Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

The paper claims that repeated short-horizon optimal control, with a terminal equality constraint, makes a quantum target exponentially stable and cuts computational cost.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Splitting quantum optimal control into repeated short-horizon MPC problems, with terminal constraints or optimized setpoints, gives faster and more robust qubit state preparation in simulations.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection The modular MPC framework is useful, but Theorem IV.1 rests on an assumption that is false for the pure-state representation used in all the numerics. the 4 major comments →

arxiv 2509.05167 v1 pith:AXTFDK6B submitted 2025-09-05 eess.SY cs.SYmath.OCquant-ph

Model predictive quantum control: A modular approach for efficient and robust quantum optimal control

Eya Guizani , Julian Berberich This is my paper
classification eess.SY cs.SYmath.OCquant-ph
keywords quantum optimal controlmodel predictive controlreceding-horizon controlclosed-loop quantum controlstability guaranteesterminal equality constraintsetpoint optimizationstate transfer
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

The paper tries to establish that model predictive control—solve a short-horizon quantum optimal control problem, apply only the first few input steps, remeasure, and repeat—can be applied to quantum systems with both practical and theoretical benefits. It claims this receding-horizon approach improves efficiency in open-loop use and robustness in closed-loop use, while remaining compatible with existing quantum control solvers. Its central theoretical result is that, when each short problem is required to end exactly at the target state, the scheme drives the fidelity to one exponentially fast. A setpoint-optimization variant relaxes that hard constraint and, in the numerical examples, achieves perfect fidelity with much shorter horizons and far lower runtime.

Core claim

The paper's central claim is that model predictive control—solve a short-horizon optimal control problem, apply only the first few input steps, reinitialize from the current state, and repeat—can be applied to quantum systems and delivers both practical and theoretical benefits. Its main theorem states that when each short problem is required to end exactly at the target state (a terminal equality constraint), the receding-horizon scheme makes the target exponentially stable: 1−F(X_t,X_ref) ≤ C γ^t (1−F(X_0,X_ref)). The proof uses the optimal cost of the short problem as a Lyapunov function; the key bound is that this cost is quadratically bounded by the distance to the target. The paper als

What carries the argument

The carrying mechanism is the receding-horizon loop: at each step, solve a shorter QOC problem of horizon L, apply only the first M steps, update the state by simulation (open loop) or measurement (closed loop), and repeat. For the stability guarantee, the machinery is the optimal value function J*(X) of the terminal-constrained problem, used as a Lyapunov function. The contraction factor γ = 1 − α/(c_u c_1) comes from combining the quadratic upper bound J*(X) ≤ c_u ||X−X_ref||^2 with the norm-fidelity inequality ||X1−X2||^2 ≤ c_1(1−F(X1,X2)) and the positive-definite stage cost. The setpoint-optimization variant replaces the hard terminal constraint by an artificial steady-state setpoint, p

Load-bearing premise

The proof assumes the cheapest way to exactly reach the target from a nearby state costs at most a constant times the squared distance to it; the experiments never verify this for the systems they simulate.

What would settle it

For the single-qubit Hamiltonian in Section VII B, compute the optimal terminal-constrained cost J*(X) for initial states approaching the target and plot J*/||X-X_ref||^2; if that ratio grows without bound as X approaches the target, Assumption IV.1.5 fails and the exponential bound of Theorem IV.1 cannot hold for that system.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • A long-horizon QOC problem can be decomposed into short subproblems with only minor performance loss; numerical results show runtime drops of roughly an order of magnitude while final fidelity stays at 1.
  • With feedback, MPQC becomes robust: for a single-qubit model with unknown drift error up to ±1, closed-loop terminal-constrained MPQC keeps final fidelity high where open-loop QOC degrades.
  • The exponential-stability theorem gives a quantitative convergence rate for state preparation whenever the terminal-constrained subproblem is feasible and the quadratic cost bound holds.
  • Setpoint optimization reduces the minimal prediction horizon from L=15-20 (terminal equality) to L=2 in the examples, cutting runtime to about one second while preserving unit fidelity.
  • Because the framework is modular, any existing QOC solver can be plugged into the basic scheme; only the guaranteed schemes require solvers that handle terminal constraints.

