Quantum Optimal Control at Intermediate Times: Controlling Revivals in Spin Chains

Ignacio R. Sol\'a; Juan J. Omiste

REVIEW 2 major objections 6 minor 48 references

Quantum Optimal Control at Intermediate Times: Controlling Revivals in Spin Chains

T0 review · 2 major / 6 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Mid-pulse targets become reachable in quantum optimal control

desk verdict Useful incremental extension of the costate-jump method to multiple intermediate times, with credible numerics; the zero-width convergence proof is a gap but not a fatal one. read the letter →

arxiv 2607.28783 v1 pith:KASQMUYO submitted 2026-07-30 quant-ph physics.comp-ph

Juan J. Omiste , Ignacio R. Solá This is my paper · ORCID

classification quant-phphysics.comp-ph

keywords quantumoptimalcontrolintermediate-timeobservablescostatejumpKrotovmethodDickestatesHeisenbergspinchainrevivalsfinite-widthmeasurementwindows

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper extends quantum optimal control so that a single driving pulse can maximize expectation values of observables at several chosen times during the evolution, not only at the final time. The key move is to treat each intermediate measurement as an infinitesimally narrow time window; the variational equations then acquire a jump in the costate at each such time. The authors implement the jump in an iterative optimal-control algorithm, argue that the iteration is monotonic and converges to the instantaneous-measurement limit, and demonstrate on a five-spin Heisenberg chain that one pulse can simultaneously shape population at an intermediate and a final time. If correct, this gives a practical recipe for tracking, protecting, and reviving quantum states on demand.

What carries the argument

The central object is the costate jump condition: each instantaneous measurement inserts a discontinuity into the Lagrange multiplier trajectory, with the jump update χ(T_k) ← χ(T_k) + O_k ψ(T_k). This jump is embedded in the standard backward-forward iterative optimal-control loop, while the finite-width window regularization O_τ(t) supplies the convergence rationale and the practical route to the zero-width limit.

What would settle it

Run the iterative algorithm with a positive-definite observable and record the objective after each iteration in the τ→0 limit: if the objective ever decreases from one iteration to the next, the monotonic-convergence claim is false. A direct numerical check on a system where the finite-width optima do not approach the delta-limit optimum would also test the limit-exchange assumption.

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Extended reading notes

Core claim

The paper claims that imposing an observable at an intermediate time T_k acts as a Dirac-delta term in the variational objective, and that this term forces the Lagrange multiplier (costate) to jump: the value just before T_k equals the value just after plus O_k times the state at T_k (Eq. 7). Inserting this jump into the standard backward-forward iteration makes the algorithm monotonic for positive-definite observables, and the delta limit is recovered by narrowing finite-width measurement windows. On a five-spin Heisenberg XXX chain restricted to one excitation, the method reaches about 0.999 population of the target Dicke state at the final time and about 0.993 at the intermediate time, an

Load-bearing premise

The claim that the iterative algorithm converges in the zero-window limit transfers the proof of monotonic convergence from each finite-width window to the discontinuous limit without proving the exchange of limit and optimization.

Editorial extensions

If this is right

A single optimized pulse can maximize the same observable at two or more times, so intermediate and final objectives can be balanced rather than prioritized.
The instantaneous-measurement protocol is the limit of finite-width measurement windows, so finite temporal resolution can be modeled explicitly and the protocol inherits a convergence guarantee in that limit.
Post-pulse field-free revivals can be enhanced or created at chosen times; creating revivals at low-baseline times costs final fidelity, while enhancing existing revivals is cheaper.
After an intermediate measurement, the optimization exploits transitions among all populated eigenstates, not only ground-state pathways, as seen in the field power spectrum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same jump-condition formalism should admit distinct observables at different times, not just repeated measurements of one operator; this would let a pulse track one quantity early and another late, a step the paper gestures at but does not test.
A formal proof of monotonic convergence in the zero-width limit is not given; one can test convergence numerically on more complex systems where the optimization landscape is not as forgiving.
Finite-width windows may serve as a built-in robustness knob: broader windows deliberately sacrifice peak value to tolerate timing jitter, which could be exploited in experiments with limited time resolution.
Applied to larger chains or interacting many-body systems, the revival-control scheme could be used to engineer dynamical decoupling or to prepare non-equilibrium states at prescribed times.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a clean, practical extension of the authors' earlier single-time costate-jump formalism (Ref. [26]) to multiple intermediate observables, implemented in Krotov and demonstrated on a 5-spin XXX chain. The variational derivation of Eq. (7) is internally consistent, and the numerics in Sec. III.A show the expected smooth approach to the Dirac-delta limit. The physical trade-off between intermediate targets and final population is clearly illustrated, and the post-pulse revival control is a nice application. Credit where due: the method works in practice, and the authors are honest about the cost of forcing revivals at low-baseline times.

