Power-optimized amplitude modulation for robust trapped-ion entangling gates: a study of gate-timing errors

Luke Ellert-Beck; Wenchao Ge

REVIEW 1 major objections 4 minor 1 cited by

Power-optimized amplitude modulation for robust trapped-ion entangling gates: a study of gate-timing errors

T0 review · 1 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper establishes that amplitude-modulated Mølmer–Sørensen gates can reduce leading-order gate-timing error from quadratic to sixth- or tenth-order scaling with only linear constraints on the Fourier coefficients and near-constant…

desk verdict Solid analytical derivation of timing-error robustness for Fourier-AM MS gates, with a power-optimality claim that needs tightening but is probably correct. read the letter →

arxiv 2412.17789 v1 pith:W5BCYF36 submitted 2024-12-23 quant-ph

Luke Ellert-Beck , Wenchao Ge This is my paper · ORCID

classification quant-ph MSC 81P68 PACS 03.67.Lx

keywords trapped-ionquantumcomputingMølmer–Sørensengateamplitude-modulatedpulseFouriershapinggate-timingerrorpoweroptimizationfidelitylinearconstraints

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 studies amplitude-modulated Mølmer–Sørensen gates on two trapped ions, writing the laser amplitude as a truncated Fourier series. It shows that requiring the first few derivatives of the phase-space trajectory to vanish at the gate time reduces to simple linear moment conditions on the Fourier coefficients. With one such constraint the leading infidelity from a gate-timing error $\Delta t$ drops from $O(\Delta t^2)$ to $O(\Delta t^6)$, and with two constraints to $O(\Delta t^{10})$. The pulse satisfying those constraints can be chosen to minimize average laser power by solving a generalized eigenvalue problem, and the extra power cost shrinks toward zero as more Fourier terms are included. The paper's point is that robust, power-efficient entangling gates need only a modest Fourier basis.

What carries the argument

The machinery is the Fourier amplitude envelope $\Omega(t)=a_0/2+\sum_{n=1}^N [a_n\cos(n\xi_0 t)+b_n\sin(n\xi_0 t)]$ together with exact integral expressions for the phase-space couplings $F(t)$ and $G(t)$. Because every derivative $F^{(i)}(T)$ and $G^{(i)}(T)$ becomes a sum over $n^i$ times the Fourier coefficients, the robustness conditions are linear moment constraints on those coefficients. Average laser power and geometric phase are both quadratic forms in the coefficients, so after projecting out the linear constraints the minimum-power pulse is the eigenvector of $P^{-1/2}AP^{-1/2}$ with the largest absolute eigenvalue, with $A(T)$ fixed to $\pi/2$ by rescaling.

What would settle it

Run the same constrained power optimization with $b_n$ allowed to be nonzero for a fixed $N$ (say $N=10$), subject to the same one- or two-constraint system, and compare the maximum attainable geometric phase per unit power with the $b_n=0$ result; any strictly larger ratio would disprove the claim that cosine-only pulses are power-optimal. Experimentally, one could measure infidelity versus gate-time offset for an $N=5$ or $N=10$ pulse and check whether the predicted $\Delta t^6$ or $\Delta t^{10}$ plateau appears over the claimed stability window.

Watch

Extended reading notes

Core claim

The central claim is that a Fourier-expanded amplitude envelope for the MS gate converts robustness to gate-timing errors into a short list of homogeneous linear equations in the Fourier coefficients, and that the minimal-power solution to those equations is found by diagonalizing one matrix. The paper derives closed-form amplitude-modulated trajectories $F(t)$ and $G(t)$, and shows that $F^{(i)}(T)=0$ and $G^{(i)}(T)=0$ collapse to the moment conditions $\sum_n a_n n^{i-1}=0$ for odd $i$ and $\sum_n b_n n^{i-1}=0$ for even $i$, with $a_1=b_1=0$ enforced by closure. Enforcing one constraint makes $1-F_{\rm MS}$ scale as $\Delta t^6$; enforcing two makes it scale as $\Delta t^{10}$. The power-optimized pulses use only cosine coefficients ($b_n=0$), reducing the constrained optimization to the Rayleigh quotient of a pair of quadratic forms, and numerical results for $N$ up to 100 show the power overhead falling to $0.51\%$ for one constraint and $1.2\%$ for two.

