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Analytic gradients for low-rank quantum optimal control

T0 review · 0 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Low-rank optimal control delivers analytic gradients at the cost of a low-rank simulation, enabling pulse design for open quantum systems that are too large for the full master equation.

desk verdict A genuinely useful methods paper: the closed-form adjoint for low-rank quantum dynamics is new and cleanly derived, with honest numerical verification; the main caveat is that the gradient is for the low-rank surrogate, not the full Lindblad loss, though the paper is upfront about this. read the letter →

arxiv 2607.14217 v1 pith:6EL6IWGZ submitted 2026-07-15 quant-ph

Leo Goutte , Vincenzo Savona This is my paper · ORCID
classification quant-ph MSC 81P6881S2249K15
keywords low-rankapproximationquantumoptimalcontroladjoint-statemethodLindbladmasterequationGRAPEpulseoptimizationsuperconductingqubitsanalyticgradients
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

The paper introduces LROC, combining a low-rank factorization of the density matrix with the adjoint-state method to compute gradients of any differentiable control objective at the same reduced cost as the low-rank forward simulation. For high-purity states, this gives a quadratic improvement in time and memory over the full Lindblad master equation, avoiding the memory explosion of reverse-mode automatic differentiation. The authors demonstrate the method on four superconducting-circuit tasks, reaching fidelities consistent with the intrinsic dissipation limits. The central claim is that quantum computing protocols, which are designed to preserve purity, keep the density matrix effectively low-rank, making this compressed optimization both accurate and scalable.

What carries the argument

The key object is the low-rank factorization ρ ≈ ΨΨ†, where Ψ is an N×M matrix, together with the low-rank master equation (6). The dissipative term O[Ψ] involves the Moore-Penrose pseudo-inverse Ψ⁺ = (Ψ†Ψ)⁻¹ Ψ†, which couples the columns and introduces nonlinearity; its derivative produces the B_s terms in the adjoint equation, corresponding to on-manifold jumps and off-manifold leakage. The analytic adjoint (9)–(10) and the GRAPE-like gradient formula (8) convert the low-rank forward simulation into a complete backpropagation at the same reduced cost, with the stored forward trajectory as the only memory overhead.

What would settle it

Compute the gradient from Eq. (8) for a candidate pulse and compare it against a finite-difference gradient of the same objective evaluated on the full Lindblad master equation, sweeping the rank M in a strongly dissipative regime. If the normalized gradient error does not decrease toward zero as M grows, or saturates at a large value when dissipation dominates the unitary dynamics, the method's central claim fails in that regime.

Watch

Extended reading notes

Core claim

The central discovery is a complete optimization loop built around the low-rank master equation Ψ̇ = -i H Ψ + O[Ψ] - μ Ψ. The forward pass propagates the N×M factor Ψ; the adjoint state λ obeys a linear backward equation with terminal condition λ(T) = ∂Φ/∂Ψ*, and the gradient with respect to piecewise-constant pulse amplitudes is given by the simple formula ∂C/∂u = 2Δt Im tr(λ† H_k Ψ) + O(Δt²). The adjoint is explicit and closed-form, including the derivative of the pseudo-inverse, and any differentiable loss enters only through the terminal derivative and a running source term, leaving the adjoint dynamics fixed. This makes gradient-based pulse optimization feasible for large open quantum s

Load-bearing premise

The whole construction presumes that, throughout the pulse, the true density matrix is well approximated inside a fixed, small M-dimensional subspace—that purity stays high—and if significant mixedness develops, the low-rank evolution and its adjoint gradient describe a different, wrong model.

Editorial extensions

If this is right

  • Pulse optimization for open quantum systems with high-purity states becomes tractable at system sizes where full Lindblad simulation is prohibitive.
  • The memory footprint of gradient computation scales as O(N M N_T), removing the substep factor that plagues reverse-mode automatic differentiation.
  • The adjoint equation is task-independent: new objectives require only the two derivatives ∂Φ/∂Ψ* and ∂ϕ/∂Ψ*, making the optimization modular.
  • The demonstrated fidelities for a 5-qubit GHZ state, cross-resonance CNOT, balanced cross-Kerr readout, and a parity-check primitive are consistent with intrinsic dissipation limits, validating the approach on realistic models.
  • The gradient accuracy is explicitly controlled by the time step Δt (quadrature error) and the rank M (truncation error), allowing straightforward convergence checks.

Reading between the lines

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

  • If the low-rank assumption holds broadly, the same analytic adjoint could be extended to tensor-network or locally purified representations, compressing both mixedness and entanglement and reaching even larger registers.
  • The method's practical range is tied to the hardware's tendency to preserve purity; a dynamically adaptive rank that tracks instantaneous mixedness would widen the regime of applicability beyond the fixed-M scheme used here.
  • The augmented-state formulation, which reduces quadrature error at the cost of dense storage, suggests a natural path to continuous pulse parametrizations (Fourier or spline coefficients) where the gradient is accumulated along the adjoint pass.
  • The explicit closed-form pseudo-inverse avoids the numerical instabilities that automatic differentiation encounters when the density matrix becomes nearly rank-deficient, a subtle advantage that grows with system size.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper introduces LROC, a gradient-based optimal control method for open quantum systems that avoids full density-matrix simulation. The approach combines a low-rank factorization of the density matrix with the adjoint-state method and a piecewise-constant GRAPE-like pulse parametrization. The forward evolution is governed by the low-rank master equation (6), and the main contribution is the analytic adjoint equation (9)–(10) and the resulting pulse gradient (8). The cost scales as O(N M^2) in time and O(N M) in memory, versus O(N^2) for the full Lindblad master equation. The method is demonstrated on four superconducting-circuit applications: five-qubit GHZ state preparation, a cross-resonance CNOT gate, dispersive readout with a nonlinear running cost, and a parity-check primitive for error correction. Final fidelities are verified with full master-equation simulations via QuantumToolbox.jl, and the code is openly available.