Where Pith is reading between the lines

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

  • If the efficiency gains persist at larger system sizes, MPQC could serve as a warm-start or approximation engine for pulse compilation, converting one hard long-horizon nonconvex problem into several easier short-horizon ones.
  • The Lyapunov argument suggests that under small measurement noise the scheme should inherit a form of input-to-state stability, but the paper does not prove this; a testable extension is to bound final infidelity by a function of the noise level.
  • For targets that are not eigenstates of the controlled Hamiltonian, the setpoint scheme can fail, as the paper notes; a rotating-frame or time-dependent setpoint variant might extend guarantees to that case.
  • Closed-loop MPQC currently assumes full state tomography, which is costly; the paper's suggestion of shadow tomography points to a concrete way to make the feedback loop practical, and one could test MPQC with very few measurement samples.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper introduces a modular model predictive control framework for quantum optimal control (MPQC). The basic idea is to replace a long-horizon quantum optimal control problem with a sequence of shorter-horizon problems, applying only the first M steps of each solution. Three schemes are presented: a basic scheme (Section III), a terminal-equality-constrained (TEC) scheme for which an exponential stability theorem is stated (Section IV), and a setpoint-optimization scheme (Section V). The paper also discusses how standard QOC algorithms such as GRAPE, Krotov's method, and CRAB can be used within the framework, and it reports single-qubit numerical experiments illustrating efficiency gains in open-loop use and robustness gains in closed-loop use.

Significance. If the stability theorem were fully valid for the systems treated in the numerical sections, the paper would be a useful bridge between classical MPC theory and quantum optimal control, and the modularity claim is attractive: existing QOC solvers could be reused in a receding-horizon scheme with formal guarantees. The paper also makes a fair effort to situate itself in the existing literature on MPC for quantum systems. However, the central theoretical result is currently stated for a state representation that is inconsistent with the pure-state formulation used in the examples, and one of the key numerical demonstrations violates a stated assumption of the theorem. The framework is promising, but the gap between the theorem and the evidence needs to be closed before the claims in the abstract can be accepted.

major comments (4)
  1. [Section IV, Assumption IV.1.1 and proof of Theorem IV.1] Assumption IV.1.1 is false for the pure-state representation used throughout the paper. In Section II A, X(t) is taken to be |ψ(t)> for pure states, and the fidelity is defined in Eq. (6) as |<ψ_ref|ψ>|^2. For any X1 and X2 = e^{iθ} X1 with θ not a multiple of 2π, we have F(X1,X2)=1 but ||X1-X2||^2 = |1-e^{iθ}|^2 ||X1||^2 > 0, so no finite c1 can satisfy (17). This inequality is used in Eq. (22) to turn the quadratic controllability bound (18) into a bound on J*(X_t) in terms of infidelity; without (17), the Lyapunov argument collapses. The statement after Assumption IV.1.1 that (17) holds 'for pure states with the diamond norm' does not fix this, because the diamond norm is not the norm induced by the state-vector representation and the numerical implementation in Appendix A works with complex vectors a+ib. A consistent repair would require reformulating the theory for density matrices
  2. [Section VII B, TEC experiment] The terminal-equality-constrained experiment in Section VII B transfers |+> to |-> for the Hamiltonian H(t)=ωσ_z + u(t)σ_x. The paper itself notes that this target is not an eigenstate of H(u_ref) for any constant u_ref, i.e., condition (9) of Assumption IV.1.4 is violated. Therefore Theorem IV.1 does not cover this experiment, and it cannot serve as numerical validation of the exponential stability claim. The same issue affects the corresponding rows of Table III. A valid numerical test of the theorem would need to use a target satisfying Assumption IV.1.4, or the theorem would need to be generalized to targets that are not eigenstates of the controlled Hamiltonian.
  3. [Assumption IV.1.5] The quadratic controllability-cost bound J*(X) ≤ c_u ||X-X_ref||^2 is a nontrivial assumption and is not verified for any of the systems in the paper. It is not merely a technical convenience: the constant c_u enters the contraction factor γ in Eq. (23) and hence the exponential rate in Theorem IV.1. For the terminal-constrained problem (16), feasibility already requires exact steering to X_ref (up to phase), and the cost depends on the stage cost weights and the numerical optimizer used. The citation to classical MPC literature is not by itself evidence that this bound holds for bilinear quantum dynamics. The authors should either prove the bound for a class of systems, verify it numerically for the reported examples, or explicitly state that the exponential-rate conclusion is conditional on an unverified controllability-with-cost assumption.
  4. [Proof of Theorem IV.1] The proof of Theorem IV.1 is a sketch. The key inequality J*(X_{t+1}) - J*(X_t) ≤ -ℓ(X_t,u_t) is asserted with the comment 'Using classical MPC arguments [16]' and no derivation. For a rigorous theorem in a quantum setting, the reader needs to see how recursive feasibility follows from Assumption IV.1.4, especially given that the terminal constraint (16d) only enforces F=1 and hence allows a global-phase difference between X_L and X_ref. It should be shown explicitly that the tail of the optimal input, augmented by the holding input u_ref, is feasible at the next sampling time and yields the stated cost decrease. This is standard in MPC but should be written out for the fidelity-based stage cost and the phase ambiguity, since those are the nonstandard parts.
minor comments (5)
  1. [Equation (26a)] The stage cost in the setpoint-optimization problem appears to contain a typo: the expressions α(1-F(X_t,X_s(t))) and ||u_t-u_s(t)||^2_R do not depend on the summation index k and seem intended to be α(1-F(X̄_k(t),X_s(t))) and ||ū_k(t)-u_s(t)||^2_R. As printed, the sum is L times the same term and does not penalize the predicted trajectory.
  2. [Section V, after Eq. (26e)] The sentence 'we allow S=0' is unclear notation; S is a set defined in (27), so it should be something like 'we allow S = {0}' or 'we allow the constraint (26e) to be dropped.' Please rephrase.
  3. [Appendix A] There is a typo: 'expresssed' should be 'expressed.' Also, the real-valued reformulation (A2) is stated for H(u) with a real part H_r and imaginary part H_i; for the Schrödinger equation including the factor -i, the signs should be checked carefully. The current text says the representation is 'mathematically equivalent,' but it would be helpful to state the precise correspondence to Eq. (2).
  4. [After Assumption IV.1.1] The claim that (17) holds 'for pure states with the diamond norm' is misleading. The diamond norm is defined for quantum operations and is not a natural norm on pure state vectors. For density matrices, a trace-norm or Hilbert-Schmidt norm bound can be appropriate, but the paper should be consistent about whether X denotes a state vector or a density matrix.
  5. [Section VII C, last paragraph] The statement that setpoint optimization 'can fail to stabilize the target setpoint' when condition (9) is violated is not quantified or illustrated. Since Section VII C otherwise reports perfect fidelity for all targets, this caveat deserves either a supporting experiment or a reference to a known counterexample.

