Pith. sign in

REVIEW 4 major objections 5 minor 50 references

Iterative linear quadratic regulator on SU(N) for multi-qubit gate synthesis

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper claims that Lie-group iLQR on SU(N) reaches infidelities below 10^-15 in one optimization iteration in the full-drive setting and converges faster early under 2-local drives, at the cost of more sensitivity to initialization.

desk verdict A clean, incremental adaptation of Lie-group iLQR to SU(N) gate synthesis; the central convergence claims survive scrutiny, and the main weaknesses are reproducibility and benchmark-specific tuning, not the BCH truncation worry. read the letter →

arxiv 2608.03656 v1 pith:57B7RYZB submitted 2026-08-04 quant-ph

Dirk Heimann , Felix Wiebe , Elie Mounzer , Shivesh Kumar This is my paper · ORCID
classification quant-ph
keywords iterativelinearquadraticregulatorquantumgatesynthesisLie-groupcontrolSU(N)pulseoptimizationtrajectory2-localHamiltonianoptimal
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 adapts the iterative linear quadratic regulator (iLQR), a trajectory-optimization workhorse, from flat Euclidean coordinates to the curved manifold SU(N) that quantum gates naturally evolve on. It claims that when every Lie-algebra basis element is available as a drive Hamiltonian, the Lie-group formulation reaches gate infidelities below $10^{-15}$ in a single optimization iteration for two- through five-qubit gates, while standard Euclidean iLQR needs many iterations. Under physically realistic 2-local drive restrictions, the Lie-group version converges faster in early optimization iterations but is more sensitive to initialization and more prone to local minima. The practical stake is pulse synthesis: a geometry-aware optimizer that warms up quickly could shorten the pulse-design pipeline for quantum hardware without changing worst-case complexity.

What carries the argument

The load-bearing object is the Lie-algebraic error state eta_k^\wedge = -i log(U_k^\dagger \tilde U_k), which measures the geodesic gap between the current unitary and the reference unitary on SU(N). Around this error state the paper linearizes the equation of motion as eta_{k+1} \approx Ad_{\exp(-i(H_0+H_d u_k)\$\sigma$ dt)}(\eta_k) + dt\, H_d\, dexp_{-i(H_0+H_d u_k)\$\sigma$ dt}(\$\sigma$ \delta u_k), using the adjoint representation of the group and the differential of the matrix exponential, with the dexp series kept to third order. This linearization turns the Bellman backward pass into a standard quadratic regulator solve while keeping the state on the manifold, and it is what lets controls be initialized at zero and still reach machine-precision infidelity in a single step when the full algebra is accessible.

What would settle it

Take an optimized control sequence from a five-qubit WZ4 run and compare, step by step, the linearized state update of Eq. (10) with the exact discrete update eta_{k+1} = -i log(exp(-\xi_k dt) exp(i eta_k) exp((\xi_k + \delta \xi_k) dt)) from Eq. (9): if the accumulated normed difference grows to the same order as the reported final infidelity, the single-iteration convergence claim is an artifact of the linearization rather than a property of the true dynamics. A simpler variant is to propagate the optimized pulses under the exact Schr\"odinger equation and compare the resulting fidelity to the value reported by the optimizer.

Watch

Extended reading notes

Core claim

The central claim is that treating the quantum state as an element of SU(N) rather than as a flat vector changes the optimization landscape enough to matter. The authors build the trajectory on Lie-algebra elements eta_k = -i log(U_k^\dagger \tilde U_k), the geodesic displacement from the reference to the candidate unitary, and linearize the discrete Schr\"odinger update through the adjoint representation Ad and the differential-of-exponential map dexp. In the idealized full-algebra setting, this makes the optimizer converge to infidelities below $10^{-15}$ in one iteration on CNOT, CCZ, Fredkin, Toffoli, and the X/Z weighted-parity gates, whereas Euclidean iLQR requires 10-22 iterations; when only 1- and 2-local drives are allowed, the Lie-group median converges faster at early iterations for three- and four-qubit gates but plateaus at worse final fidelities for five-qubit gates and exhibits broader run-to-run spread.