Significance. If the central derivation is correct, this is a valuable contribution: it provides a practical route to pulse-level optimization in open systems whose Hilbert-space dimension makes full density-matrix simulation prohibitive, under the explicit assumption that the state remains close to a rank-M manifold. The adjoint derivation in Appendix A is explicit and self-contained, and the paper goes beyond a pure formalism by validating final fidelities against full master-equation simulations. The four applications cover distinct objective structures—linear terminal, multi-state average, nonlinear running, and nonlinear terminal—which demonstrates the modularity of the adjoint framework. The availability of the code and the reproducible verification procedure are additional strengths. The main caveat, which the paper partially acknowledges in Sec. IV, is that Eq. (8) is the gradient of the low-rank surrogate objective, not of the exact Lindblad loss; this should be stated more prominently. Overall, the central claim is sound and the paper is likely to be useful to the quantum control community.

minor comments (6)
  1. [Abstract; Sec. II.B, Eq. (8)] The phrase 'gradient of any differentiable objective' should be qualified: Eq. (8) is the exact gradient of the low-rank surrogate objective, not of the full Lindblad loss. The authors are aware of this (Sec. IV), but the abstract and introduction can be misread as claiming the stronger statement. Add one sentence clarifying that the gradient is exact for the low-rank dynamics and that its accuracy for the physical loss is controlled by the forward model's accuracy, verified by M-convergence and full master-equation simulation.
  2. [Sec. III] All application results are single optimization runs; no repeated-seed statistics or random initializations are reported. Since the optimization landscapes are nonconvex, the reported fidelities may depend on the initial pulse. Reporting mean and standard deviation over a few independent runs (e.g., 5–10 seeds) for at least the GHZ and CNOT tasks would strengthen the claim that the results are typical rather than exceptional.
  3. [Sec. II.C] The quadratic scaling advantage is stated for fixed M. For completeness, briefly discuss how M should be chosen and whether it is expected to grow with system size in the intended applications. The M-convergence checks in Fig. 2(d) are helpful, but a short comment on the practical selection of M would improve the presentation.
  4. [Fig. 4 caption] Typo: 'Optimization of a the cross-resonance CNOT gate' should read 'Optimization of the cross-resonance CNOT gate'.
  5. [Fig. 5 caption] The sentence 'optimized two-step pulse of the sole .' is incomplete; it should probably read 'optimized two-step pulse of the sole drive.' Also, there are missing spaces in the main text, e.g., 'the optimization withM= 2over' in Sec. III.D.
  6. [Sec. III.D] Minor notation issue: 'withM= 2over a total time ofT= 100ns' should have spaces ('with M = 2 over a total time of T = 100 ns'). Please proofread for consistent spacing, especially in the applications sections and figure captions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the adjoint gradient is derived from the low-rank dynamics by the Lagrangian method and is benchmarked against full master-equation simulation.

full rationale

The claimed new result is the analytic gradient Eq. (8), obtained from the Lagrange functional Eq. (A1) and the adjoint equation Eqs. (A3)-(A5). This is a derivation from the low-rank equation of motion Eq. (6), not a fit: no parameter is tuned to the final fidelities, and the loss enters only through the terminal/source derivatives ∂Φ/∂Ψ* and ∂ϕ/∂Ψ*. The low-rank model itself is imported as prior work [15,20,21] and is openly an approximation with an adjustable rank M; convergence in M and comparisons with QuantumToolbox.jl full master-equation simulations are external benchmarks, so the reported fidelities are not used as inputs to the model. The self-citations [20,21] support the forward low-rank dynamics but not the new adjoint construction, and the prior low-rank equation is independently stated with assumptions (high purity, M≪N) that do not include the present gradient result. The skeptic's concern—that Eq. (8) is the gradient of the low-rank surrogate, not exactly of the full Lindblad loss—is a real approximation-accuracy limitation, explicitly acknowledged in Sec. IV ('In dissipation-dominated regimes, the backward pass instead suffers gradient inaccuracy...'), but it is not a circularity: the derivative is of the stated model, the model's validity is checked externally, and no prediction reduces by construction to an input.

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

The method introduces no new physical entities. The free parameters are numerical hyperparameters (rank M, regularization, pulse smoothing) rather than fitted physical constants. The central load-bearing axiom is the low-rank assumption, which is stated and bounded by M-convergence checks.