The soft spots are real but not fatal. The convergence claim in Sec. II.B.2 is asserted, not proved: convergence for each finite-width O_τ(t) does not imply the τ→0 limit inherits monotonicity, and the cited theorem [34] does not apply to a sum of Dirac deltas. This is a formal gap, and the numerics are supportive but not conclusive—no iteration counts, no systematic τ-scaling of errors, no code released. The 'tracking' language in the abstract and conclusions is an overreach; these are open-loop optimizations at a handful of times, not continuous tracking. The citation to Ref. [26] is appropriate; self-citation is not a problem here.

The paper is not a paradigm shift, but it is a solid incremental contribution that QOC practitioners will find useful. The proof gap should be addressed in revision—either a real exchange-of-limits argument or a softened claim that the method converges in practice. With that, I'd take it as a solid paper. Deserves a serious referee.

Referee Report

2 major / 6 minor

Summary. The paper extends quantum optimal control (QOC) to objectives that are superpositions of instantaneous expectation values at multiple intermediate times, represented by Dirac-delta window functions O(t)=Σ_k O_k δ(t−T_k). Starting from the variational QOC functional, the authors derive a jump condition for the costate at each measurement time (Eq. (7)). They implement this in a Krotov-based scheme, applying it to a five-spin Heisenberg XXX chain in the single-excitation manifold, with the goal of maximizing the population of a target Dicke state at one intermediate time and at the final time. They compare the Dirac-delta limit with finite-width window regularizations, study the resulting control fields and their power spectra, and apply the method to post-pulse, field-free revivals, demonstrating a trade-off between intermediate and final objectives.

Significance. If the convergence claim is correct, the framework is a useful extension of QOC to multi-time tracking and to control problems in which observables must be maximized at specified intermediate instants. The variational derivation of the jump condition is internally consistent, and the numerical results are physically plausible and demonstrate a clear effect of intermediate measurements on the optimized fields and populations. The relation to the single-measurement formalism of Ref. [26] is clearly acknowledged, and the multi-time extension together with the spin-chain applications is a reasonable incremental contribution. However, the advertised rigorous validation of the τ→0 limit is not actually supplied: the proof in Sec. II.B.2 is only a heuristic argument, and the numerics in Sec. III.A are qualitative. The central claim therefore needs either a direct convergence proof or a clearly softened statement supported by quantitative diagnostics.

major comments (2)

[Sec. II.B.2, Eq. (11)] The transfer of Krotov convergence from the regularized objective O_τ(t) to the τ→0 discontinuous problem is asserted rather than proved. The sentence 'Krotov algorithm converges for O_τ(t) for all τ, including τ→0' does not follow from the cited theorem [34], since that theorem's hypotheses (smoothness/boundedness of the time-dependent objective) fail for a finite sum of Dirac deltas, and monotone convergence for each τ>0 does not imply that the limiting algorithm with the jump update (Eq. (10)) inherits monotonicity. In the standard Krotov monotonicity proof the cross-term involves the time-dependent objective; with delta sources that term becomes a discrete sum, and its non-negativity is exactly what needs to be established. Because the abstract and Introduction advertise rigorous convergence, this gap is load-bearing. Please provide a direct proof of monotone convergence for the jump
[Sec. III.A, Figs. 1–2] The numerical evidence for the τ→0 limit is qualitative. The paper reports that even for τ=0.5 and 2 the population dynamics 'converge closely but do not coincide exactly,' and attributes the discrepancy to numerical instabilities. No iteration counts, no per-iteration monotonicity data for the optimized functional, and no quantitative error measure (e.g., sup-norm or L2 difference between finite-τ and τ→0 results as a function of τ) are given. Without these diagnostics, the claim of convergence in the zero-window limit rests on visual closeness in selected plots. Adding such metrics would also help distinguish genuine convergence from numerical artifacts.