Load-bearing premise

The load-bearing premise is that the first-order Baker-Campbell-Hausdorff linearization of the Lie-group dynamics, which drops terms of order ||(eta_k, \$\sigma$ \delta u_k)||^2 and truncates the dexp series at third order, accurately describes the real Schr\"odinger evolution at the control amplitudes the optimizer produces; if that approximation breaks down, the reported convergence curves do not describe the true quantum dynamics.

Editorial extensions

If this is right

  • If correct, the Lie-group iLQR can serve as a one-shot pulse synthesizer for full-algebra benchmarks, producing machine-precision gates from zero-initialized controls.
  • For 2-local constrained gates, the faster early convergence means useful moderate-fidelity pulses can be obtained with fewer iterations for three- and four-qubit gates, though not for the five-qubit parity gates tested.
  • Because the worst-case complexity is O(T N^6) per iteration for both formulations, the Lie-group version's smaller state dimension (N^2-1 versus 2N^2) gives it a wall-clock edge for three- and four-qubit gates, a lead that reverses at five qubits.
  • The paper's suggestion that Lie-group solutions warm-start the Euclidean variant would combine fast early progress with the Euclidean method's better late-stage five-qubit behavior.

Reading between the lines

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

  • One untested extension is to initialize 2-local restricted runs with the single-iteration, full-algebra solution; because the full-algebra solve is near-instant, this could sidestep the sensitivity to initialization that the paper observes.
  • A second-order DDP variant on SU(N) would likely reduce the local-minima tendency, since the paper identifies the first-order linearization as the source of its approximation error and DDP is the natural next rung.
  • The dexp-order scan suggests an adaptive policy: choose the truncation order per gate based on the size of higher-order ad-terms encountered along the trajectory, rather than fixing order three for everything.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper adapts the iterative linear quadratic regulator (iLQR) to the special unitary group SU(N) for multi-qubit gate synthesis. It derives Lie-algebraic equations of motion for the error state via the adjoint representation and a first-order Baker-Campbell-Hausdorff linearization, defines a geodesic final cost, extends the formulation to smoothed controls and 2-local drive restrictions, and provides a worst-case time-complexity analysis. Numerically, it compares the Lie-group iLQR against a Euclidean iLQR on two- to five-qubit benchmark gates. The central claims are that, with full Lie-algebra basis access, the Lie-group formulation reaches infidelities below 10^-15 in a single optimization iteration, and that under 2-local restrictions it converges faster in early iterations but exhibits greater sensitivity to initialization and a stronger tendency toward local minima. The paper also reports per-iteration wall-clock runtimes and tests the effect of the dexp series truncation order.

Significance. If the numerical claims hold, the paper makes a useful contribution to quantum optimal control: it provides a concrete SU(N) adaptation of an established Lie-group trajectory optimization framework, with analytic derivative expressions, a complexity analysis, and a systematic comparison against the Euclidean iLQR on physically motivated multi-qubit gates. The derivation in Section III and Appendix A is plausible and follows standard Lie-group calculus, and the paper honestly acknowledges limitations in the Discussion, including the unsatisfactory five-qubit constrained results and the fact that increasing time steps or dexp order does not cure them. The dexp-order experiment in Section IV.B4 is a good example of probing a secondary approximation. However, the evidence for the headline early-convergence advantage is currently provisional: the first-order BCH truncation is not directly validated, the hyperparameter selection is performed on the same benchmark problems used for the final comparison, and the code and exact hyperparameter values are not available for independent audit.