free parameters (4)
  • Rank M = M=2 for GHZ, CNOT, QEC; M=4 for readout
    Truncation rank of the low-rank factorization. Chosen per task to balance accuracy and cost, not fitted to target fidelities; convergence in M is verified (Fig. 2d).
  • Auxiliary column weight ε = 1e-5
    Small weight placed on auxiliary columns of Ψ to keep the Gram matrix full rank and represent small mixedness. Numerical regularization, not physics.
  • Tikhonov regularization δ = 1e-8 tr(Ψ†Ψ)/M
    Added to Ψ†Ψ before inversion to stabilize the pseudo-inverse. Numerical hyperparameter, not fitted to data.
  • Pulse-smoothing width s = not stated
    Width of the Gaussian smoothing kernel in Eq. (C2). Controls pulse bandwidth; no explicit value is given in the text, so exact reproduction requires inspecting the code.
assumptions (6)
  • domain assumption The open-system dynamics is governed by the Markovian Lindblad master equation with zero-temperature collapse operators.
    The entire formalism starts from Eq. (1). Non-Markovian environments or finite-temperature baths are outside the scope.
  • domain assumption The density matrix admits an accurate low-rank factorization ρ ≈ ΨΨ† with M≪N because quantum computing protocols preserve purity.
    Central premise introduced in the abstract and Sec. II.A. If violated, the low-rank evolution and its adjoint describe the wrong dynamics.
  • domain assumption The low-rank master equation Eq. (6) accurately approximates the Lindblad dynamics within the rank-M manifold.
    Equation (6) is taken from prior low-rank methods [20,21]. It is not exact for arbitrary Lindblad dynamics; its accuracy is controlled by M and checked numerically.
  • domain assumption Transmon qubits are modeled as five-level nonlinear oscillators with no RWA and a fixed T1 decay channel.
    Sec. III describes the device model. Results depend on this truncation and on neglecting higher transmon levels, dephasing, and temperature effects.
  • standard math A 2-design of input states accurately approximates the Haar-average gate fidelity.
    Sec. III.B uses the 2-design state set to estimate average gate fidelity, a standard result when the fidelity is a polynomial of degree ≤ 2 in the input state.
  • standard math The weak Pontryagin maximum principle and Wirtinger calculus justify the adjoint equation and gradient formula.
    Appendix A applies these standard variational tools to the low-rank evolution. Treating Ψ and Ψ* as independent is standard for non-holomorphic objectives.

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Cite this review

Pith. "Pith review of Analytic gradients for low-rank quantum optimal control." pith.science (2026). https://pith.science/paper/6EL6IWGZ

@misc{pith2026260714217,
  author       = {Pith},
  title        = {Pith review of: Analytic gradients for low-rank quantum optimal control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6EL6IWGZ}},
  note         = {Machine review of arXiv:2607.14217}
}
read the original abstract

We introduce low-rank optimal control (LROC), a method for designing control pulses in open quantum systems whose full density-matrix simulation is prohibitively expensive. The method exploits a feature of quantum computing itself: because protocols are designed to preserve purity, the density matrix is dominated by a few pure states and admits an accurate low-rank factorization. LROC propagates only this factorized form and, by deriving the corresponding adjoint equation, obtains the gradient of any differentiable objective at the same reduced cost as the simulation, leading to a quadratic improvement in time and memory compared to the full master equation. We illustrate the breadth of the method on four superconducting-circuit tasks: preparation of a five-qubit GHZ state, a CNOT gate, qubit readout, and an error correction primitive, modeled with realistic multilevel transmons, decay, and strong drives, in each case reaching fidelities consistent with the intrinsic dissipation limits. LROC thereby extends pulse-level optimization to system sizes beyond the reach of existing gradient-based methods.

Figures

Figures reproduced from arXiv: 2607.14217 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the LROC algorithm. A forward pass [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Memory footprint and (b) total runtime as a func [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Preparation of a 5-qubit GHZ state. (a) Evolution of the loss function (13) as a function of the optimization epoch. (b) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Optimization of a the cross-resonance CNOT gate. (a) Loss function (18) during training. (b) Optimized pulses on [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Optimal control of the balanced cross-Kerr transmon readout. (a) Convergence of the loss function Eq. (23) and (b) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Optimization of the mutual parity measurement. (a) Evolution of the loss function (26). (b)-(d) Optimal pulses. The [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

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Works this paper leans on

95 extracted references · 10 canonical work pages

  1. [20]

    Goutte and V

    L. Goutte and V. Savona, Physical Review Research 8, 013085 (2026), ISSN 2643-1564, URLhttps://link. aps.org/doi/10.1103/v383-2vss

    work page doi:10.1103/v383-2vss 2026

  2. [1]

    Blais, A

    A. Blais, A. L. Grimsmo, S. Girvin, and A. Wallraff, Re- views of Modern Physics93, 025005 (2021), ISSN 0034- 6861, 1539-0756, URLhttps://link.aps.org/doi/10. 1103/RevModPhys.93.025005

    2021

  3. [2]

    Krantz, M

    P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, Applied Physics Reviews6, 021318 (2019), ISSN 1931-9401, URLhttps: //pubs.aip.org/apr/article/6/2/021318/570326/ A-quantum-engineer-s-guide-to-superconducting

    2019

  4. [3]

    C. P. Koch, U. Boscain, T. Calarco, G. Dirr, S. Fil- ipp, S. J. Glaser, R. Kosloff, S. Montangero, T. Schulte- Herbrüggen, D. Sugny, et al., EPJ Quantum Technology 9, 19 (2022), ISSN 2196-0763, URLhttps://doi.org/ 10.1140/epjqt/s40507-022-00138-x

    doi:10.1140/epjqt/s40507-022-00138-x 2022

  5. [4]

    Khaneja, T

    N. Khaneja, T. Reiss, C. Kehlet, T. Schulte- Herbrüggen, and S. J. Glaser, Journal of Magnetic Resonance172, 296 (2005), ISSN 1090-7807, URL https://www.sciencedirect.com/science/article/ pii/S1090780704003696

    2005

  6. [5]

    Machnes, U

    S. Machnes, U. Sander, S. J. Glaser, P. De Fouquières, A. Gruslys, S. Schirmer, and T. Schulte-Herbrüggen, Physical Review A84, 022305 (2011), ISSN 1050-2947, 1094-1622, URLhttps://link.aps.org/doi/10.1103/ PhysRevA.84.022305