Circularity Check

0 steps flagged

No circularity: the paper's stability theorem is a conditional MPC Lyapunov argument, and the numerical benchmarks are independent of the theorem's assumptions.

full rationale

The paper's central theoretical result (Theorem IV.1, Section IV) is conditional: under Assumption IV.1, the terminal-equality-constrained MPQC scheme is shown to make the infidelity decay exponentially. The proof uses the standard MPC value-function argument: recursive feasibility gives the decrease inequality (20), Assumptions IV.1.1 and IV.1.5 give the upper bound (22), and combining them gives the contraction (23). The constants c1 and cu are existential constants in assumptions, not fitted parameters, and the theorem's conclusion is not used to define or calibrate them. No quantity in the numerical results (final fidelity, runtime) is generated from the theorem's own constants; comparisons are made against GRAPE, direct QOC, and basic MPQC, which are independent baselines. The only self-citation is [25], a tutorial on bilinear control systems by one of the authors, cited to motivate the unified system class (1); it is not used to prove Theorem IV.1 or to exclude alternative control schemes, so it is not load-bearing. The paper explicitly acknowledges scope limits (Section VIII: deriving guarantees for setpoint optimization is future work; Section VII B: the |+> -> |-> example violates condition (9)), which further indicates the authors are not masking assumptions as results. A separate correctness concern, not a circularity, is that Assumption IV.1.1 (Eq. (17)) is incompatible with the pure-state vector representation used in Section II A and Appendix A: for X2 = e^{iθ} X1, the fidelity (6) equals 1 while ||X1-X2||^2 > 0, so no c1 satisfies (17). This undermines the theorem's coverage of the pure-state examples as written, but it is an assumption-validity problem, not a reduction of the conclusion to its inputs.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The central theorem is conditional on five assumptions plus initial feasibility, rather than being a free-standing derivation. Numerical results depend on hand-chosen weights (alpha, R, eta, S) and horizons L, M. No new physical entities are introduced.

