Quantum optimal control using phase-modulated driving fields

Fedor Jelezko; Haibin Liu; Jianming Cai; Jiazhao Tian; Pengcheng Yang; Ralf Betzholz; Ressa S. Said; Yu Liu

REVIEW 3 major objections 6 minor 82 references

Quantum optimal control using phase-modulated driving fields

T0 review · 3 major / 6 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read Phase-modulated driving fields make constrained quantum optimal control more efficient, reaching the best ensemble fidelity with three parameters where a Fourier basis needs twenty.

desk verdict A useful new basis for gradient-free quantum control, but the order-of-magnitude speedup claim may be partly an artifact of the evaluation-cap design. read the letter →

arxiv 2009.10275 v1 pith:YRQGS5TY submitted 2020-09-22 quant-ph

Jiazhao Tian , Haibin Liu , Yu Liu , Pengcheng Yang , Ralf Betzholz , Ressa S. Said , Fedor Jelezko , Jianming Cai This is my paper · ORCID

classification quant-ph

keywords phase-modulateddrivingfieldsquantumoptimalcontrolensembleinhomogeneousbroadeninggradient-freeoptimizationrobustnesscoherencetimedynamicaldecouplingnitrogen-vacancycenters

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 proposes a gradient-free quantum optimal control scheme in which the driving field is built from phase-modulated rather than plain Fourier basis functions. In numerical experiments on an inhomogeneously broadened ensemble of two-level systems, a single phase-modulated term with three parameters reaches the best ensemble fidelity in the shortest search time, while a phase-extended Fourier basis needs twenty parameters and about an order of magnitude more evaluations to match it. The optimized fields are also more robust to detuning spread, control-amplitude variations, and dephasing, and the paper shows that an XY-8 dynamical decoupling sequence built from such gates prolongs simulated coherence from roughly 24 to 39 microseconds. A sympathetic reader would care because the method attacks the parameter-count bottleneck that makes optimal control of ensembles and many-body systems computationally expensive.

What carries the argument

The central object is the phase-modulated driving field $g_{pm}(t)=\sum_{j=1}^{N}a_j\cos[\omega_0 t+(b_j/\nu_j)\sin(\nu_j t)]$, whose modulation converts, via the Jacobi–Anger identity, into an infinite set of sidebands at frequencies $\omega_0+l\nu_j$ with Bessel-function weights. A single PM basis element therefore contains multiple Fourier components, so the optimizer can cover the ensemble's frequency spread with very few parameters; in the interaction picture the same term produces both $\sigma_x$ and $\sigma_y$ control components, which the paper identifies as another source of its advantage over plain SFB. This basis, not the specific solver, is what carries the efficiency and robustness claims.

What would settle it

A direct check is an experiment on a spin ensemble with independently characterized Gaussian broadening: implement the PM-optimized X and Y gates in an XY-8 sequence and compare $T_2$ with rectangular pulses of the same maximal amplitude; the paper's central quantitative prediction is a ratio of about $39/24\approx1.6$, and a result far from that would falsify the claim.

Watch

Extended reading notes

Core claim

The central claim is that the choice of basis for parameterizing a control field matters more than the number of parameters: a phase-modulated basis packs many frequency components into a single basis function, so few parameters can shape a rich, robust control field. In the constrained ensemble-control problem with fixed maximal amplitude, the PM method with $N=1$ and three parameters obtains the highest objective value $F_{\rm obj}$ with the shortest average search time; the SFB-P2 method needs $N=5$ and twenty parameters to give comparable results, at about ten times the search cost. The same PM fields enlarge the fidelity region in the detuning-versus-amplitude-error plane by a factor of about 1.98 without dephasing and about 1.74 with dephasing, compared with SFB-P2 at the same parameter count. Applied to gate synthesis, PM-optimized pulses yield a simulated XY-8 coherence time $T_2\approx 39\,\mu\mathrm{s}$, versus roughly $24\,\mu\mathrm{s}$ for rectangular pulses.