    2011

  7. [6]

    M. H. Goerz, F. Motzoi, K. B. Whaley, and C. P. Koch, npj Quantum Information3, 37 (2017), ISSN 2056-6387, URLhttps://www.nature.com/articles/ s41534-017-0036-0

    2017

  8. [7]

    George, J

    N. George, J. L. Allen, R. Kosut, and E. Gi- nossar, Scientific Reports16, 2773 (2025), ISSN 2045-2322, URLhttps://www.nature.com/articles/ s41598-025-32747-8

    2025

Show all 95 references
  1. [8]

    Abdelhafez, D

    M. Abdelhafez, D. I. Schuster, and J. Koch, Phys- ical Review A99, 052327 (2019), ISSN 2469-9926, 2469-9934, URLhttps://link.aps.org/doi/10.1103/ PhysRevA.99.052327

    2019

  2. [9]

    Gautier, E

    R. Gautier, E. Genois, and A. Blais, Phys. Rev. Lett. 134, 070802 (2025), URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.134.070802

    2025 doi

  3. [10]

    Y. Lu, S. Joshi, V. San Dinh, and J. Koch, Jour- nal of Physics Communications8, 025002 (2024), ISSN 2399-6528, URLhttps://doi.org/10.1088/2399-6528/ ad22e5

    2024 doi

  4. [11]

    Dalibard, Y

    J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. 68, 580 (1992), URLhttps://link.aps.org/doi/10. 1103/PhysRevLett.68.580

    1992

  5. [12]

    Mølmer and Y

    K. Mølmer and Y. Castin, Quantum and Semiclassical Optics: Journal of the European Optical Society Part B 8, 49 (1996), ISSN 1355-5111, URLhttps://doi.org/ 11 10.1088/1355-5111/8/1/007

    1996 doi

  6. [13]

    Verstraete, J

    F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Phys- ical Review Letters93, 207204 (2004), ISSN 0031-9007, 1079-7114, URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.93.207204

    2004

  7. [14]

    Weimer, A

    H. Weimer, A. Kshetrimayum, and R. Orús, Reviews of Modern Physics93, 015008 (2021), ISSN 0034-6861, 1539-0756, URLhttps://link.aps.org/doi/10.1103/ RevModPhys.93.015008

    2021

  8. [15]

    Le Bris and P

    C. Le Bris and P. Rouchon, Physical Review A87, 022125 (2013), URLhttps://link.aps.org/doi/10. 1103/PhysRevA.87.022125

    2013

  9. [16]

    Joubert-Doriol and A

    L. Joubert-Doriol and A. F. Izmaylov, The Jour- nal of Chemical Physics142, 134107 (2015), ISSN 0021-9606, 1089-7690, URLhttps://pubs. aip.org/jcp/article/142/13/134107/901966/ Problem-free-time-dependent-variational-principle

    2015

  10. [17]

    McCaul, K

    G. McCaul, K. Jacobs, and D. I. Bondar, Physical Review Research3, 013017 (2021), ISSN 2643- 1564, URLhttps://link.aps.org/doi/10.1103/ PhysRevResearch.3.013017

    2021

  11. [18]

    Santos, X

    S. Santos, X. Song, and V. Savona, Quantum9, 1620 (2025), URLhttps://quantum-journal.org/papers/ q-2025-02-04-1620/

    2025

  12. [19]

    Gravina and V

    L. Gravina and V. Savona, Physical Review Research 6, 023072 (2024), ISSN 2643-1564, URLhttps://link. aps.org/doi/10.1103/PhysRevResearch.6.023072

    2024 doi

  13. [21]

    Joubert-Doriol, I

    L. Joubert-Doriol, I. G. Ryabinkin, and A. F. Iz- maylov, The Journal of Chemical Physics141, 234112 (2014), ISSN 0021-9606, 1089-7690, URLhttps: //pubs.aip.org/jcp/article/141/23/234112/194230/ Non-stochastic-matrix-Schrodinger-equation-for

    2014

  14. [22]

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. Du- venaud, inProceedings of the 32nd International Confer- ence on Neural Information Processing Systems(Curran Associates Inc., Red Hook, NY, USA, 2018), NIPS’18, p. 6572–6583

    2018

  15. [23]

    Doria, T

    P. Doria, T. Calarco, and S. Montangero, Physical Review Letters106, 190501 (2011), ISSN 0031-9007, 1079-7114, URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.106.190501

    2011

  16. [24]

    Petruhanov and A

    V. Petruhanov and A. Pechen, Journal of Physics A: Mathematical and Theoretical56, 305303 (2023), ISSN 1751-8113, 1751-8121, arXiv:2307.08479 [quant- ph], URLhttps://iopscience.iop.org/article/10. 1088/1751-8121/ace13f

    2023 arXiv

  17. [25]

    L.-X. Xie, L. Z. d. Paula, W. Cai, Q.-X. Jie, L. Sun, C.- L. Zou, G.-C. Guo, Z.-J. Chen, and X.-B. Zou,Optimal control with flag qubits(2026), arXiv:2603.12162 [quant- ph], URLhttp://arxiv.org/abs/2603.12162

    2026

  18. [26]

    Schmidt, A

    R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, Physical Review Letters107, 130404 (2011), ISSN 0031-9007, 1079-7114, URLhttps://link. aps.org/doi/10.1103/PhysRevLett.107.130404

    2011 doi

  19. [27]

