |00…⟩ to target state) once the neglected off-block terms are restored; without this verification the derived controls may satisfy the reduced dynamics yet fail to be feasible or optimal in the full Hilbert space.

    Authors: The block-diagonalization is exact in the perfect Rydberg blockade limit (infinite interaction strength), where off-block matrix elements vanish identically; the reduced dynamics therefore coincide with the full dynamics inside the computational subspace, and the initial |00…⟩ to target boundary conditions are preserved by construction. For finite but large blockade strengths the derived controls serve as high-quality initial guesses that are subsequently refined by numerical optimization on the full Hamiltonian, as already illustrated in the numerical examples of the manuscript. We will revise the text to state this limit explicitly, add a short perturbation argument showing that small off-block terms do not alter the leading-order boundary matching, and include a brief numerical check confirming that the PMP-derived schedules remain near-optimal when the full Hamiltonian is restored. revision: yes

  2. Referee: [Classification of normal and abnormal extremals for N=2] The classification that abnormal extremals are absent or suboptimal for the N=2 case rests on the reduced Hamiltonian; it is not shown whether reintroducing the blockade-approximation error terms can promote an abnormal extremal to optimality or alter the normal/abnormal distinction.

    Authors: The normal/abnormal classification is performed on the exact block-diagonalized Hamiltonian that governs the perfect-blockade regime. Within this model abnormal extremals are either absent or yield longer transfer times than the normal ones we retain. Because the neglected off-block terms are suppressed by the large blockade energy, they act as a small perturbation that cannot change the leading-order optimality ranking; any candidate abnormal control would still be penalized by the same cost functional. We will add a short robustness paragraph that invokes this perturbative argument and supports it with a numerical comparison of normal versus abnormal candidates on the full Hamiltonian for representative finite-blockade strengths. revision: yes

Circularity Check

0 steps flagged

PMP derivation on block-diagonalized Hamiltonian is self-contained

full rationale

The paper derives the correspondence between laser detuning and quartic-potential motion by applying the Pontryagin Maximum Principle directly to the block-diagonalized N-qubit Rydberg Hamiltonian. No quoted step reduces a claimed prediction or extremal classification to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the reduced semi-analytic formulation is obtained from the simplified dynamics and boundary conditions of the original control problem. The approach is therefore independent of its target results and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the physical validity of the Rydberg blockade regime and the mathematical simplification achieved by block-diagonalization; no free parameters or new entities are mentioned in the abstract.

axioms (1)
  • domain assumption The Rydberg blockade regime permits block-diagonalization of the multi-qubit Hamiltonian that simplifies the dynamics enough for semi-analytic PMP application.
    Invoked in the abstract as the foundation for studying neutral-atom processors.

pith-pipeline@v0.9.0 · 5677 in / 1263 out tokens · 51094 ms · 2026-05-21T16:49:48.598075+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

81 extracted references · 81 canonical work pages

  1. [1]

    semi- analytic

    and [31]. The combination of analytic and numeri- cal methods justifies referring to our approach as “semi- analytic” and thus provides an interesting alternative to the fully-numerical solutions. Our analysis shows how the Pontryagin Maximum Principle, through its standard classification of candidate extremals, provides a powerful framework for uncoverin...

    work page

  2. [2]

    Any other excitation is forbidden. More generally, forNRydberg atoms, the new effec- tivek-th two-level system possesses a ground state|0⟩ k, realized when the firstkatoms occupy the|1⟩state and the remainingN−katoms remain in|0⟩. In Eq. (1), this effective ground state|0⟩ k is coupled to its correspond- ing excited state|1⟩ k by the laser field. Consiste...

    work page

  3. [3]

    the followingAdjoint Equationshold: ˙λ∗ =− ∂H ∂q∗

    work page

  4. [4]

    the followingMaximum Principleholds: H(t,q ∗,u ∗, λ∗ 0, λ∗) = max u∈U H(t,q ∗,u, λ ∗ 0, λ∗)

    work page

  5. [5]

    the pseudo-Hamiltonian vanishes along the optimal trajectory: H(t,q ∗,u ∗, λ∗ 0, λ∗) = 0

    work page

  6. [6]

    the costatesλ ∗ relative to some candidate trajectory are perpendicular at the final timeTto the target manifoldN

    the followingTransversality Conditionshold: ⟨λ∗(t), v⟩Tq∗ (T) N = 0 for anyv∈T q∗(T) N, i.e. the costatesλ ∗ relative to some candidate trajectory are perpendicular at the final timeTto the target manifoldN. It is standard [3, 4, 27, 60] to defineextremala trajectoryq ∗ satisfying the necessary conditions for optimality as stated above. Furthermore, extre...