Load-bearing premise

The whole comparison assumes the ensemble inhomogeneity is exactly a zero-mean Gaussian distribution with a given FWHM, that control-amplitude errors enter as one constant multiplicative factor, and that every spin has the same dephasing rate; if real ensembles have non-Gaussian, correlated, or spatially varying inhomogeneity, the reported robustness ratios and the coherence-time gain may not transfer.

Editorial extensions

If this is right

In the constrained ensemble-control problem with fixed maximal amplitude, the PM basis with $N=1$ and three parameters reaches the largest objective value $F_{\rm obj}$ in the shortest average search time, while SFB-P2 needs $N=5$ and twenty parameters to match it at about an order of magnitude more function evaluations.
Optimal PM fields widen the region of high fidelity in the detuning–amplitude-variation plane: the area with fidelity above 0.9 is about 1.98 times larger than SFB-P2 with the same parameter count, and about 1.74 times larger when dephasing at $\gamma=2\pi\times2\,\mathrm{MHz}$ is included.
The PM method also yields robust single-qubit gates for Hadamard, Pauli-X, Pauli-Y, and Pauli-Z under inhomogeneous broadening, though for Pauli-Y it no longer shows absolute superiority over SFB-P2.
In a simulated XY-8 dynamical-decoupling sequence, PM-optimized gates prolong the coherence time from roughly $T_2=24\,\mu\mathrm{s}$ with rectangular pulses to $T_2=39\,\mu\mathrm{s}$, an increase of about one half.
The relative advantage is attributed to the form of the basis rather than the specific solver, so the paper expects similar gains when the PM basis is used with gradient-based solvers.

Reading between the lines

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

The authors do not test non-Gaussian or spatially correlated inhomogeneity; because the Jacobi–Anger expansion itself is generic, the parameter-efficiency advantage may persist, but the concrete robustness ratios (1.98:1, 1.74:1) are conditional on the idealized Gaussian, constant-scaling, common-dephasing model.
A testable extension is to reuse the PM basis in bandwidth-limited control of many-body or multi-qubit systems, where the objective evaluation is expensive; the paper's mechanism predicts that the few-parameter frequency coverage should lower the number of required function evaluations, but the ruggedness of those landscapes could alter the picture.
The $T_2$ gain is a numerical prediction from one Ornstein-Uhlenbeck noise model; an experimental implementation should hold maximal pulse amplitude fixed and account for the unequal pulse lengths (100 ns PM vs 50 ns rectangular) before attributing the gain to phase modulation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

The thing to know about this paper: the phase-modulated (PM) ansatz g(t)=sum_j a_j cos[omega0 t + (b_j/nu_j) sin(nu_j t)] is a genuine new element in the CRAB family, and the numerical evidence that it needs fewer parameters than a standard Fourier basis is solid enough to take seriously. The paper is honest: SFB-P2 with 20 parameters catches up to PM with 3, and they built SFB-P2 specifically for a fair comparison. The Jacobi-Anger expansion showing infinite Floquet sidebands from few parameters is a nice way to see why this could work.

Where I have reservations is the headline 'one order of magnitude less search time.' The default cap on function evaluations is 200 x N_p. For the main comparison (PM N=1 vs SFB-P2 N=5) that gives caps of 600 vs 4000, a ratio of 6.7. If a large fraction of runs terminate by hitting the cap, the reported average search times are predetermined by the experimental design, not by the basis. The paper never says how many of the 120 runs hit the cap. The trajectory comparison in Fig. 2 (both N=1) supports the qualitative claim, but the quantitative order-of-magnitude statement needs a target-based comparison, e.g., time to reach a fixed Fobj, or at least a histogram of n_f with cap hits marked.

The other soft spots are less severe. Everything is single-realization numerics with no error bars; the T2 approx 39 microseconds vs 24 microseconds comes from one noise model with fixed parameters. The Gaussian-detuning assumption is standard and stated, but the 50% coherence gain might not transfer to non-Gaussian broadening. No code or data is provided, so I can't verify the simulations directly. These are minor-to-moderate issues; none looks load-bearing.