    Z.-J. Chen, H. Huang, L. Sun, Q.-X. Jie, J. Zhou, Z. Hua, Y. Xu, W. Wang, G.-C. Guo, C.-L. Zou, et al., Sci- ence Advances11, eadr0875 (2025), URLhttps://www. science.org/doi/10.1126/sciadv.adr0875

    2025 doi

  20. [28]

    Breuer and F

    H.-P. Breuer and F. Petruccione,The theory of open quantum systems(Oxford University Press, Oxford ; New York, 2002), ISBN 978-0-19-852063-4, oCLC: ocm49872077

    2002

  21. [29]

    Shillito, A

    R. Shillito, A. Petrescu, J. Cohen, J. Beall, M. Hauru, M. Ganahl, A. G. Lewis, G. Vidal, and A. Blais, Phys. Rev. Appl.18, 034031 (2022), URLhttps://link.aps. org/doi/10.1103/PhysRevApplied.18.034031

    2022 doi

  22. [30]

    Le Bris, P

    C. Le Bris, P. Rouchon, and J. Roussel, Physi- cal Review A92, 062126 (2015), ISSN 1050-2947, 1094-1622, URLhttps://link.aps.org/doi/10.1103/ PhysRevA.92.062126

    2015

  23. [31]

    Campaioli, J

    F. Campaioli, J. H. Cole, and H. Hapuarachchi, PRX Quantum5, 020202 (2024), ISSN 2691-3399, URL https://link.aps.org/doi/10.1103/PRXQuantum.5. 020202

    2024 doi

  24. [32]

    Werner, D

    A. Werner, D. Jaschke, P. Silvi, M. Kliesch, T. Calarco, J. Eisert, and S. Montangero, Physical Review Letters116, 237201 (2016), ISSN 0031-9007, 1079-7114, URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.116.237201

    2016

  25. [33]

    Guo and S

    Y. Guo and S. Yang, Chinese Physics Letters 41, 120302 (2024), ISSN 0256-307X, 1741-3540, URLhttps://iopscience.iop.org/article/10.1088/ 0256-307X/41/12/120302

    2024

  26. [34]

    D. P. Kingma and J. Ba, inInternational Conference on Learning Representations (ICLR)(2015), URLhttps: //arxiv.org/abs/1412.6980

    2015 arXiv

  27. [35]

    Boscain, M

    U. Boscain, M. Sigalotti, and D. Sugny, PRX Quantum 2, 030203 (2021), ISSN 2691-3399, URLhttps://link. aps.org/doi/10.1103/PRXQuantum.2.030203

    2021 doi

  28. [36]

    M. H. Goerz, S. C. Carrasco, A. Marshall, and V. S. Malinovsky, Journal of Open Source Software10, 8813 (2025), ISSN 2475-9066, URLhttps://joss.theoj. org/papers/10.21105/joss.08813

    2025 doi

  29. [37]

    Moses and V

    W. Moses and V. Churavy, inAdvances in Neural In- formation Processing Systems, edited by H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin (Cur- ran Associates, Inc., 2020), vol. 33, pp. 12472–12485, URLhttps://proceedings.neurips.cc/paper/2020/ file/9332c513ef4...

    2020

  30. [38]

    W. S. Moses, V. Churavy, L. Paehler, J. Hückelheim, S. H. K. Narayanan, M. Schanen, and J. Doerfert, in Proceedings of the International Conference for High Per- formance Computing, Networking, Storage and Analy- sis(Association for Computing Machinery, New York, NY, USA, 2021...

    2021

  31. [39]

    Swiadek, R

    F. Swiadek, R. Shillito, P. Magnard, A. Remm, C. Hellings, N. Lacroix, Q. Ficheux, D. C. Zanuz, G. J. Norris, A. Blais, et al., PRX Quantum5, 040326 (2024), ISSN 2691-3399, URLhttps://link.aps.org/doi/10. 1103/PRXQuantum.5.040326

    2024

  32. [40]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Bran- dao, D. A. Buell, et al., Nature574, 505 (2019), ISSN 1476-4687, URLhttps://www.nature.com/articles/ s41586-019-1666-5

    2019

  33. [41]

    D. Sank, Z. Chen, M. Khezri, J. Kelly, R. Barends, B. Campbell, Y. Chen, B. Chiaro, A. Dunsworth, A. Fowler, et al., Physical Review Letters117, 190503 (2016), ISSN 0031-9007, 1079-7114, URLhttps://link. aps.org/doi/10.1103/PhysRevLett.117.190503

    2016 doi

  34. [42]

    Kohler, Physical Review A98, 023849 (2018), ISSN 2469-9926, 2469-9934, URLhttps://link.aps

    S. Kohler, Physical Review A98, 023849 (2018), ISSN 2469-9926, 2469-9934, URLhttps://link.aps. org/doi/10.1103/PhysRevA.98.023849. 12

    2018 doi

  35. [43]

    Khezri, A

    M. Khezri, A. Opremcak, Z. Chen, K. C. Miao, M. McEwen, A. Bengtsson, T. White, O. Naaman, D. Sank, A. N. Korotkov, et al., Physical Review Applied 20, 054008 (2023), ISSN 2331-7019, URLhttps://link. aps.org/doi/10.1103/PhysRevApplied.20.054008

    2023 doi

  36. [44]

    M. F. Dumas, B. Groleau-Paré, A. McDonald, M. H. Muñoz-Arias, C. Lledó, B. D’Anjou, and A. Blais, Physical Review X14, 041023 (2024), ISSN 2160-3308, URLhttps://link.aps.org/doi/10.1103/ PhysRevX.14.041023

    2024

  37. [45]