free parameters (6)
  • stage cost weight alpha = 1
    User-chosen weight in (8); chosen as alpha=1 in all numerical experiments. Theorem IV.1 only needs alpha>0.
  • control cost weight R = 10^-4 I
    Penalty on control deviation in (8); set to 10^-4 times identity. Tunes smoothness and affects all reported costs.
  • setpoint penalty weights eta, S = eta=5, S=I
    Chosen for setpoint optimization in Section VII C; no sensitivity analysis is given.
  • prediction horizon L = L=10 basic, L=30 TEC, L_min=2 setpoint
    The key design parameter. Small L gives speedup; L_min was selected per target by requiring convergence/tolerance in at least 80% of optimization steps (Table III caption).
  • applied steps M = M=1
    Number of control steps applied per MPC iteration. Theorem IV.1 assumes M=1.
  • discretization parameters N, T, Delta t = N=100, T=5 ns, Delta t=0.05 ns; N=40 in closed loop
    Time grid chosen in the numerical examples; the runtime and efficiency comparisons depend on this.
axioms (7)
  • standard math Classical MPC feasibility/cost-decrease argument: with terminal equality constraint and an admissible steady-state input, optimal cost J* satisfies J*(X_{t+1}) - J*(X_t) <= -ell(X_t,u_t), used in the proof of Theorem IV.1 via (20).
    Imported from Rawlings/Mayne/Diehl [16]; the paper does not prove it. Requires recursive feasibility and an input u_ref in U that keeps X_ref fixed, which is not explicitly stated as an assumption.
  • domain assumption Assumption IV.1.1: there exists c1>0 such that ||X1-X2||^2 <= c1(1-F(X1,X2)).
    A norm-fidelity bound; holds for common pure-state/unitary fidelities but restricts the metric.
  • domain assumption Assumption IV.1.2: input constraint set U is compact.
    Given as nonrestrictive in Section IV; needed for well-posedness of the optimal control subproblem.
  • domain assumption Assumption IV.1.3: stage cost weight alpha > 0.
    User-chosen and easy to satisfy; used to extract infidelity decay from the cost decrease.
  • domain assumption Assumption IV.1.4: target is an eigenstate, A(u_ref) X_ref = lambda X_ref, for a constant input u_ref (and implicitly u_ref in U).
    Needed to hold the state at the target after the horizon; not satisfied by the TEC example in Section VII B.
  • domain assumption Assumption IV.1.5: J*(X) <= c_u ||X-X_ref||^2 for all feasible X.
    A quadratic controllability-with-cost bound that drives the exponential decay in (22)-(23); unverified numerically and not derived from bilinear controllability results.
  • domain assumption Feasibility of the terminal-constrained problem (16) at t=0.
    Explicit condition in Theorem IV.1; requires ability to drive the initial state to the target within L steps.

reviewed 2026-08-05 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Model predictive quantum control: A modular approach for efficient and robust quantum optimal control." pith.science (2026). https://pith.science/paper/AXTFDK6B

@misc{pith2026250905167,
  author       = {Pith},
  title        = {Pith review of: Model predictive quantum control: A modular approach for efficient and robust quantum optimal control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AXTFDK6B}},
  note         = {Machine review of arXiv:2509.05167}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Model predictive control (MPC) is one of the most successful modern control methods. It relies on repeatedly solving a finite-horizon optimal control problem and applying the beginning piece of the optimal input. In this paper, we develop a modular framework for improving efficiency and robustness of quantum optimal control (QOC) via MPC. We first provide a tutorial introduction to basic concepts of MPC from a QOC perspective. We then present multiple MPC schemes, ranging from simple approaches to more sophisticated schemes which admit stability guarantees. This yields a modular framework which can be used 1) to improve efficiency of open-loop QOC and 2) to improve robustness of closed-loop quantum control by incorporating feedback. We demonstrate these benefits with numerical results, where we benchmark the proposed methods against competing approaches.

Figures

Figures reproduced from arXiv: 2509.05167 by Eya Guizani, Julian Berberich.

Figure 1
Figure 1. Figure 1: FIG. 1: System evolution under the MPQC schemes for [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Single-qubit state transfer from [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Robustness comparison between closed-loop [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Model Predictive Control-Inspired Quantum Algorithm

    quant-ph 2026-07 conditional novelty 7.0

    A model-predictive-control-inspired hybrid algorithm optimizes quantum circuit layers over a receding horizon and is proven to at least match FALQON while sometimes outperforming it in practice.