What the paper does well: the physics is clear, the baselines are fair, and the robustness maps (fidelity vs detuning and amplitude variation) are a useful way to present the advantage. The XY-8 demonstration is a good practical hook.

I'd send this to review. The method is simple, clearly presented, and likely useful for people working on spin-ensemble control. But I'd ask the authors to release code or data and to redo the efficiency comparison in a way that is insensitive to the evaluation cap, or at minimum report cap-hit fractions.

Referee Report

3 major / 6 minor

Summary. The paper proposes a phase-modulated (PM) driving field basis for gradient-free quantum optimal control, in which each basis element carries an infinite set of frequency sidebands through a single modulation index. The method is applied to the control of an inhomogeneously broadened ensemble of two-level systems, and compared numerically with the standard Fourier basis (SFB) and two phase-introduced variants (SFB-P and SFB-P2) in terms of the ensemble objective function, the search time (number of objective-function evaluations), and robustness against detuning and control-amplitude variations. The authors report that PM with only three parameters (N=1) reaches the best objective value with the shortest average search time, that SFB-P2 needs twenty parameters to match it, that PM yields larger fidelity contours in the detuning–amplitude plane, and that an XY-8 sequence built from PM-optimized gates extends the simulated coherence time from about 24 µs to about 39 µs. The paper also discusses gate synthesis for Hadamard, Pauli-X, Pauli-Y, and Pauli-Z gates and places PM within the CRAB family.

Significance. If the efficiency claim survives scrutiny, the PM basis is a useful and physically motivated addition to the truncated-basis toolbox: the Jacobi–Anger expansion shows that a single basis element already covers many frequencies, which is a clear conceptual advantage over ordinary Fourier terms. The robustness maps, the gate-fidelity study, and the XY-8 coherence-time simulation indicate practical potential for quantum sensing and ensemble control. The manuscript is careful to include a fairer comparison basis (SFB-P2) and to test robustness against unoptimized factors; the work also explicitly discusses when randomization is not beneficial. However, because the central performance claims are numerical and the search-time claim may be affected by the per-parameter evaluation cap, the quantitative conclusions need additional verification before the paper can be recommended for publication.

major comments (3)

[Sec. III.B, Figs. 1(c), 5(d-f)] The headline claim that PM achieves the same objective value with 'one order of magnitude less average search time' than SFB-P2 is not established by the reported data, because the default termination cap is proportional to the number of parameters (200×Np, i.e., 600 for PM N=1 and 4000 for SFB-P2 N=5), and the manuscript does not report how many of the 120 runs terminate by hitting the cap. If most runs of both methods exhaust their respective budgets, the ratio of mean evaluation counts is largely predetermined by the experimental design rather than by the control basis. Please provide the distribution of nf across runs, the fraction of cap-limited runs, and a target-based comparison (e.g., evaluations needed to reach a fixed Fobj threshold, or a common evaluation budget for both methods) to support the order-of-magnitude claim.
[Sec. III.B, Sec. IV.A (constraint handling)] The implementation of the amplitude constraint max|g(t)| ≤ Ωmax in the Nelder–Mead search is not described. For PM with N=1 the constraint can be checked and enforced on a single amplitude parameter, whereas for SFB-P2 with N=5 the constraint couples all twenty parameters nontrivially through the sum of cosine terms. If the constraint is enforced by penalty or rejection methods, the comparison between PM and SFB-P2 conflates basis expressiveness with constraint-handling difficulty. Please specify exactly how the constraint is imposed for each basis and whether the reported Fobj values and evaluation counts are affected by constraint handling.
[Sec. IV.A, Sec. V (Figs. 6, 8, 11)] Several quantitative robustness and performance statements are reported as single numbers without uncertainty or sensitivity analysis: the f>0.9 area ratios (1:1.98 and 1:1.74), the Fobj values in Figs. 1 and 9, and the coherence-time improvement T2≈39 µs versus ≈24 µs. Figure 11 averages 1200 evolutions per point, so a standard error on T2 should be computable, and the contour-area ratios depend on the detuning/amplitude grid resolution. Please add error bars, confidence intervals, or a sensitivity analysis (e.g., varying M and K, or bootstrapping over initial points) to support these quantitative claims, and state explicitly how the contour areas are computed.