major comments (4)
  1. [Section III.B, Eq. (10) / Appendix A3 / Section IV.B4] The 2-local numerical comparison rests on the first-order BCH linearization in Eq. (10) and Eq. (A14), which drops O(||(eta, sigma delta u)||^2) terms. Section IV.B4 tests only the dexp series truncation order, not the validity of this more fundamental first-order truncation at the control and error magnitudes produced by the optimizer. The full-algebra single-iteration demonstration with zero initial controls cannot validate the approximation away from the identity. Please add a direct check: compare the linearized eta_{k+1} from Eq. (10) with the exact eta_{k+1} from Eq. (9) along representative optimizer trajectories for the 2-local cases, and report the sizes of the neglected terms. Without such a check, the faster early decrease of the Lie-group curves in Figs. 2, 3, 5, and 6 could reflect the surrogate linearization rather than a genuine geometric advantage of the true SU(N) dynamics. The Discussion's observation in Section V that more time steps and higher dexp orders do not help the five-qubit gates is consistent with this concern, but it is not itself a measurement of the truncation error.
  2. [Section IV.B1-IV.B5, Eqs. (20)-(21)] The hyperparameters for the 2-local comparison are selected by Bayesian optimization using cost functions evaluated on the same benchmark gates and the same N_s=25 seed protocol that later produce the reported medians. The manuscript does not state whether the 25 evaluation seeds are disjoint from the seeds used by the Bayesian optimizer. This creates a risk of in-sample hyperparameter selection that can bias the comparison between the Lie-group and Euclidean variants. Please either evaluate on a holdout seed set or explicitly state and justify the overlap, and report the chosen hyperparameter values for each method and gate. Without this, the claim that the Lie-group formulation consistently converges faster in early iterations is not yet fully supported.
  3. [Section IV.B1 and Ref. [29]] The central evidence of the paper is numerical, but the code is not available with the submission and the exact hyperparameter configurations (initial amplitude scale u_init, running cost R_u, final cost Q_f, and the Bayesian-optimization settings) are not reported. Reference [29] points to a generic organization page rather than a specific archived repository. To make the comparison auditable, please include a table of the best hyperparameters for every reported experiment, all random seed metadata, and make the code available as a review artifact or a persistent DOI in the revision.
  4. [Section V / Table I] The single-iteration full-algebra results in Table I are reported without any seed variation, confidence intervals, or statement of how many independent runs were performed. Since the claim is about reaching machine-precision infidelities in one iteration, a few repetitions with different seeds (at least for the multi-qubit gates) are needed to rule out numerical accidents and to make the comparison with the two-iteration GEOPE result meaningful. If the observed behavior is deterministic, state so explicitly.
minor comments (5)
  1. [Section IV.A2] The phrase "single-time-step setting" is ambiguous: it should be clarified whether this means a single optimization iteration, a discretization with T=1, or both, especially because Eq. (6) and the single-qubit example in Section IV.A1 use T=80.
  2. [Appendix C, Eq. (C2)] The Fredkin gate expansion lists the term |101><110| twice and appears to omit |110><101|; please check the formula.
  3. [Section IV.B4] The text states that the implementation retains terms up to third order in the dexp series, while Fig. 4 uses orders {0,1,3,7}. Please clarify how these order labels are counted so that the implementation and the experiment are consistent.
  4. [Eqs. (20)-(21)] The symbol T denotes both the number of time steps and the final-state notation, while the target unitary is U_g; please rename one of these to avoid confusion in the definitions of the Bayesian optimization scores.
  5. [Section III.B] The sentence "the dexp series truncation is a secondary approximation that catches some higher-order terms but not all" is imprecise; it would be helpful to state exactly which terms are retained in the implemented Jacobians and which are dropped.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Lie-group iLQR derivation is self-contained from standard BCH/algebra identities, and the numerical claims are supported by direct simulation rather than by fitted inputs or load-bearing self-citations.