Reference graph

Works this paper leans on

58 extracted references · 55 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Based on the stateX t, solve (14)

    work page

  2. [2]

    , M−1, to the system (4)

    Apply the firstMsteps of the optimal input ut+i = ¯u∗ i (t),i= 0, . . . , M−1, to the system (4)

    work page

  3. [3]

    Algorithm III.1 yields a control input sequence {ut}N−1 t=0 for the QOC problem (11) when stopping the iteration as soon ast≥N

    Sett=t+Mand go back to Step 1. Algorithm III.1 yields a control input sequence {ut}N−1 t=0 for the QOC problem (11) when stopping the iteration as soon ast≥N. It tackles the QOC prob- lem (11) with horizonNby repeatedly solving the smaller QOC problem (14) with horizonL. Note that, except for the different time horizons, the two problems are identi- cal. ...

    work page

  4. [4]

    The input constraint setUis compact

    work page

  5. [5]

    The stage cost(8)is positive definite in the state, i.e., it holds thatα >0

    work page

  6. [6]

    , N−1, and it satisfies(9)

    The input reference is constant¯u ref,t =u ref ,t= 0, . . . , N−1, and it satisfies(9)

    work page

  7. [7]

    The optimal cost of the MPQC problem(16)satis- fies J ∗(X)≤c u∥X−X ref ∥2 (18) with somec u >0and for allXfor which(16)is feasible. The first assumption involving the inequality (17) is satisfied for most relevant QOC scenarios, e.g., for pure states with the diamond norm and the fidelity (6), for unitaries with the Frobenius norm and the fidelity (7), 5 ...

    work page

  8. [8]

    Quantum control theory and applications: a survey,

    D. Dong and I. R. Petersen, “Quantum control theory and applications: a survey,”IET Control Theory Appl., vol. 4, no. 12, pp. 2651–2671, 2010

    work page 2010

  9. [9]

    Modeling and control of quantum systems: an introduction,

    C. Altafini and F. Ticozzi, “Modeling and control of quantum systems: an introduction,”IEEE Transactions on Automatic Control, vol. 57, no. 8, pp. 1898–1917, 2012

    work page 1917

  10. [10]

    Training schr¨ odinger’s cat: quantum optimal control, strategic report on current status, visions and goals for research in Europe,

    S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. K¨ ockenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Her¨ uggen, D. Sugny, and F. K. Wilhelm, “Training schr¨ odinger’s cat: quantum optimal control, strategic report on current status, visions and goals for research in Europe,”The European Physical Journal D, vol. 69, p. 279, 2015

    work page 2015

  11. [11]

    Quantum estimation, con- trol and learning: opportunities and challenges,

    D. Dong and I. R. Petersen, “Quantum estimation, con- trol and learning: opportunities and challenges,”Annual Reviews in Control, vol. 54, pp. 243–251, 2022

    work page 2022

  12. [12]

    Quantum optimal control in quantum technologies. strategic report on current status, visions and goals for research in Eu- rope,

    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 optimal control in quantum technologies. strategic report on current status, visions and goals for research in Eu- rope,”EPJ Quantum Technology, vol. 9, p. 19, 2022

    work page 2022

  13. [13]

    Quantum optimal control theory,

    J. Werschnik and E. K. U. Gross, “Quantum optimal control theory,”J. Phys. B: At. Mol. Opt. Phys., vol. 40, p. R175, 2007

    work page 2007

  14. [14]

    Bonnard and D

    B. Bonnard and D. Sugny,Optimal control with applica- tions in space and quantum dynamics, ser. AIMS on Ap- plied Mathematics. American Institute of Mathematical Sciences, 2012, vol. 5

    work page 2012

  15. [15]

    Introduction to quantum opti- mal control for quantum sensing with nitrogen-vacancy centers,

    P. Rembold, N. Oshnik, M. M. M¨ uller, S. Montangero, T. Calarco, and E. Neu, “Introduction to quantum opti- mal control for quantum sensing with nitrogen-vacancy centers,”A VS Quantum Sci., vol. 2, p. 024701, 2020

    work page 2020

  16. [16]

    Introduction to the Pontryagin maximum principle for quantum optimal control,

    U. Boscain, M. Sigalotti, and D. Sugny, “Introduction to the Pontryagin maximum principle for quantum optimal control,”PRX Quantum, vol. 2, p. 030203, 2021

    work page 2021

  17. [17]