Load-bearing premise

The load-bearing assumption is that zeroing the sine Fourier coefficients $b_n$ sacrifices no optimality in the ratio of geometric phase to average laser power; if a nonzero-$b_n$ solution under the same constraints beats the $b_n=0$ one, the reported power costs would be underestimates and the power-optimal claim would be unsupported.

Editorial extensions

If this is right

One added linear constraint changes the leading infidelity from timing error from $O(\Delta t^2)$ to $O(\Delta t^6)$; two constraints change it to $O(\Delta t^{10})$.
The linear constraints are homogeneous and can in principle be extended to arbitrarily high order, since each additional constraint removes four powers of $\Delta t$ from the leading error.
The additional average laser power is small and shrinks with $N$: $0.51\%$ overhead at $N=100$ for one constraint and $1.2\%$ for two.
The amplitude-modulated pulses develop a soft start, which leaves smaller residual phase-space displacement and a wider region of near-unit fidelity around the nominal gate time.
The same strategy of expanding fidelity in a control parameter and imposing linear coefficient constraints can be applied to other error sources such as laser frequency fluctuations.

Reading between the lines

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

If the $b_n=0$ ansatz is wrong, the reported power overheads are optimistic; a numerical search over the full coefficient space for fixed $N$ would settle this without new physics.
The two-ion single-mode model omits carrier transitions and higher Lamb-Dicke terms that the paper itself estimates contribute coherent errors near $10^{-4}$, so the asymptotic scaling may not dominate in a full multi-mode device.
An experimental test only needs an arbitrary waveform generator and a two-ion chain; measuring the width of the flat infidelity region for increasing $N$ would directly check whether the stability region narrows as predicted.
Because the method only reshapes the amplitude envelope, it can likely be stacked with frequency or phase modulation and with standard error-mitigation techniques, though the paper does not demonstrate such combinations.

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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

Short version: the O(Δt^6) and O(Δt^10) scaling for timing errors is derived cleanly and I believe it holds within the stated single-mode model. The power-optimized label is not fully earned: the paper restricts to the cosine (a_n) subspace without a proof that sine coefficients can't do better, though my own spot-check suggests they likely can't in the parameter range shown.

What's genuinely new: the reduction of timing-error robustness to a compact set of linear constraints on the Fourier coefficients of an AM-only pulse, and the numerical demonstration that the extra power cost tends to zero as the number of coefficients grows. The derivation in Appendix C is straightforward and correct, and the generalized-eigenvalue optimization is standard but neatly executed. The plots in Fig. 7 clearly show the improved scaling.

Soft spots, in order of importance. First, the b_n=0 argument in Appendix D is heuristic. They note that a sine-only solution has geometric phase per power at most 1/3, and they suggest the optimized cosine solution beats that, but they never verify it for every N. For 2 LC with N=4, for example, the cosine-only ratio is about 0.36, above 1/3, so it's plausibly fine; but for N=3 with 2 LC the cosine ratio is well below 1/3, though there are no nontrivial sine solutions there. The missing proof matters because the title says 'power-optimized' and the reported overheads are minima only if the restriction is optimal. If sine coefficients can help, those numbers are upper bounds. This should be either proven or the language softened to 'power-optimized within the cosine subspace.'

Second, Eq. (14) misdefines ΔA: from the periodicity, A(T+Δt) = π/2 + A(Δt), so ΔA should be A(Δt), not A(Δt)-π/2. The text and Appendix A effectively use the correct definition, so it's a typo, but it's confusing. Appendix A also has a wrong coefficient (and sign) on the squared term in the fidelity expansion; it's higher order and drops out, so the main argument survives.

Third, the model is deliberately simple—single mode, no carrier—and the paper honestly says timing errors aren't the leading error source in real gates. So the practical significance is moderate; the value is in the clear analytical framework.