full rationale

The paper's central derivation is the linearized equation of motion Eq. (10)/Eq. (A14) for the Lie-algebra error state eta. This is derived in Appendix A 3 from the Baker-Campbell-Hausdorff formula and standard adjoint/dexp identities, with the neglected terms stated explicitly as O(||(eta_k, sigma delta u_k)||^2). No fitted parameter is used to define the dynamics; the only inputs are the target gate, the discretized Schrodinger equation, and the chosen quadratic costs. The final cost l_f(eta_T) = 1/2 eta_T^T Q_f eta_T is an explicit modeling choice (the geodesic distance in the algebra), not a hidden equivalent of the output claim. The full-basis single-iteration result is a consequence of the linearized dynamics being exact at zero controls with H0 = 0 and the full algebra spanning the log-direction to the target; it is a legitimate numerical finding, not a reduction of the prediction to the input. The comparison against Euclidean iLQR uses a published baseline [5] from the same group, but the comparison is executed by independent simulation rather than justified by citation; no self-citation is load-bearing for the derivation. The Discussion's admission that dexp order and time-step increases do not help the five-qubit results, and the untested first-order BCH truncation, are correctness/validation limitations, not circularity. Hyperparameters are tuned via Bayesian optimization on the same benchmarks, which is a methodological selection-bias concern, not a circular derivation. Overall the paper is self-contained against standard external mathematical identities and numerical benchmarks, so the circularity score is 0.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The derivation is mathematically standard, but the numerical conclusions depend on a set of tuned hyperparameters and on the adequacy of the first-order BCH approximation. No new physical entities are introduced. The main burden is practical: hyperparameters for every method and gate are chosen via Bayesian optimization on the benchmark itself, and the linearization error is not bounded.

free parameters (6)
  • R_u (running control cost weight) = Not reported; selected by Bayesian optimization per method and gate
    Controls the magnitude of control updates and convergence behavior; hyperparameter search in Sec. IV B 1 varies R_u diagonal entries over 1e-5 to 1e5.
  • Q_f (final geodesic cost weight) = Not reported; selected by Bayesian optimization per method and gate
    Sets the weight of the final infidelity proxy; varied in the same Bayesian search in Sec. IV B 1.
  • u_init (initial control amplitude scale) = Not reported; selected by Bayesian optimization
    Initial control amplitudes strongly affect the Lie-group variant; the paper notes higher sensitivity to initialization in Sec. IV B 3.
  • T (number of discrete time steps) = 12 for 2-local benchmarks, 80 for the one-qubit example
    Chosen by hand and fixed uniformly across qubit sizes to limit the hyperparameter search space, as stated in Sec. IV B 1.
  • dexp series truncation order = 3
    Set to 3 in all main runs; the choice is justified only by the numerical comparison in Sec. IV B 4, not by an error bound.
  • H_d (drive coefficient) = 1/(2*pi) in the one-qubit example
    A scalar drive coefficient in Eq. (7); value for multi-qubit runs is not explicitly reported.
assumptions (4)
  • domain assumption Quantum evolution is exactly time-discretized as U_{k+1} = exp(i(H0 + Hd u_k)sigma dt) U_k.
    Eq. (6) assumes piecewise-constant controls, no noise, and a drive Hamiltonian entering linearly with scalar H_d.
  • ad hoc to paper First-order Taylor/BCH linearization of the Lie-algebra dynamics, Eq. (10) with O(||(eta,delta_xi)||^2) neglected, is sufficiently accurate for the iLQR updates.
    The method linearizes dynamics at each iteration; no error bound is given, and the dexp truncation order is set to 3 based on numerical tests rather than theory.
  • domain assumption The geodesic final cost vanishes iff g_T equals the target up to a global phase, and this is an adequate proxy for gate fidelity.
    Sec. III C defines the final cost via eta_T = -i log(g_T^dagger g_goal); relative phase is ignored, which may matter for some hardware encodings.
  • domain assumption Restricting drive terms to one- and two-body Pauli interactions models hardware-constrained controls.
    Sec. III D and Sec. IV B use this restriction; the paper itself notes in Sec. V that specific native gate sets such as cross-resonance are harder to solve.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Iterative linear quadratic regulator on SU(N) for multi-qubit gate synthesis." pith.science (2026). https://pith.science/paper/57B7RYZB