    Introduc- tion to theoretical and experimental aspects of quantum optimal control,

    Q. Ansel, E. Dionis, F. Arrouas, B. Peaudecerf, S. Gu´ erin, D. Gu´ ery-Odelin, and D. Sugny, “Introduc- tion to theoretical and experimental aspects of quantum optimal control,”J. Phys. B: At. Mol. Opt. Phys., vol. 57, p. 133001, 2024

    work page 2024

  18. [18]

    Optimal control of coupled spin dy- namics: design of NMR pulse sequences by gradient as- cent algorithms,

    N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen, and S. J. Glaser, “Optimal control of coupled spin dy- namics: design of NMR pulse sequences by gradient as- cent algorithms,”Journal of Magnetic Resonance, vol. 172, pp. 296–305, 2005

    work page 2005

  19. [19]

    Monotoni- cally convergent optimization in quantum control using Krotov’s method,

    D. M. Reich, M. Ndong, and C. P. Koch, “Monotoni- cally convergent optimization in quantum control using Krotov’s method,”J. Chem. Phys., vol. 136, p. 104103, 2012

    work page 2012

  20. [20]

    Chopped random-basis quantum optimization,

    T. Caneva, T. Calarco, and S. Montangero, “Chopped random-basis quantum optimization,”Phys. Rev. A, vol. 84, p. 022326, 2011

    work page 2011

  21. [21]

    Optimal con- trol technique for many-body quantum dynamics,

    P. Doria, T. Calarco, and S. Montangero, “Optimal con- trol technique for many-body quantum dynamics,”Phys. Rev. Lett., vol. 106, p. 190501, 2011

    work page 2011

  22. [22]

    One decade of quantum optimal control in the chopped random basis,

    M. M. M¨ uller, R. S. Said, F. Jelezko, T. Calarco, and S. Montangero, “One decade of quantum optimal control in the chopped random basis,”Reports on Progress in Physics, vol. 85, no. 7, p. 076001, 2022

    work page 2022

  23. [23]

    J. B. Rawlings, D. Q. Mayne, and M. M. Diehl,Model Predictive Control: Theory, Computation, and Design. Nob Hill Pub, 2020, 3rd printing

    work page 2020

  24. [24]

    Efficient quantum state tomography,

    M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. L.-C., D. Poulin, and Y.- K. Liu, “Efficient quantum state tomography,”Nature Communications, vol. 1, no. 149, 2010

    work page 2010

  25. [25]

    Linear model predic- tive control for schr¨ odinger equation,

    J.-P. Humaloja and S. Dubljevic, “Linear model predic- tive control for schr¨ odinger equation,” inProc. American Control Conf. (ACC), 2018, pp. 2569–2574

    work page 2018

  26. [26]

    Robust quan- tum control via a model predictive control strategy,

    Y. Lee, I. R. Petersen, and D. Dong, “Robust quan- tum control via a model predictive control strategy,” arXiv:2402.07396, 2024

    Pith/arXiv arXiv 2024

  27. [27]

    Model predictive control of two-level open quan- tum systems,

    ——, “Model predictive control of two-level open quan- tum systems,” inProc. 63rd IEEE Conf. Decision and Control, 2024, pp. 25–30

    work page 2024

  28. [28]

    Time-Optimal Control of Finite Dimensional Open Quantum Systems via a Model Predictive Strategy

    ——, “Time-optimal control of finite dimensional open quantum systems via a model predictive strategy,” arXiv:2508.16205, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  29. [29]

    Model- predictive quantum control via Hamiltonian learning,

    M. Clouˆ atre, M. J. Khojasteh, and M. Z. Win, “Model- predictive quantum control via Hamiltonian learning,” IEEE Trans. Quantum Engineering, vol. 3, p. 4100623, 2022

    work page 2022

  30. [30]

    Model predictive control for robust quantum state preparation,

    A. Goldschmidt, J. L. DuBois, S. L. Brunton, and J. N. Kutz, “Model predictive control for robust quantum state preparation,”Quantum, vol. 6, p. 837, 2022

    work page 2022

  31. [31]

    Comparing, optimizing, and benchmarking quantum- control algorithms in a unifying programming frame- work,

    S. Machnes, U. Sander, S. J. Glaser, P. de Fouquieres, A. Gruslys, S. Schirmer, and T. Schulte-Herbr¨ uggen, “Comparing, optimizing, and benchmarking quantum- control algorithms in a unifying programming frame- work,”Physical Review A, vol. 84, no. 2, p. 022305, 2011

    work page 2011

  32. [32]