    Ferrari, L

    F. Ferrari, L. Gravina, D. Eeltink, P. Scarlino, V. Savona, and F. Minganti, Physical Review Research7, 013276 (2025), ISSN 2643-1564, URLhttps://link.aps.org/ doi/10.1103/PhysRevResearch.7.013276

    2025 doi

  38. [46]

    Mercurio, Y.-T

    A. Mercurio, Y.-T. Huang, L.-X. Cai, Y.-N. Chen, V. Savona, and F. Nori, Quantum9, 1866 (2025), ISSN 2521-327X, URLhttps://doi.org/10.22331/ q-2025-09-29-1866

    2025

  39. [47]

    D. Zhu, Z. P. Cian, C. Noel, A. Risinger, D. Biswas, L. Egan, Y. Zhu, A. M. Green, C. H. Alderete, N. H. Nguyen, et al., Nature Communications13, 6620 (2022), ISSN 2041-1723, URLhttps://www.nature. com/articles/s41467-022-34279-5

    2022

  40. [48]

    Rigetti and M

    C. Rigetti and M. Devoret, Physical Review B81, 134507 (2010), ISSN 1098-0121, 1550-235X, URLhttps: //link.aps.org/doi/10.1103/PhysRevB.81.134507

    2010 doi

  41. [49]

    J. M. Chow, A. D. Córcoles, J. M. Gambetta, C. Rigetti, B. R. Johnson, J. A. Smolin, J. R. Rozen, G. A. Keefe, M. B. Rothwell, M. B. Ketchen, et al., Physi- cal Review Letters107, 080502 (2011), ISSN 0031-9007, 1079-7114, URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.107.080502

    2011

  42. [50]

    Tripathi, M

    V. Tripathi, M. Khezri, and A. N. Korotkov, Phys- ical Review A100, 012301 (2019), ISSN 2469-9926, 2469-9934, URLhttps://link.aps.org/doi/10.1103/ PhysRevA.100.012301

    2019

  43. [51]

    E.MagesanandJ.M.Gambetta, PhysicalReviewA101, 052308 (2020), ISSN 2469-9926, 2469-9934, URLhttps: //link.aps.org/doi/10.1103/PhysRevA.101.052308

    2020 doi

  44. [52]

    M. A. Nielsen, Physics Letters A303, 249 (2002), ISSN 0375-9601, URLhttps://www.sciencedirect. com/science/article/pii/S0375960102012720

    2002

  45. [53]

    Dankert, R

    C. Dankert, R. Cleve, J. Emerson, and E. Livine, Phys- ical Review A80, 012304 (2009), ISSN 1050-2947, 1094-1622, URLhttps://link.aps.org/doi/10.1103/ PhysRevA.80.012304

    2009

  46. [55]

    A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez- Soto, Journal of Physics A: Mathematical and Theo- retical40, 3987 (2007), ISSN 1751-8121, URLhttps: //doi.org/10.1088/1751-8113/40/14/014

    2007 doi

  47. [56]

    A. B. Klimov, J. L. Romero, G. Björk, and L. L. Sánchez- Soto, Annals of Physics324, 53 (2009), ISSN 0003- 4916, URLhttps://www.sciencedirect.com/science/ article/pii/S0003491608001541

    2009

  48. [57]

    Sheldon, E

    S. Sheldon, E. Magesan, J. M. Chow, and J. M. Gam- betta, Physical Review A93, 060302 (2016), ISSN 2469- 9926, 2469-9934, URLhttps://link.aps.org/doi/10. 1103/PhysRevA.93.060302

    2016

  49. [58]

    Caneva, T

    T. Caneva, T. Calarco, and S. Montangero, Phys. Rev. A84, 022326 (2011), URLhttps://link.aps.org/doi/ 10.1103/PhysRevA.84.022326

    2011 doi

  50. [59]

    Y. Baum, M. Amico, S. Howell, M. Hush, M. Liuzzi, P. Mundada, T. Merkh, A. R. Carvalho, and M. J. Biercuk, PRX Quantum2, 040324 (2021), URLhttps: //link.aps.org/doi/10.1103/PRXQuantum.2.040324

    2021 doi

  51. [60]

    Cohen, A

    J. Cohen, A. Petrescu, R. Shillito, and A. Blais,Rem- iniscence of classical chaos in driven transmons(2022), arXiv:2207.09361 [quant-ph], URLhttp://arxiv.org/ abs/2207.09361

    2022 arXiv

  52. [61]

    Boutin, C

    S. Boutin, C. K. Andersen, J. Venkatraman, A. J. Fer- ris, and A. Blais, Physical Review A96, 042315 (2017), ISSN 2469-9926, 2469-9934, URLhttps://link.aps. org/doi/10.1103/PhysRevA.96.042315

    2017 doi

  53. [62]

    Chatterjee, J

    A. Chatterjee, J. Schwinger, and Y. Y. Gao, Physical Review Applied23, 054057 (2025), ISSN 2331-7019, URLhttps://link.aps.org/doi/10.1103/ PhysRevApplied.23.054057

    2025

  54. [63]

    Genois, N

    E. Genois, N. J. Stevenson, N. Goss, I. Siddiqi, and A. Blais, Phys. Rev. Appl.24, 034073 (2025), URL https://link.aps.org/doi/10.1103/d9yg-d3qr

    2025 doi

  55. [64]

    Bengtsson, A

    A. Bengtsson, A. Opremcak, M. Khezri, D. Sank, A. Bourassa, K. J. Satzinger, S. Hong, C. Erickson, B. J. Lester, K. C. Miao, et al., Phys. Rev. Lett. 132, 100603 (2024), URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.132.100603