@misc{pith2026260803656,
  author       = {Pith},
  title        = {Pith review of: Iterative linear quadratic regulator on SU(N) for multi-qubit gate synthesis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/57B7RYZB}},
  note         = {Machine review of arXiv:2608.03656}
}
read the original abstract

In quantum optimal control theory, gradient-based trajectory optimization techniques have proven versatile in designing multi-qubit quantum gates. Furthermore, incorporating the underlying Lie-group structure can accelerate the optimization process. In this work, we adapt the Lie-group formulation of the iterative linear quadratic regulator (iLQR) to the special unitary group SU(N) and apply it to quantum gate synthesis, systematically comparing it against the standard Euclidean iLQR formulation across multiple two- to five-qubit gates. We find that in the idealized, unconstrained setting, where all Lie-algebra basis elements are available as drive Hamiltonian terms, the Lie-group formulation converges faster than the Euclidean iLQR formulation. If drive terms are constrained to 2-local Hamiltonian terms, the Lie-group variant converges faster in early optimization iterations, but exhibits greater sensitivity to initialization and a stronger tendency towards local minima. These results demonstrate that incorporating Lie-group geometry into iLQR substantially improves convergence and highlight important next steps for improvements in constrained control settings.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 43 canonical work pages

  1. [29]

    G. I. Boutselis and E. Theodorou, Discrete-time differ- ential dynamic programming on lie groups: Derivation, convergence analysis, and numerical results, IEEE T. Au- tomat. Contr.66, 4636 (2021)

    work page 2021

  2. [1]

    Both the Lie-group iLQR formu- lation with and without smoothed controls achieve an infidelity below 10−15 after the first optimization itera- tion

    One-qubit example We begin with the simplest case of a single qubit with two energy levels and chooseT= 80 time-steps together withH d = 1/(2π). Both the Lie-group iLQR formu- lation with and without smoothed controls achieve an infidelity below 10−15 after the first optimization itera- tion. The resulting control amplitudes are depicted in Fig. 1; the sm...

    work page

  3. [2]

    Therefore, we instead com- pute the corresponding matrix forf η,k at each optimiza- tion iteration and at each time step by explicitly com- puting its columns according to Eq

    Multi-qubit examples Beyond SU(2), deriving closed-form matrix expressions for the adjoint representation of the group becomes sub- stantially more challenging. Therefore, we instead com- pute the corresponding matrix forf η,k at each optimiza- tion iteration and at each time step by explicitly com- puting its columns according to Eq. (14). As in the prev...

    work page

  4. [3]

    Hyperparameter search with Bayesian optimization On one hand, the entire hyperparameter space of iLQR is too large to exhaustively optimize all relevant param- eters over a broad range of values. On the other hand, individual hyperparameters have well-defined purposes, enabling the manual selection of reasonable initial values to serve as starting points ...

    work page

  5. [4]

    Three-qubit gates Figure 2 shows the median infidelity over 25 random seeds as a function of optimization iteration for both the Lie-group and Euclidean iLQR formulations applied to three-qubit benchmark gates with 2-local drive Hamil- tonian terms for the best hyperparameter set obtained by Bayesian optimization with cost function Eq. (20). The shaded ba...

    work page

  6. [5]

    For the three- qubit WX 2 and WZ 2 gates, the faster convergence rate of the median run is most pronounced

    Scaling from three-qubit to five-qubit gates Similar results hold for the X and Z parity gates that we test for three, four, and five qubits. For the three- qubit WX 2 and WZ 2 gates, the faster convergence rate of the median run is most pronounced. The Lie-group formulation depends more on the initial amplitudes as represented by the broader whiskers and...

    work page

  7. [6]

    (1) for the optimization of the four-qubit parity X gate, which does not achieve machine-precision infidelities

    Example for varying the orders of thedexpseries We assess the impact of varying the dexp series trunca- tion order in Eq. (1) for the optimization of the four-qubit parity X gate, which does not achieve machine-precision infidelities. The plots in Fig. 4 show infidelities across 25 optimization runs, with the best hyperparameter set obtained via independe...

    work page

  8. [7]