minor comments (6)

[Eq. (4) and throughout] The symbol N is used both for the number of basis functions (e.g., Eq. (5)) and for the normalization constant in Eq. (4); this is confusing and should be disambiguated (e.g., by using a script N or a different letter for the normalization).
[Sec. III.B] The parameter-count notation Np is used in figure captions but is not defined in the main text; please define it at first use (e.g., Np = 3N for PM and Np = 4N for SFB-P2).
[Fig. 1 caption] The caption contains a typo ('An comparison') and Fig. 4 uses 'SFB=P2' instead of 'SFB-P2'; these should be corrected.
[Sec. IV.B] The Lindblad simulations assign the same dephasing rate γ to every spin, while real ensembles typically have a distribution of T2*; this limitation should be acknowledged explicitly when interpreting Figs. 7 and 8.
[Sec. V] The statement in the conclusion that 'similar results can also be obtained using a gradient-based solver' is not demonstrated anywhere in the manuscript; either remove this claim or provide a supporting numerical example.
[General] The manuscript does not include a data or code availability statement; since the results are purely numerical, making the simulation code or data available would strengthen reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PM ansatz is introduced and then benchmarked against SFB/SFB-P2 on the same objective function, and the reported improvements are empirical search results rather than recycled fit parameters.

full rationale

The central claims are that PM with three parameters reaches a comparable Fobj to SFB-P2 with twenty parameters in fewer function evaluations, and that PM-optimized XY-8 pulses yield T2 approximately 39 microseconds versus 24 microseconds for rectangular pulses. Neither claim reduces to an input. Fobj is a common objective (Eq. 4) evaluated for each basis, and the comparisons are numerical benchmarks, not derivations from fitted quantities. The PM basis does have infinite frequency sidebands by construction via the Jacobi-Anger identity (Eq. 7), but the paper presents this as a designed feature of the ansatz, not as a derived prediction, and the optimization advantage is established by simulation rather than by definition. The search-time comparison uses a maximum evaluation count of 200 times the number of parameters, which might raise benchmarking fairness questions, but that cap is not a fitted parameter and the reported times are not identities; any concern there belongs to experimental design or correctness, not circularity. Several self-citations (e.g., Refs. 11, 23, 37, 64, 78) are used as contextual examples of CRAB implementations, Floquet techniques, and noise parameters; they carry no load-bearing uniqueness theorem and do not force the PM choice. The T2 result uses the same static-broadening width W as an optimization input, but the simulated coherence time is produced by a full time-dependent noise model (Eqs. 21-22), so it is an independent dynamical output. No equation in the paper is equivalent by construction to the claimed prediction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced; the phase-modulated basis is a mathematical ansatz, not a physical entity. The free parameters listed are hand-chosen simulation settings that the numerical comparisons depend on, including the detuning sample count, frequency bounds, evaluation cap, and noise parameters.

free parameters (5)

Number of objective-function detuning samples M = 15
Chosen in Sec. III A to represent the Gaussian distribution with delta_k in [-W, W]; the optimization landscape and resulting pulses depend on this hand-picked discretization.
Frequency parameter bound for omega_k, nu_k = 2*pi*[0, 5/T]
Chosen in Sec. III A to reject high frequencies; restricts the search space and shapes the achievable controls.
Maximum function evaluations cap = 200 * N_p
Chosen in Sec. III B; the comparison of search time is partly determined by this cap, which is smaller for the lower-parameter PM method.
Noise parameters for XY-8 simulation = tau=20 microsec, (c*tau/2)^(1/2)=2*pi*50 kHz, static FWHM 2*pi*26.5 MHz
Chosen in Sec. V to target T2* approx 20 ns; the claimed 50% coherence-time improvement is specific to this noise regime.
Number of final fidelity samples K = 10^5
Chosen in Sec. III A to approximate the ensemble fidelity; affects the reported F values and contour areas.