    Bringing quantum systems under control: a tutorial in- vitation to quantum computing and its relation to bilin- ear control systems,

    J. Berberich, R. L. Kosut, and T. Schulte-Herbr¨ uggen, “Bringing quantum systems under control: a tutorial in- vitation to quantum computing and its relation to bilin- ear control systems,” inProc. 63rd IEEE Conf. Decision and Control (CDC), 2024, pp. 5231–5247

    work page 2024

  33. [33]

    Gradient- ascent pulse engineering with feedback,

    R. Porotti, V. Peano, and F. Marquardt, “Gradient- ascent pulse engineering with feedback,”PRX Quantum, vol. 4, p. 030305, 2023

    work page 2023

  34. [34]

    Improving ro- bustness of quantum feedback control with reinforcement learning,

    M. Guatto, G. A. Susto, and F. Ticozzi, “Improving ro- bustness of quantum feedback control with reinforcement learning,”Physical Review A, vol. 110, p. 012605, 2024

    work page 2024

  35. [35]

    Nonlin- ear model predictive control of a four tank system: An experimental stability study,

    T. Raff, S. Huber, Z. K. Nagy, and F. Allg¨ ower, “Nonlin- ear model predictive control of a four tank system: An experimental stability study,” inProc. IEEE Int. Conf. Control Applications (CCA), 2006, pp. 237–242

    work page 2006

  36. [36]

    NMPC without terminal constraints,

    L. Gr¨ une, “NMPC without terminal constraints,” in Proc. IF AC Conf. Nonlinear Model Predictive Control, 2012, pp. 1–13

    work page 2012

  37. [37]

    Dis- tance measures to compare real and ideal quantum pro- cesses,

    A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Dis- tance measures to compare real and ideal quantum pro- cesses,”Physical Review A, vol. 71, p. 062310, 2005

    work page 2005

  38. [38]

    D’Alessandro,Introduction to quantum control and dynamics, 2nd ed

    D. D’Alessandro,Introduction to quantum control and dynamics, 2nd ed. Chapman and Hall/CRC, 2021

    work page 2021

  39. [39]

    MPC for tracking piecewise constant references for con- strained linear systems,

    D. Lim´ on, I. Alvarado, T. Alamo, and E. F. Camacho, “MPC for tracking piecewise constant references for con- strained linear systems,”Automatica, vol. 44, no. 9, pp. 2382–2387, 2008

    work page 2008

  40. [40]

    Nonlinear MPC for tracking piece-wise constant refer- ence signals,

    D. Lim´ on, A. Ferramosca, I. Alvarado, and T. Alamo, “Nonlinear MPC for tracking piece-wise constant refer- ence signals,”IEEE Trans. Automat. Control, vol. 63, 12 no. 11, pp. 3735–3750, 2018

    work page 2018

  41. [41]

    A nonlin- ear tracking model predictive control scheme for dynamic target signals,

    J. K¨ ohler, M. A. M¨ uller, and F. Allg¨ ower, “A nonlin- ear tracking model predictive control scheme for dynamic target signals,”Automatica, vol. 118, p. 109030, 2020

    work page 2020

  42. [42]

    Model predictive control for tracking using artificial references: Funda- mentals, recent results and practical implementation,

    P. Krupa, J. K¨ ohler, A. Ferramosca, I. Alvarado, M. N. Zeilinger, T. Alamo, and D. Limon, “Model predictive control for tracking using artificial references: Funda- mentals, recent results and practical implementation,” inProc. 63rd IEEE Conf. Decision and Control (CDC), 2024, pp. 2977–2991

    work page 2024

  43. [43]

    A nonlinear MPC scheme for output tracking without terminal ingre- dients,

    R. Soloperto, J. K¨ ohler, and F. Allg¨ ower, “A nonlinear MPC scheme for output tracking without terminal ingre- dients,”IEEE Trans. Automat. Control, vol. 68, no. 4, pp. 2368–2375, 2022

    work page 2022

  44. [44]

    CasADI - A software framework for nonlinear optimization and optimal control,

    J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawl- ings, and M. Diehl, “CasADI - A software framework for nonlinear optimization and optimal control,”Math. Program. Comput., vol. 11, no. 1, pp. 1–36, 2019

    work page 2019

  45. [45]