    Optimization with area under the curve Bayesian optimization score We take the same initial hyperparameter sets as in Secs. IV B 2 and IV B 3 but perform the Bayesian opti- mization according to the following logarithmic area un- 10 0 20 40 60 80 100 Iteration number 100 10−3 10−6 10−9 10−12 10−15 dexp 0 dexp 1 dexp 3 dexp 7 1 (a) Median dexp 0 dexp 1 dex...

    work page

Show all 50 references
  1. [8]

    Wall-clock runtime For the simulations in Secs. IV B 2 and IV B 3, we record wall-clock time for gate optimization with 2-local drive Hamiltonian terms across 25 independent initializa- tions, using the optimal hyperparameter set, on a dedi- cated cluster node with an AMD EPYC...

  2. [9]

    Denote the vector space of smooth vector fields onGbyX(G) :=X

    Basic definitions A Lie groupGis an abstract group which is also a smooth manifold on which the operations of group multiplication (g7→hg, forg,h∈G) and inversion (g7→g −1, forg∈G) are smooth diffeomorphisms ofGontoG.L g :G→Gwith Lg(h) =hgdenotes left-translation byg∈G. Denote...

  3. [10]

    Taylor expansion on Lie groups With the definitions and identities from the previous section, we calculate the first and second derivatives of the smooth, real functionf(g ϵ) by following the proof of Theorem 2.5 in Ref. [38]: f gϵ =f gexp(tξ∧) c(s) :=gexp(sξ ∧) andc(0) =g γ(s...

  4. [11]

    (9), the analogue of Eq

    Equation of motion for su(N) elements We can combine the previously defined equations to derive η∧ k+1 =−ilog U† k+1Uϵ,k+1 (8) =−ilog exp(ξ∧ k ∆t)† ˜Uk†Uϵ,k exp ∆tξ∧ ϵ,k (4) =−ilog exp(ξ∧ k ∆t)†U† kUk exp(iη∧ k ) exp(∆tξϵ,k) |ξ∧ k is skew (5) =−ilog(exp(−ξ ∧ k ∆t) exp(iη∧ k ) ...

  5. [12]

    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 op- timal control in quantum technologies. strategic report on current status, visions and goals for research in eu...

    2022

  6. [13]

    Khaneja, T

    N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen, and S. J. Glaser, Optimal control of coupled spin dynam- ics: design of nmr pulse sequences by gradient ascent algorithms, J. Magn. Reson.172, 296 (2005)

    2005

  7. [14]

    J. Shao, J. Combes, J. Hauser, and M. M. Nicotra, Projection-operator-based newton method for the trajec- tory optimization of closed quantum systems, Phys. Rev. A105, 032605 (2022)

    2022

  8. [15]

    Trowbridge, A

    A. Trowbridge, A. Bhardwaj, K. He, D. I. Schuster, and Z. Manchester, Direct collocation for quantum op- timal control, in2023 IEEE International Conference on Quantum Computing and Engineering (QCE)(IEEE Computer Society, Los Alamitos, CA, USA, 2023) pp. 1278–1285

    2023

  9. [16]

    Heimann, F

    D. Heimann, F. Wiebe, T. Abad, E. Mounzer, T. Huang, F. Kirchner, and S. Kumar, Iterative Linear Quadratic Regulator for Quantum Optimal Control , in2025 IEEE International Conference on Quantum Computing and Engineering (QCE)(IEEE Computer Society, Albu- querque, NM, USA, 202...