    work page

  7. [7]

    (χ 0,|χ(t)⟩)̸= (0,0)

    work page

  8. [8]

    they are steered by the same time dependent Schr¨ odinger equation of|ψk⟩

    The costates satisfy |˙χk⟩=− ∂H ∂|ψk⟩ =−iH|χ k⟩, i.e. they are steered by the same time dependent Schr¨ odinger equation of|ψk⟩

    work page

  9. [9]

    The Maximum principle H |ψ(t)⟩,|χ(t)⟩, χ 0, φ(t) = max ξ H ψ(t), χ(t), χ0, ξ holds for every timet. In order to solve forφ, we impose dH dφ = 0 and find that cosφ= P k √ kIm⟨χ k|σx|ψk⟩ ω (12) and sinφ=− P k √ kIm⟨χ k|σy|ψk⟩ ω ,(13) with ω= vuut X k √ kIm⟨χ k|σx|ψk⟩ !2 + X k √ kIm⟨χ k|σy|ψk⟩ !2 (14) wheneverω̸= 0 (whenω= 0, the controlφcannot be expressed ...

    work page

  10. [10]

    (15) vanishes over the extremals and the associated costates: H |ψ(t)⟩,|χ(t)⟩, χ 0, φ(t) = 0 (16) at each timet

    The pseudo-Hamiltonian in Eq. (15) vanishes over the extremals and the associated costates: H |ψ(t)⟩,|χ(t)⟩, χ 0, φ(t) = 0 (16) at each timet. As mentioned previously in Sec. III, we define abnormal extremal an extremal for which χ0 = 0 and normal extremal one such thatχ 0 ̸= 0. Therefore, normal extremals (Sec. IV C) lead to ω̸= 0: the aforementioned des...

    work page

  11. [11]

    As an explicit example, we consider N= (eiαk |1⟩k)k :α k ∈[0,2π] ,(17) which is theN-TLSs generalization of the target manifold (5) ofCase (i)withN= 2

    The projection of the costates over the tangent space of the target manifoldNis null at the final timeT. As an explicit example, we consider N= (eiαk |1⟩k)k :α k ∈[0,2π] ,(17) which is theN-TLSs generalization of the target manifold (5) ofCase (i)withN= 2. The re- spective generalization forN-TLSs ofCase (ii) is similar and henceforth we omit here its ex-...

    work page

  12. [12]

    + √ 8 q r2 1 −z 2 1 q r2 2 −z 2 2 cos(ξ1 −ξ 2). 10 (a)∆ 0 = 1.26,V 0 =−1.17 (b)∆ + = 0.67, ∆− =−0.84,V 0 =−0.39 (c)∆ 0 = 0.77,V 0 = 0.02 (d)∆ + = 3.07, ∆− =−0.13,V 0 =−1.00 (e)∆ 0 = 1.01,V 0 =−0.15 (f )∆ + = 6.86, ∆− =−1.86,V 0 =−35.02 (g)1.26≤∆ 0 ≤1.50,V 0 =−1.17 (h)∆ 0 = 1.26,−1.50≤V 0 ≤ −0.8 FIG. 3: Trajectories on the Bloch sphere corresponding to dif...

    work page

  13. [13]

    (49) and (50), we obtainz 1 + 2z2 = 2∆ and z1 +z 2 =C

    = 16 ˙z2 1 +[4−(r 2 1 −z 2 1)−2(r 2 2 −z 2 1)]2.(55) From Eqs. (49) and (50), we obtainz 1 + 2z2 = 2∆ and z1 +z 2 =C. Hence, z1 =−2∆ + 2C,(56) z2 = 2∆−C.(57) Substituting this into Eq. (55) and using that ˙C= 0 (from Eq. (49)) gives 1 2 ˙∆2 +V(∆) = 0.(58) where V(∆) =c 4∆4 +c 3∆3 +c 2∆2 +c 1∆ +c 0 (59) with c4 = 1 8 , c 3 = 0, c 2 = −2C2 +r 2 1 −2r 2 2 + ...

    work page

  14. [14]

    that ∆(0) = ∆(T) = 0. Furthermore, the potential admits two real roots ∆ + >0 and ∆ − <0 such that the potential can be expressed as V(∆) = (∆−∆ +)(∆−∆ −) 1 8∆2 + 1 8 ∆+ + ∆− ∆ + V0 ∆+∆− .(66) We determine the extremals by proceeding as forCase (i)above and we find the same results of [31]. In par- ticular, in Fig. 4 (b) we confront the extremal controlφ ...