    Sequential quadratic pro- gramming,

    P. T. Boggs and J. W. Tolle, “Sequential quadratic pro- gramming,”Acta Numerica, vol. 4, pp. 1–51, 1995

    work page 1995

  46. [46]

    On the implementation of an interior-point filter line-search algorithm for large- scale nonlinear programming,

    A. W¨ achter and L. T. Biegler, “On the implementation of an interior-point filter line-search algorithm for large- scale nonlinear programming,”Math. Program., vol. 106, pp. 25–57, 2005

    work page 2005

  47. [47]

    Nominal stability of real-time iteration scheme for nonlinear model predictive control,

    M. Diehl, R. Findeisen, F. Allgower, H. G. Bock, and J. P. Schloder, “Nominal stability of real-time iteration scheme for nonlinear model predictive control,”IEE Pro- ceedings - Control Theory and Applications, vol. 152, no. 3, pp. 296–308, 2005

    work page 2005

  48. [48]

    Efficient nu- merical methods for nonlinear MPC and moving horizon estimation,

    M. Diehl, H. J. Ferreau, and N. Haverbeke, “Efficient nu- merical methods for nonlinear MPC and moving horizon estimation,” inLecture notes in control and information sciences: Vol. 483. Nonlinear model predictive control. Springer, 2009, pp. 391–417

    work page 2009

  49. [49]

    Time-distributed optimization for real-time model pre- dictive control: stability, robustness, and constraint sat- isfaction,

    D. Liao-McPherson, M. M. Nicotra, and I. Kolmanovsky, “Time-distributed optimization for real-time model pre- dictive control: stability, robustness, and constraint sat- isfaction,”Automatica, vol. 117, p. 108973, 2020

    work page 2020

  50. [50]

    A Lyapunov function for the combined system-optimizer dynamics in inexact model predictive control,

    A. Zanelli, Q. Tran-Dinh, and M. Diehl, “A Lyapunov function for the combined system-optimizer dynamics in inexact model predictive control,”Automatica, vol. 134, p. 109901, 2021

    work page 2021

  51. [51]

    Quan- tum optimal control with geodesic pulse engineering,

    D. Lewis, R. Wiersema, and S. Bose, “Quan- tum optimal control with geodesic pulse engineering,” arXiv:2508.16029, 2025

    Pith/arXiv arXiv 2025

  52. [52]

    Unitary gate synthesis via polynomial optimization,

    L. B. Gaggioli, D. I. Bondar, J. Vala, R. Ovsiannikov, and J. Marecek, “Unitary gate synthesis via polynomial optimization,”arXiv:2508.01356, 2025

    arXiv 2025

  53. [53]

    Low-rank optimal control of quantum devices,

    L. Goutte and V. Savona, “Low-rank optimal control of quantum devices,”arXiv:2508.18114, 2025

    work page arXiv 2025

  54. [54]

    Open and closed loop approaches for energy efficient quantum optimal control,

    S. Fauquenot, A. Sarkar, and S. Feld, “Open and closed loop approaches for energy efficient quantum optimal control,”Adv. Quantum Technol., p. 2400690, 2025

    work page 2025

  55. [55]

    On the stability of constrained MPC without terminal con- straint,

    D. Lim´ on, T. Alamo, F. Salas, and E. F. Camacho, “On the stability of constrained MPC without terminal con- straint,”IEEE Trans. Automat. Control, vol. 51, no. 5, pp. 832–836, 2006

    work page 2006

  56. [56]

    A survey of quantum Lyapunov control methods,

    S. Cong and F. Meng, “A survey of quantum Lyapunov control methods,”The Scientific World Journal, vol. 2013, p. 967529, 2013

    work page 2013

  57. [57]

    Predicting many properties of a quantum system from very few mea- surements,

    H.-Y. Huang, R. Kueng, and J. Preskill, “Predicting many properties of a quantum system from very few mea- surements,”Nat. Phys., vol. 16, no. 10, pp. 1050–1057, 2020

    work page 2020

  58. [58]

    ˙A(t) ˙B(t) # =

    H. M. Wiseman and G. J. Milburn,Quantum measure- ment and control. Cambridge, UK: Cambridge Univer- sity Press, 2009. Appendix A: Details on the numerical implementation The exact optimization problems introduced in Sec- tions III–V cannot be passed directly to standard non- linear optimization toolboxes such as CasADi. The main obstacle is that the discr...

    work page 2009

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.