minor comments (6)

[Throughout] There are typos: 'fullﬁlled' and 'fullﬁlls' should be 'fulfilled' and 'fulfills'; 'aknowledged' should be 'acknowledged' in the Acknowledgments.
[Eq. (11)] The functions f_k(t−T_k,τ) are introduced in Sec. II.B.2 but only defined later in Sec. III.A, Eq. (16). Define them earlier or state explicitly that they are positive nascent deltas with unit integral.
[Sec. II.B.2] The text says O_τ(t) is 'positive definite.' For projectors, the appropriate term is 'positive semidefinite.'
[Sec. III.A] The sentence 'Specifically, we maximize the population of |5⟩ at T_int=100 and T_end=200 for FWHM τ=0.5, 2, 5, 10 at each measurement time' appears twice, once before and once after Eq. (15). Remove the duplication.
[Eqs. (8)–(10)] The one-sided limits in Eq. (7) are clear, but the update in Eq. (10) would benefit from explicit left/right notation, e.g., χ(T_k^-) ← χ(T_k^+) + O_k ψ(T_k), to remove any ambiguity about the direction of backward propagation.
[Figs. 2 and 6] The captions could state more explicitly that the plotted quantity is the optimal control field. The y-axis ranges differ between panels in Fig. 6; a note explaining this would avoid misleading comparisons.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the multi-time costate jump is derived from the stated variational principle, and the τ→0 claim is a proof gap rather than a circular reduction.

full rationale

I walked the derivation chain. The objective (Eq. 6) is a sum of Dirac-delta observables; inserting it into the Euler-Lagrange equation (4b) and integrating around T_k yields the jump condition (7). Equation (10) is the direct implementation of that condition in the backward costate propagation, not an independent predictive claim. The single-measurement boundary condition (9) follows from transversality χ(T_end)=0, not from a fitted or predefined ansatz. No parameter is fitted to reproduce the numerical targets: α and S(t) are control parameters, and the finite-width comparison in Sec. III.A is a self-consistency check between O_τ(t) and the delta-limit algorithm. The convergence argument invokes the external Krotov theorem [34] for positive-definite time-dependent observables. The sentence in Sec. II.B.2, 'Hence, Krotov algorithm converges for O_τ(t) for all τ, including τ→0', does assert an unproved exchange of limits, and I flag that as a proof gap, but convergence for each finite τ is not, by construction, equivalent to convergence of the discontinuous limit algorithm, so it is not a circularity. Ref. [26] is the authors' own predecessor for the variational method and the single-measurement result, but the multi-time jump is re-derived here and the extension has independent content; the self-citation is not load-bearing. Overall: no circular reduction found.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The framework introduces no new physical entities, particles, forces, or dimensions. The only hand-chosen quantities are numerical parameters (α, envelope times) that shape the demonstration but are not fitted to reproduce the target results. The key assumptions are the standard variational calculus of QOC, the monotonic-convergence property of Krotov for positive-definite objectives, and the limit-transfer from finite-width windows to delta functions.

free parameters (2)

penalty factor α = 0.1
Hand-chosen in Eq. (3) to weight field fluence against the objective. It affects the specific optimal fields and trade-off magnitudes in Sec. III, but not the general framework.
pulse envelope times t_on, t_off = 10, 5
Chosen in Eq. (14) to ramp the control field smoothly. They are simulation settings, not fitted to the target results.