    2025

  10. [17]

    Propson, B

    T. Propson, B. E. Jackson, J. Koch, Z. Manchester, and D. I. Schuster, Robust quantum optimal control with trajectory optimization, Phys. Rev. Appl.17, 014036 (2022)

    2022

  11. [18]

    T. A. Howell, B. E. Jackson, and Z. Manchester, Altro: A fast solver for constrained trajectory optimization, in 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)(2019) pp. 7674–7679

    2019

  12. [19]

    M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Quantum computation as geometry, Science311, 1133 (2006)

    2006

  13. [20]

    M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Do- herty, Optimal control, geometry, and quantum comput- ing, Phys. Rev. A73, 062323 (2006)

    2006

  14. [21]

    Bhattacharyya, P

    A. Bhattacharyya, P. Nandy, and A. Sinha, Renormalized circuit complexity, Phys. Rev. Lett.124, 101602 (2020)

    2020

  15. [22]

    D. L. Goodwin and I. Kuprov, Modified newton-raphson grape methods for optimal control of spin systems, J. Chem. Phys.144, 204107 (2016)

    2016

  16. [23]

    Foroozandeh and P

    M. Foroozandeh and P. Singh, Optimal control of spins by analytical lie algebraic derivatives, Automatica129, 109611 (2021)

    2021

  17. [24]

    D. L. Goodwin and M. S. Vinding, Accelerated newton- raphson grape methods for optimal control, Phys. Rev. Res.5, L012042 (2023)

    2023

  18. [25]

    Kaiser, Solving the matrix exponential function for the lie groups su(3), su(4) and sp(2), Eur

    N. Kaiser, Solving the matrix exponential function for the lie groups su(3), su(4) and sp(2), Eur. Phys. J. A58 (2022)

    2022

  19. [26]

    Lewis, R

    D. Lewis, R. Wiersema, J. Carrasquilla, and S. Bose, Geodesic algorithm for unitary gate design with time- independent hamiltonians, Phys. Rev. A111(2025)

    2025

  20. [27]

    Lewis, R

    D. Lewis, R. Wiersema, and S. Bose, Quantum optimal control with geodesic pulse engineering, arXiv:2508.16029 (2025)

    2025 arXiv

  21. [28]

    Jacobson and D

    D. Jacobson and D. Mayne,Differential Dynamic Pro- gramming(Elsevier, New York, NY, 1970)

    1970

  22. [30]

    S. Teng, W. Clark, A. Bloch, R. Vasudevan, and M. Ghaf- fari, Lie algebraic cost function design for control on lie groups, in2022 IEEE 61st Conference on Decision and Control (CDC)(IEEE Computer Society, Cancun, Mex- ico, 2022) pp. 1867–1874

    2022

  23. [31]

    Kumar, A

    S. Kumar, A. Mueller, P. Wensing, and F. Kirchner, To- wards Continuous Time Finite Horizon LQR Control in SE(3), inIEEE International Conference on Robotics and Automation (ICRA) 2023 Workshop on Geometric Rep- resentations The Roles of Modern Screw Theory, Lie al- gebra, and...

    2023

  24. [32]

    Iserles, H

    A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett, and A. Zanna, Lie-group methods, Acta Numer.9, 215–365 (2000)

    2000

  25. [33]

    Bellman, The theory of dynamic programming, Bul- letin of the American Mathematical Society60, 503 (1954)

    R. Bellman, The theory of dynamic programming, Bul- letin of the American Mathematical Society60, 503 (1954)

    1954

  26. [34]

    Mayne, A second-order gradient method for determin- ing optimal trajectories of non-linear discrete-time sys- tems, International Journal of Control3, 85 (1966)

    D. Mayne, A second-order gradient method for determin- ing optimal trajectories of non-linear discrete-time sys- tems, International Journal of Control3, 85 (1966)

    1966

  27. [35]

    Li and E

    W. Li and E. Todorov, Iterative linear quadratic regu- lator design for nonlinear biological movement systems, inFirst International Conference on Informatics in Con- trol, Automation and Robotics, Vol. 2 (SciTePress, 2004) pp. 222–229

    2004

  28. [36]

    Todorov and W

    E. Todorov and W. Li, A generalized iterative lqg method for locally-optimal feedback control of constrained non- linear stochastic systems, inProceedings of the 2005, American Control Conference, Vol. 1 (IEEE Computer Society, Portland, OR, USA, 2005) pp. 300–306

    2005

  29. [37]