    work page 2021

  15. [15]

    Santambrogio,A Course in the Calculus of Variations: Optimization, Regularity, and Modeling(Springer, Uni- versitext, 2023)

    F. Santambrogio,A Course in the Calculus of Variations: Optimization, Regularity, and Modeling(Springer, Uni- versitext, 2023)

    work page 2023

  16. [16]

    Kaufman, Canadian Mathematical Bulletin7, 500 (1964)

    H. Kaufman, Canadian Mathematical Bulletin7, 500 (1964)

    work page 1964

  17. [17]

    Agrachev and Y

    A. Agrachev and Y. Sachkov,Control Theory from the Geometric Viewpoint(Springer-Verlag, Berlin Heidel- berg, 2004)

    work page 2004

  18. [18]

    Sch¨ attler and U

    H. Sch¨ attler and U. Ledzewicz,Geometric Optimal Con- trol: Theory, Methods and Examples(Springer, 2012)

    work page 2012

  19. [19]

    C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Fil- ipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte- Herbr¨ uggen, D. Sugny, and F. K. Wilhelm, EPJ Quan- tum Technology9, 19 (2022)

    work page 2022

  20. [20]

    Casalino and L

    L. Casalino and L. Mascolo, inNew Trends and Chal- lenges in Optimization Theory Applied to Space Engi- neering, Springer Optimization and Its Applications, Vol. 221 (Springer, Cham, 2025) pp. 9–22

    work page 2025

  21. [21]

    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, C. K¨ uhn, I. Kuprov, B. Luy, S. G. Schirmer, T. Schulte- Herbr¨ uggen, D. Sugny, and F. K. Wilhelm, European 14 Physical Journal D69, 279 (2015)

    work page 2015

  22. [22]

    M. H. Goerz, T. Calarco, and C. P. Koch, J. Phys. B: At. Mol. Opt. Phys.44, 154011 (2011)

    work page 2011

  23. [23]

    M. M. M¨ uller, D. M. Reich, M. Murphy, H. Yuan, J. Vala, K. B. Whaley, T. Calarco, and C. P. Koch, Phys. Rev. A 84, 042315 (2011)

    work page 2011

  24. [24]

    M. H. Goerz, E. J. Halperin, J. M. Aytac, C. P. Koch, and K. B. Whaley, Phys. Rev. A90, 032329 (2014)

    work page 2014

  25. [25]

    Omran, H

    A. Omran, H. Levine, A. Keesling, G. Semeghini, T. T. Wang, S. Ebadi, H. Bernien, A. S. Zibrov, H. Pichler, S. Choi, J. Cui, M. Rossignolo, P. Rembold, S. Mon- tangero, T. Calarco, M. Endres, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Science365, 570 (2019)

    work page 2019

  26. [26]

    J. Cui, R. van Bijnen, T. Pohl, S. Montangero, and T. Calarco, Quantum Sci. Technol.2, 035006 (2017)

    work page 2017

  27. [27]

    Smith, B

    A. Smith, B. E. Anderson, H. Sosa-Martinez, C. A. Ri- ofr´ ıo, I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett. 111, 170502 (2013)

    work page 2013

  28. [28]

    B. E. Anderson, H. Sosa-Martinez, C. A. Riofr´ ıo, I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett.114, 240401 (2015)

    work page 2015

  29. [29]

    Nebendahl, H

    V. Nebendahl, H. Haffner, and C. F. Roos, Phys. Rev. A 79, 012312 (2009)

    work page 2009

  30. [30]

    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

  31. [31]

    D. J. Egger and F. K. Wilhelm, Supercond. Sci. Technol. 27, 014001 (2014)

    work page 2014

  32. [32]

    Kelly, R

    J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, I.-C. Hoi, E. Jef- frey, A. Megrant, J. Mutus, C. Neill, P. J. J. O’Malley, C. Quintana, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, A. N. Cleland, and J. M. Mar- tinis, Phys. Rev. Lett.112, 240504 (2014)

    work page 2014

  33. [33]

    Huang and H.-S

    S.-Y. Huang and H.-S. Goan, Phys. Rev. A90, 012318 (2014)

    work page 2014

  34. [34]

    Werninghaus, D

    M. Werninghaus, D. J. Egger, F. Roy, S. Machnes, F. K. Wilhelm, and S. Filipp, Phys. Rev. A90, 032330 (2014)

    work page 2014

  35. [35]

    D’Alessandro,Introduction to Quantum Control and Dynamics, Applied Mathematics and Nonlinear Science Series (Chapman & Hall/CRC, Boca Raton, FL, 2007)