    2024 doi

  56. [65]

    Blais, R.-S

    A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Physical Review A69, 062320 (2004), ISSN 1050-2947, 1094-1622, URLhttps://link.aps. org/doi/10.1103/PhysRevA.69.062320

    2004 doi

  57. [66]

    Didier, J

    N. Didier, J. Bourassa, and A. Blais, Physical Re- view Letters115, 203601 (2015), ISSN 0031-9007, 1079-7114, URLhttps://link.aps.org/doi/10.1103/ PhysRevLett.115.203601

    2015

  58. [67]

    A. A. Chapple, O. Benhayoune-Khadraoui, S. Richer, and A. Blais, Phys. Rev. Lett.135, 256002 (2025), URL https://link.aps.org/doi/10.1103/r4v5-wyyt

    2025 doi

  59. [68]

    Wang, F.-M

    C. Wang, F.-M. Liu, H. Chen, Y.-F. Du, C. Ying, J.-W. Wang, Y.-H.Huo, C.-Z.Peng, X.Zhu, M.-C.Chen, etal., Physical Review Letters135, 060803 (2025), ISSN 0031- 9007, 1079-7114, URLhttps://link.aps.org/doi/10. 1103/98n9-13y4

    2025

  60. [69]

    C. Mori, V. Milchakov, F. D’Esposito, L. Ruela, S. Ku- mar, V. N. Suresh, W. Ardati, D. Nicolas, G. Cap- pelli, A. Ranadive, et al.,High-power readout of a transmon qubit using a nonlinear coupling(2025), arXiv:2507.03642 [quant-ph], URLhttp://arxiv.org/ abs/2507.03642

    2025 arXiv

  61. [70]

    Beaulieu, J.-Z

    G. Beaulieu, J.-Z. Chen, M. Scigliuzzo, O. Benhayoune- Khadraoui, A. A. Chapple, P. A. Spring, A. Blais, and P. Scarlino,Fast, high-fidelity Transmon readout with intrinsic Purcell protection via nonperturbative cross- Kerr coupling(2026), arXiv:2601.04975 [quant-ph], URL http...

    2026

  62. [71]

    G. P. Gehér, M. Jastrzebski, E. T. Campbell, and O. Crawford, npj Quantum Information11, 39 (2025), ISSN 2056-6387, URLhttps://www.nature. com/articles/s41534-025-00998-y

    2025

  63. [72]

    P. D. Kurilovich, T. Connolly, C. G. L. Bøttcher, D. K. Weiss, S. Hazra, V. R. Joshi, A. Z. Ding, H. Nho, S. Diamond, V. D. Kurilovich, et al.,High-frequency readout free from transmon multi-excitation resonances (2025), arXiv:2501.09161, URLhttp://arxiv.org/abs/ 2501.09161

    2025 arXiv

  64. [73]

    R. A. Mencia, T. Imaizumi, I. A. Golovchanskiy, A. Lizzit, and V. E. Manucharyan,Raising the Cavity Frequency in cQED(2025), arXiv:2511.22764 [quant-ph], URLhttp://arxiv.org/abs/2511.22764. 13

    2025

  65. [74]

    A. V. Dixit, Z. L. Parrott, D. Chunikhin, B. Hauer, T. F. Q. Larson, and J. D. Teufel,Millimeter Wave Read- out of a Superconducting Qubit(2026), arXiv:2603.13837 [quant-ph], URLhttp://arxiv.org/abs/2603.13837

    2026

  66. [75]

    W. Dai, S. Hazra, D. K. Weiss, P. D. Kurilovich, T. Connolly, H. K. Babla, S. Singh, V. R. Joshi, A. Z. Ding, P. D. Parakh, et al.,Spectroscopy of drive-induced unwanted state transitions in superconducting circuits (2025), arXiv:2506.24070 [quant-ph] version: 1, URL http://ar...

    2025

  67. [76]

    Saira, J

    O.-P. Saira, J. Groen, J. Cramer, M. Meretska, G. De Lange, and L. DiCarlo, Physical Review Letters 112, 070502 (2014), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett. 112.070502

    2014 doi

  68. [77]

    Kjaergaard, M

    M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, Annual Review of Condensed Matter Physics 11, 369 (2020), ISSN 1947-5454, 1947-5462, URL https://www.annualreviews.org/content/journals/ 10.1146/annurev-conmatphys-031119-050605

    2020 doi

  69. [78]

    Ali,Surface-code Superconducting Quantum Proces- sors: From Calibration To Logical Performance(2025), arXiv:2504.17082 [quant-ph], URLhttp://arxiv.org/ abs/2504.17082

    H. Ali,Surface-code Superconducting Quantum Proces- sors: From Calibration To Logical Performance(2025), arXiv:2504.17082 [quant-ph], URLhttp://arxiv.org/ abs/2504.17082

    2025 arXiv

  70. [79]

    C. K. Andersen, A. Remm, S. Lazar, S. Krinner, J. Heinsoo, J.-C. Besse, M. Gabureac, A. Wallraff, and C. Eichler, npj Quantum Information5, 69 (2019), ISSN 2056-6387, URLhttps://www.nature.com/articles/ s41534-019-0185-4

    2019

  71. [80]

    Remm, Ph.D

    A. Remm, Ph.D. thesis, ETH Zurich (2023), artwork Size: 188 p. Medium: application/pdf Pages: 188 p., URLhttp://hdl.handle.net/20.500.11850/652933

    2023

  72. [81]