assumptions (6)

domain assumption Rotating-wave approximation: counter-rotating terms are neglected in the interaction-picture Hamiltonians (Eqs. 10-14, Sec. III A).
Standard in spin control when omega0 is much larger than control frequencies and coupling; for the NV-center parameters targeted (omega0 about 2.8 GHz) it is a good approximation, but it is an unproved reduction.
domain assumption Ensemble detunings delta follow a zero-mean Gaussian distribution p(delta) with FWHM W (Eq. 2, Sec. II).
A model assumption for inhomogeneous broadening; the optimized pulses are tailored to this distribution and may not be robust to other shapes.
domain assumption Ensemble spins are spatially sparse and direct dipole-dipole interactions are neglected (Sec. II).
Justified for dilute NV ensembles by rapid distance decay, but it defines the model.
domain assumption Lindblad master equation for pure dephasing: d rho/dt = -i[H,rho] + (gamma/2)(sigma_z rho sigma_z - rho) (Eq. 16, Sec. IV B).
Standard Markovian dephasing model; ignores longitudinal relaxation and non-Markovian memory.
domain assumption Dynamic noise is modeled as an Ornstein-Uhlenbeck process with the discretized recurrence of Eq. (22) (Sec. V).
A phenomenological noise model with chosen tau and c; the T2 result is specific to these parameters.
standard math Jacobi-Anger identity expands the phase-modulated field into Bessel-weighted sidebands (Eq. 7, Sec. III A).
Mathematically exact; used to argue that the PM basis covers many frequencies.

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Pith. "Pith review of Quantum optimal control using phase-modulated driving fields." pith.science (2026). https://pith.science/paper/YRQGS5TY

@misc{pith2026200910275,
  author       = {Pith},
  title        = {Pith review of: Quantum optimal control using phase-modulated driving fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YRQGS5TY}},
  note         = {Machine review of arXiv:2009.10275}
}

abstract

Quantum optimal control represents a powerful technique to enhance the performance of quantum experiments by engineering the controllable parameters of the Hamiltonian. However, the computational overhead for the necessary optimization of these control parameters drastically increases as their number grows. We devise a novel variant of a gradient-free optimal-control method by introducing the idea of phase-modulated driving fields, which allows us to find optimal control fields efficiently. We numerically evaluate its performance and demonstrate the advantages over standard Fourier-basis methods in controlling an ensemble of two-level systems showing an inhomogeneous broadening. The control fields optimized with the phase-modulated method provide an increased robustness against such ensemble inhomogeneities as well as control-field fluctuations and environmental noise, with one order of magnitude less of average search time. Robustness enhancement of single quantum gates is also achieved by the phase-modulated method. Under environmental noise, an XY-8 sequence constituted by optimized gates prolongs the coherence time by $50\%$ compared with standard rectangular pulses in our numerical simulations, showing the application potential of our phase-modulated method in improving the precision of signal detection in the field of quantum sensing.

Figures

Figures reproduced from arXiv: 2009.10275 by the authors.

**Figure 2.** FIG. 2. Infidelity of the final state as a function of the number of ob [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Optimal fidelity as a function of the FWHM ( [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Fidelity of the final state under different values of the detuning and the relative variations of the control-field amplitude. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Average fidelity [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: FIG. 8. Fidelity of the final state with dephasing rate [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9. Optimal results of the robust gate fidelity given by PM [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. Gate fidelity under different values of the detuning and the relative variations of the control-field amplitude. [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11. Simulation of the coherence time [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

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