    D. D’Alessandro,Introduction to Quantum Control and Dynamics, Applied Mathematics and Nonlinear Science Series (Chapman & Hall/CRC, Boca Raton, FL, 2007)

    work page 2007

  36. [36]

    Ansel, E

    Q. Ansel, E. Dionis, F. Arrouas, B. Peaudecerf, S. Gu´ erin, D. Gu´ ery-Odelin, and D. Sugny, J. Phys. B: At. Mol. Opt. Phys.57, 133001 (2024)

    work page 2024

  37. [37]

    Garon, S

    A. Garon, S. J. Glaser, and D. Sugny, Phys. Rev. A88, 043422 (2013)

    work page 2013

  38. [38]

    Albertini and D

    F. Albertini and D. D’Alessandro, Journal of Mathemat- ical Physics56, 012106 (2015)

    work page 2015

  39. [39]

    Fresse-Colson, S

    O. Fresse-Colson, S. Gu´ erin, X. Chen, and D. Sugny, Physical Review A112, 022618 (2025)

    work page 2025

  40. [40]

    Romano, Phys

    R. Romano, Phys. Rev. A90, 062302 (2014)

    work page 2014

  41. [41]

    Boscain, M

    U. Boscain, M. Sigalotti, and D. Sugny, PRX Quantum 2, 030203 (2021)

    work page 2021

  42. [42]

    Khaneja, T

    N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen, and S. J. Glaser, Journal of Magnetic Resonance172, 296 (2005)

    work page 2005

  43. [43]

    D. M. Reich, M. Ndong, and C. P. Koch, Journal of Chemical Physics136, 104103 (2012)

    work page 2012

  44. [44]

    Werschnik and E

    J. Werschnik and E. K. U. Gross, J. Phys. B: At. Mol. Opt. Phys.40, R175 (2007)

    work page 2007

  45. [45]

    Jandura and G

    S. Jandura and G. Pupillo, Quantum6, 712 (2022)

    work page 2022

  46. [46]

    Fromonteil, R

    C. Fromonteil, R. Tricarico, F. Cesa, and H. Pichler, Phys. Rev. Research6, 033333 (2024)

    work page 2024

  47. [47]

    Bergonzoni, S

    M. Bergonzoni, S. Jandura, and G. Pupillo, Physical Re- view A112(2025)

    work page 2025

  48. [48]

    Mohan, R

    M. Mohan, R. de Keijzer, and S. Kokkelmans, Phys. Rev. Res.5, 033052 (2023)

    work page 2023

  49. [49]

    R. J. P. T. de Keijzer, L. Y. Visser, O. Tse, and S. J. J. M. Kokkelmans, Phys. Rev. Res.7, 023063 (2025)

    work page 2025

  50. [50]

    R. J. P. T. de Keijzer, L. Y. Visser, O. Tse, and S. J. J. M. F. Kokkelmans, Phys. Rev. A111, 052625 (2025)

    work page 2025

  51. [51]

    Delakouras, G

    A. Delakouras, G. Doultsinos, and D. Petrosyan, arXiv preprint arXiv:2507.16602 10.48550/arXiv.2507.16602 (2025), arXiv:2507.16602 [quant-ph]

    work page doi:10.48550/arxiv.2507.16602 2025

  52. [52]

    T. F. Gallagher,Rydberg Atoms, Cambridge Monographs on Atomic, Molecular and Chemical Physics (Cambridge University Press, Cambridge, 1994)

    work page 1994

  53. [53]

    Morgado and S

    M. Morgado and S. Whitlock, AVS Quantum Science3, 023501 (2021)

    work page 2021

  54. [54]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod. Phys.82, 2313 (2010)

    work page 2010

  55. [55]

    Saffman, Journal of Physics B: Atomic, Molecular and Optical Physics49, 202001 (2016)

    M. Saffman, Journal of Physics B: Atomic, Molecular and Optical Physics49, 202001 (2016)

    work page 2016

  56. [56]

    Levine, A

    H. Levine, A. Keesling, G. Semeghini, A. Omran, T. T. Wang, S. Ebadi, H. Bernien, M. Greiner, V. Vuleti´ c, H. Pichler, and M. D. Lukin, Phys. Rev. Lett.123, 170503 (2019)

    work page 2019

  57. [57]

    S. Ma, G. Liu, P. Peng, S. Jandura, G. Pupillo,et al., Nature622, 279 (2023)

    work page 2023

  58. [58]

    T. M. Graham, M. Kwon, B. Grinkemeyer, Z. Marra, X. Jiang, M. T. Lichtman, Y. Sun, M. Ebert, and M. Saffman, Physical Review Letters123, 230501 (2019)

    work page 2019

  59. [59]