    J. F. Marques, B. M. Varbanov, M. S. Moreira, H. Ali, N. Muthusubramanian, C. Zachariadis, F. Battistel, M. Beekman, N. Haider, W. Vlothuizen, et al., Nature Physics18, 80 (2022), ISSN 1745-2481, URLhttps: //www.nature.com/articles/s41567-021-01423-9

    2022

  73. [82]

    J. M. Chow, J. M. Gambetta, E. Magesan, D. W. Abra- ham, A. W. Cross, B. R. Johnson, N. A. Masluk, C. A. Ryan, J. A. Smolin, S. J. Srinivasan, et al., Nature Communications5, 4015 (2014), ISSN 2041-1723, URL https://www.nature.com/articles/ncomms5015

    2014

  74. [83]

    R. L. Kosut, M. D. Grace, and C. Brif, Physi- cal Review A88, 052326 (2013), ISSN 1050-2947, 1094-1622, URLhttps://link.aps.org/doi/10.1103/ PhysRevA.88.052326

    2013

  75. [84]

    Goutte,Lroc.jl: Low-rank optimal control(2026), URLhttps://doi.org/10.5281/zenodo.21372277

    L. Goutte,Lroc.jl: Low-rank optimal control(2026), URLhttps://doi.org/10.5281/zenodo.21372277

    2026 doi

  76. [85]

    K. Koor, Y. Qiu, L. C. Kwek, and P. Rebentrost,A short tutorial on wirtinger calculus with applications in quantum information(2023), 2312.04858, URLhttps: //arxiv.org/abs/2312.04858

    2023 arXiv

  77. [86]

    Tsitouras and View Profile, Computers & Mathe- matics with Applications62, 770 (2011), URLhttps: //dl.acm.org/doi/10.1016/j.camwa.2011.06.002

    C. Tsitouras and View Profile, Computers & Mathe- matics with Applications62, 770 (2011), URLhttps: //dl.acm.org/doi/10.1016/j.camwa.2011.06.002

    2011 doi

  78. [87]

    Rackauckas and Q

    C. Rackauckas and Q. Nie, Journal of Open Research Software5, 15 (2017)

    2017

  79. [88]

    Leung, M

    N. Leung, M. Abdelhafez, J. Koch, and D. Schus- ter, Physical Review A95, 042318 (2017), ISSN 2469- 9926, 2469-9934, URLhttp://link.aps.org/doi/10. 1103/PhysRevA.95.042318

    2017

  80. [89]

    M. H. Goerz, S. C. Carrasco, and V. S. Ma- linovsky, Quantum6, 871 (2022), URLhttps:// quantum-journal.org/papers/q-2022-12-07-871/

    2022

  81. [90]

    Gautier (2024)

    R. Gautier (2024). Appendix A: Details and derivation of LROC algorithm In this appendix we collect the derivation details un- derlying the main-text results of Sec. II. The derivation follows the application of the Weak Pontryagin maximum principle [35] to the low-rank evolut...

    2024

  82. [91]

    (B1) withA=|ψ t⟩⟨ψt|andh(F) = 1−F gives the adjoint boundary condition directly, λ(T) = ∂Φ ∂Ψ∗ =− ψt ψt Ψ(T),(B3) with no running source

    State fidelity The simplest objective is the terminal state infidelity with objective|ψ t⟩, C= 1−tr Ψ†(T) ψt ψt Ψ(T) .(B2) Applying Eq. (B1) withA=|ψ t⟩⟨ψt|andh(F) = 1−F gives the adjoint boundary condition directly, λ(T) = ∂Φ ∂Ψ∗ =− ψt ψt Ψ(T),(B3) with no running source. The...

  83. [92]

    Running cost: transmon readout Transmon readout combines several features at once: a running cost, two coupled trajectories, an outer non- linearity, and a penalty. Each of these pieces reduces to 15 Contribution LossCBoundary∂Φ/∂Ψ ∗ Source∂ϕ/∂Ψ ∗ State fidelity1−tr Ψ†(T)P tΨ(...

  84. [93]

    (26) is nonlinear in the state fidelities, demonstrating that the recipe is not restricted to objectives linear or quadratic inΨ

    Nonlinear objective: parity-check The parity-check objective of Eq. (26) is nonlinear in the state fidelities, demonstrating that the recipe is not restricted to objectives linear or quadratic inΨ. For each state in the 2-design indexed byk, CQEC = 1− 1 K KX k=1 q p+ k + q p− ...

  85. [94]

    The Gram matrix is Tikhonov-regularized,Ψ †Ψ→Ψ †Ψ+δ11 withδ= 10−8 tr(Ψ†Ψ)/M, before inversion

    Practical implementation The forward (6) and adjoint (9) equations are in- tegrated with an adaptive fifth-order Tsitouras scheme 16 (Tsit5) [86, 87] at fixed tolerances, decoupling the step count fromN T. The Gram matrix is Tikhonov-regularized,Ψ †Ψ→Ψ †Ψ+δ11 withδ= 10−8 tr(Ψ†...

  86. [95]

    Pulse constraints Experimental pulses must vanish at the endpoints, re- spect bandwidth limits, and remain within amplitude bounds. Rather than penalizing violations in the loss, we enforce these constraints through the parametriza- tion: the bare optimization variablesu(j) k ...

  87. [96]

    In this sec- tion, we briefly summarize the key distinctions

    Comparison to automatic differentiation The trade-off between hard-coded gradients and au- tomatic differentiation (AD) in gradient-based quantum control is well characterized [8, 10, 88, 89]. In this sec- tion, we briefly summarize the key distinctions. The signature advantag...

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