assumptions (6)

standard math Variational calculus with ψ and ψ* as independent fields yields the Euler-Lagrange equations (4a)-(4c) and transversality condition (5).
Invoked in Sec. II.A without proof; this is the standard constrained-optimization formulation in quantum optimal control (Werschnik-Gross, Ref. [45]).
domain assumption Krotov's algorithm converges monotonically for positive-definite time-dependent observables.
Used in Sec. II.B.2, relying on Ref. [34]. The cited result covers finite-width time-dependent targets; the τ→0 extension is asserted.
ad hoc to paper The limit τ→0 of the finite-width regularized problems is the Dirac-delta problem, and optimization commutes with this limit.
This is the load-bearing premise of the convergence claim; asserted in Sec. II.B.2 and validated numerically in Sec. III.A, but not rigorously proven.
domain assumption The Heisenberg XXX spin chain with [H,S_z]=0 can be restricted to the single-excitation Dicke subspace.
Stated in Sec. II.C; reduces the Hilbert space from dimension 32 to 5 and defines the model used in all demonstrations.
domain assumption The target operator O=|5⟩⟨5| is positive definite, satisfying the monotonic-convergence condition.
Used implicitly in Sec. II.B.2; rank-one projectors are positive semidefinite, so the convergence condition holds for the chosen application.
domain assumption Optimizing expectation values at intermediate times models 'tracking' or 'measuring' the quantum state without accounting for measurement backaction.
The abstract and Sec. IV use 'tracking/measuring' language, but the computation is open-loop expectation-value optimization; no measurement collapse or feedback is modeled.

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Pith. "Pith review of Quantum Optimal Control at Intermediate Times: Controlling Revivals in Spin Chains." pith.science (2026). https://pith.science/paper/KASQMUYO

@misc{pith2026260728783,
  author       = {Pith},
  title        = {Pith review of: Quantum Optimal Control at Intermediate Times: Controlling Revivals in Spin Chains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KASQMUYO}},
  note         = {Machine review of arXiv:2607.28783}
}

read the original abstract

Standard Quantum Optimal Control (QOC) protocols typically maximize a physical objective just at a final target time. However, tracking or measuring the quantum state during the time evolution requires the control at intermediate stages of their evolution. In this work, we extend QOC to accommodate the simultaneous optimization of observables at arbitrary intermediate times. Using a variational approach, we show that intermediate observations induce discontinuities in the costate trajectory, which can be handled with a Krotov algorithm leading to monotonic optimization and convergent results in the limit of zero temporal measurement windows. We apply this multi-time formulation to a Heisenberg XXX spin chain to control the propagation, including field-free revivals, of a Dicke state excitation. Our results demonstrate that simultaneous optimization of the same observable reshapes the driving field to balance intermediate targets with final populations. Finally, we show how this framework enables dynamic tracking of spin excitations and the active manipulation of post-pulse quantum state revivals.

Figures

Figures reproduced from arXiv: 2607.28783 by the authors.

**Figure 3.** FIG. 3. Population of the eigenstates [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** FIG. 4. Power spectrum of the optimal driving field, [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 2.** Figure 2: FIG. 2. Optimal control field as a function of time for an [PITH_FULL_IMAGE:figures/full_fig_p005_2.png]

**Figure 5.** Figure 5: (solid red), where we identify several maxima. We then apply the intermediate-time QOC to maximize the revivals for selected times, specifically those at t = 109.7, 127.35, 161.5, and 165.5, and the expectation value at the final time Tend = 200. Our results show that…

**Figure 6.** Figure 6: FIG. 6. Optimal field for the dynamics shown in Fig. 5. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Population of [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

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T.-S. Ho and H. Rabitz, “Accelerated monotonic con- vergence of optimal control over quantum dynamics,” Phys. Rev. E 82, 026703 (2010)

2010
[42]

Fast-kick-oﬀ monotonically convergent al- gorithm for searching optimal control ﬁelds,

S.-L. Liao, T.-S. Ho, S.-I. Chu, and H. Rab- itz, “Fast-kick-oﬀ monotonically convergent al- gorithm for searching optimal control ﬁelds,” Phys. Rev. A 84, 031401(R) (2011)

2011
[43]

Kro- tov A Python implementation of Krotov’s method for quantum optimal control,

M. H. Goerz, D. Basilewitsch, F. Gago-Encinas, M. G. Krauss, K. P. Horn, D. M. Reich, and C. P. Koch, “Kro- tov A Python implementation of Krotov’s method for quantum optimal control,” SciPost Phys. 7, 80 (2019)

2019
[44]

Searching for quantum opti- mal controls under severe constraints,

G. Riviello, K. M. Tibbetts, C. Brif, R. Long, R. B. Wu, T. S. Ho, and H. Rabitz, “Searching for quantum opti- mal controls under severe constraints,” Phys. Rev. A 91, 043401 (2015)