    Pagano, S

    A. Pagano, S. Weber, D. Jaschke, T. Pfau, F. Meinert, S. Montangero, and H. P. B¨ uchler, Physical Review Re- search4, 033019 (2022)

    work page 2022

  60. [60]

    Jandura, J

    S. Jandura, J. D. Thompson, and G. Pupillo, PRX Quan- tum4, 020336 (2023)

    work page 2023

  61. [61]

    Evered, D

    S. Evered, D. Bluvstein, M. Kalinowski,et al., Nature 622, 268 (2023)

    work page 2023

  62. [62]

    A. G. Radnaev, W. C. Chung, D. C. Cole, D. Mason, T. G. Ballance,et al., PRX Quantum6, 030334 (2025)

    work page 2025

  63. [63]

    R. Tao, O. Lib, F. Gyger, H. Timme, M. Ammen- werth, I. Bloch, and J. Zeiher, arXiv preprint (2025), arXiv:2506.10714 [quant-ph]

    work page arXiv 2025

  64. [64]

    J. A. Muniz, M. Stone, D. T. Stack, M. Jaffe, J. M. Kindem,et al., PRX Quantum6, 020334 (2025)

    work page 2025

  65. [65]

    R. B.-S. Tsai, X. Sun, A. L. Shaw, R. Finkelstein, and M. Endres, PRX Quantum (2024)

    work page 2024

  66. [66]

    Semeghini, H

    G. Semeghini, H. Levine, A. Keesling, S. Ebadi, T. T. Wang, D. Bluvstein, R. Verresen, H. Pichler, M. Kali- nowski, R. Samajdar, A. Omran, S. Sachdev, A. Vish- wanath, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Science 374, 1242 (2021)

    work page 2021

  67. [67]

    A. W. Gl¨ atzle, M. Dalmonte, R. Nath, I. Rousochatzakis, R. Moessner, and P. Zoller, Physical Review X4, 041037 (2014)

    work page 2014

  68. [68]

    Browaeys and T

    A. Browaeys and T. Lahaye, Nature Physics16, 132 (2020)

    work page 2020

  69. [69]

    Lahaye, C

    T. Lahaye, C. Menotti, L. Santos, and M. Lewenstein, Reports on Progress in Physics72, 126401 (2009)

    work page 2009

  70. [70]

    Gonz´ alez-Cuadra, M

    D. Gonz´ alez-Cuadra, M. Hamdan, T. V. Zache, and et al., Nature642, 321 (2025)

    work page 2025

  71. [71]

    Bluvstein, A

    D. Bluvstein, A. Omran, H. Levine, A. Keesling, G. Se- meghini, S. Ebadi, T. T. Wang, A. A. Michailidis, N. Maskara, W. W. Ho, S. Choi, M. Serbyn, M. Greiner, V. Vuleti´ c, and M. D. Lukin, Science371, 1355 (2021). 15

    work page 2021

  72. [72]

    Labuhn, D

    H. Labuhn, D. Barredo, S. Ravets, S. de L´ es´ eleuc, T. Macr` ı, T. Lahaye, and A. Browaeys, Nature534, 667 (2016)

    work page 2016

  73. [73]

    A. Cao, W. J. Eckner, T. Lukin, Y. Yelin,et al., Nature 634, 315 (2024)

    work page 2024

  74. [74]

    Jandura,Optimized quantum gates for neutral atom quantum computers, phdthesis (2024), nNT: 2024STRAF027

    S. Jandura,Optimized quantum gates for neutral atom quantum computers, phdthesis (2024), nNT: 2024STRAF027

    work page 2024

  75. [75]

    Cesa and H

    F. Cesa and H. Pichler, Phys. Rev. Lett.131, 170601 (2023)

    work page 2023

  76. [76]

    Recall thatT q∗(t)M ≃R dimM

    work page

  77. [77]

    R. A. Adams and J. J. F. Fournier,Sobolev Spaces, 2nd ed. (Academic Press, 2003)

    work page 2003

  78. [78]

    Brezis,Functional Analysis, Sobolev Spaces and Par- tial Differential Equations(Springer, 2010)

    H. Brezis,Functional Analysis, Sobolev Spaces and Par- tial Differential Equations(Springer, 2010)

    work page 2010

  79. [79]

    M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

    work page 2010

  80. [80]

    See [31] for other symmetries relevant to the realization of general C-phase gates, including the CZ gate considered in the state-to-state transfer ofCase (ii)

    work page

Showing first 80 references.