    Tassa, T

    Y. Tassa, T. Erez, and E. Todorov, Synthesis and stabi- lization of complex behaviors through online trajectory optimization, in2012 IEEE/RSJ International Confer- ence on Intelligent Robots and Systems(IEEE Computer Society, Vilamoura-Algarve, Portugal, 2012) pp. 4906– 4913

    2012

  30. [38]

    Mastalli, W

    C. Mastalli, W. Merkt, J. Marti-Saumell, H. Ferrolho, J. Sol` a, N. Mansard, and S. Vijayakumar, A feasibility- driven approach to control-limited ddp, Autonomous Robots46, 985 (2022)

    2022

  31. [39]

    Bradbury, R

    J. Bradbury, R. Frostig, P. Hawkins, M. J. John- son, Y. Katariya, C. Leary, D. Maclaurin, G. Nec- ula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: composable transformations of Python+NumPy programs (2018)

    2018

  32. [40]

    Heimann, F

    D. Heimann, F. Wiebe, E. Mounzer, and S. Ku- mar, Iterative linear quadratic regulator on SU(N) for multi-qubit gate synthesis, https://github.com/dfki-ric- quantum/ (2026)

    2026

  33. [41]

    J. M. Chow, A. D. C´ orcoles, J. M. Gambetta, C. Rigetti, B. R. Johnson, J. A. Smolin, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, and M. Steffen, Simple all-microwave entangling gate for fixed-frequency super- conducting qubits, Phys. Rev. Lett.107, 080502 (2011). 21

    2011

  34. [42]

    W.-J. Lin, H. Cho, Y. Chen, M. G. Vavilov, C. Wang, and V. E. Manucharyan, 24 days-stable cnot gate on flux- onium qubits with over 99.9% fidelity, PRX Quantum6, 010349 (2025)

    2025

  35. [43]

    Marxer, J

    F. Marxer, J. Mro˙ zek, J. Andersson, L. Abdurakhi- mov, J. Adam, V. Bergholm, R. Beriwal, C. F. Chan, S. Dahl, S. R. Das, F. Deppe, O. Fedorets, Z. Gao, A. Gomez Frieiro,et al., Above 99.9% fidelity single-qubit gates, two-qubit gates, and readout in a single super- conductin...

    2026

  36. [44]

    D. J. Egger, C. Capecci, B. Pokharel, P. K. Barkoutsos, L. E. Fischer, L. Guidoni, and I. Tavernelli, Pulse vari- ational quantum eigensolver on cross-resonance-based hardware, Phys. Rev. Res.5(2023)

    2023

  37. [45]

    ¨Ust¨ un, A

    G. ¨Ust¨ un, A. Morello, and S. Devitt, Single-step parity check gate set for quantum error correction, Quantum Sci. Technol.9, 035037 (2024)

    2024

  38. [46]

    Nogueira, Bayesian Optimization: Open source constrained global optimization tool for Python, https://github.com/bayesian- optimization/BayesianOptimization (2014)

    F. Nogueira, Bayesian Optimization: Open source constrained global optimization tool for Python, https://github.com/bayesian- optimization/BayesianOptimization (2014)

    2014

  39. [47]

    R.-B. Wu, B. Chu, D. H. Owens, and H. Rabitz, Data- driven gradient algorithm for high-precision quantum control, Phys. Rev. A97, 042122 (2018)

    2018

  40. [48]

    Mahony and J

    R. Mahony and J. H. Manton, The geometry of the new- ton method on non-compact lie groups, J. Glob. Optim. 23, 309 (2002)

    2002

  41. [49]

    P. W. Michor, The cohomology of the diffeomorphism group of a manifold is a gelfand-fuks cohomology, inPro- ceedings of the 14th Winter School on Abstract Analysis (Circolo Matematico di Palermo, 1987) pp. 235–246

    1987

  42. [50]

    V. S. Varadarajan,Lie groups, Lie algebras, and their representations(Englewood Cliffs, N.J.: Prentice-Hall, 1974)

    1974

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.