2015
[45]

Frequency-domain quantum optimal control under mul- tiple constraints,

C.-C. Shu, T.-S. Ho, X. Xing, and H. Rabitz, “Frequency-domain quantum optimal control under mul- tiple constraints,” Phys. Rev. A 93, 033417 (2016)

2016
[46]

Monotonically con- vergent quantum optimal control with exact equality con- straints,

C.-C. Shu, T.-S. Ho, and H. Rabitz, “Monotonically con- vergent quantum optimal control with exact equality con- straints,” Phys. Rev. A 93, 053418 (2016)

2016
[47]

Quantum optimal control theory,

J. Werschnik and E. K. U. Gross, “Quantum optimal control theory,” J. Phys. B At. Mol. Opt. Phys. 40, R175–R211 (2007)

2007
[48]

Quantum tracking control of the orientation of symmetric-top molecules,

A. B. Magann, T.-S. Ho, C. Arenz, and H. A Rabitz, “Quantum tracking control of the orientation of symmetric-top molecules,” Phys. Rev. A 108, 033106 (2023)

2023

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [26]

Preparing spin-squeezed states in Rydberg atom arrays via quan- tum optimal control,

E. S. Carrera, H. Erbin, and G. Misguich, “Preparing spin-squeezed states in Rydberg atom arrays via quan- tum optimal control,” Phys. Rev. A 112 (2025)

2025

[2] [34]

Optimal con- trol theory for unitary transformations,

J. P. Palao and R. Kosloﬀ, “Optimal con- trol theory for unitary transformations,” Phys. Rev. A 68, 062308 (2003)

2003

[3] [1]

Inserting this observable in Eq

Single measurement time Let us consider ﬁrst the single measurement case, O(t) = Oδ(t−T ). Inserting this observable in Eq. (7), we obtain lim ǫ→0+ χ(Ω,T −ǫ) = lim ǫ→0+ χ(Ω,T +ǫ) +Oψ(Ω,T ). (8) Since χ(Ω,t ) fulﬁlls the TDSE in the interval t ∈ (T,T end),χ(Ω,T +ǫ) = 0 for ǫ> 0, Therefore, we obtain, for χ(Ω,T −ǫ), lim ǫ→0+ χ(Ω,T −ǫ) = Oψ(Ω,T ), (9) which ...

[4] [2]

(7) can be easily im- plemented in the algorithm to solve the QOC equations using Krotov’s algorithm [45]

Method implementation and numerical convergence of Krotov algorithm The discontinuities given by Eq. (7) can be easily im- plemented in the algorithm to solve the QOC equations using Krotov’s algorithm [45]. As described in Ref. [26], the propagation end time can be set to be the largest measurement time, Tend = max( Tk). Hence, an ob- servable O(t) = Oδ(...

[5] [3]

Quantum 8 optimal control in quantum technologies. Strategic re- port on current status, visions and goals for research in Europe,

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, “Quantum 8 optimal control in quantum technologies. Strategic re- port on current status, visions and goals for research in Europe,” EPJ Quantum Technol. 9, 19 (2022)

2022

[6] [4]

Op- timal control of quantum-mechanical systems: Ex- istence, numerical approximation, and applications,

A. P. Peirce, M. A. Dahleh, and H. Rabitz, “Op- timal control of quantum-mechanical systems: Ex- istence, numerical approximation, and applications,” Phys. Rev. A 37, 4950–4964 (1988)

1988

[7] [5]

Wavepacket dancing: Achiev- ing chemical selectivity by shaping light pulses,

R. Kosloﬀ, S. A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, “Wavepacket dancing: Achiev- ing chemical selectivity by shaping light pulses,” Chem. Phys. 139, 201–220 (1989)

1989

[8] [6]

Diﬀerentiating Be- tween Enantiomers with Nuclear Quadrupole Coupling Using Microwave Three-Wave Mixing,

F. E. L. Bergg¨ otz, M. Leibscher, W. Sun, C. P. Koch, and M. Schnell, “Diﬀerentiating Be- tween Enantiomers with Nuclear Quadrupole Coupling Using Microwave Three-Wave Mixing,” J. Phys. Chem. Lett. 16, 12087–12094 (2025)

2025