Who this is for: anyone working on amplitude-modulated ion gates, especially the power-robustness trade-off. It deserves review; the core result is sound and useful. I'd recommend sending it to referees, with a request to tighten the power-optimality argument and fix the small errors.

Referee Report

1 major / 4 minor

Summary. The paper analyzes amplitude-modulated Mølmer-Sørensen gates for two trapped ions, expanding the laser amplitude envelope in a Fourier series. It derives closed-form expressions for the phase-space trajectory, converts the requirement of vanishing low-order derivatives of F and G at the gate time into linear constraints on the Fourier coefficients, and selects pulses by maximizing the geometric phase per unit average power subject to those constraints via a generalized eigenvalue problem. Numerical results show infidelity scaling O(Δt^6) with one constraint and O(Δt^10) with two constraints, with laser-power overhead that decreases as the number of Fourier components increases.

Significance. The central scaling result is an analytical, parameter-free prediction within the stated single-mode model: the constraints are linear, the optimization is a Rayleigh quotient, and the asymptotic improvements in Δt are verified numerically rather than inferred from fits. This is a useful and clean demonstration that amplitude-modulated MS gates can suppress gate-timing errors with vanishing power overhead as N grows. The main limitation is that the 'power-optimized' claim is proved only for the cosine-only subspace; the global optimality over all Fourier coefficients is asserted rather than proven. With that point repaired or appropriately qualified, the paper would be a solid contribution to pulse-shaping methods for trapped-ion gates.

major comments (1)

[Section III.D and Appendix D] The claim that the optimized pulse is 'minimum-power' (abstract and Section III.D) is not established for the full Fourier space. Appendix D argues that b_n should be zero because nonzero b_n add positive contributions to both P and A, and because a b-only solution has |A|/P ≤ 1/3. This is heuristic: it does not rule out a mixed a_n,b_n solution with larger |A|/P, and the '≤1/3' bound is not compared with the actual optimized a-only ratio for each N reported in Fig. 8. The missing step is a convexity argument: for the optimized a-only solution with negative A_a, any nonzero b vector adds A_b>0 and P_b>0, which strictly lowers |A_a|/P_a; if such a proof is supplied (or an explicit check that the chosen eigenvector has A_a<0 for the relevant N), the restriction is justified. Without it, the wording should be softened to 'optimal within the cosine-only subspace,' and the overhead percentages in Fig. 8 are upper bounds rather than minima.

minor comments (4)

[Eq. (14) and Appendix A] Equation (14) is inconsistent with Appendix A: using periodicity of F and g, A(T+Δt)=A(T)+A(Δt)=π/2+A(Δt), so ΔA=A(Δt), not A(Δt)-π/2. Appendix A in fact uses the correct relation. Please correct Eq. (14) and the sentence that refers to it.
[Appendix A, Eq. (A1)] The coefficient 1/128 on the dropped term in Eq. (A1) does not follow from the expansion of Eq. (11) as displayed, and the accompanying statement 'A=O(Δt²)' is not correct for the MS pulse (one finds ΔA=O(Δt³) for the constant-amplitude case). Please recompute the dropped term and its order; the leading-order term in Eq. (15) is unaffected.
[Section III.C and abstract] The phrase 'one linear constraint' should be qualified. If the b_n are retained, the reduced system for the first nontrivial order contains both a0/2+Σ a_n=0 and Σ b_n n=0; the latter is automatic only in the cosine-only subspace. Please make the counting explicit.
[Fig. 8] The caption should state the normalization of the plotted power (apparently P=1 for the unmodulated MS pulse) and define 'additional power' as (P-1)×100%.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the scaling and power-optimization results are derived analytically from the MS Hamiltonian and an external fidelity formula, with no fitted parameter defining the central prediction.

full rationale

The paper's central claims are not circular. The O(Δt^6) and O(Δt^10) scalings are obtained by imposing explicit linear constraints on Fourier coefficients, derived from the analytic derivative formulas in Eqs. (24)-(25) and reduced to Eqs. (28)-(29). These constraints set specific derivatives of F and G to zero at the gate time; the fidelity scaling then follows from the known external fidelity expansion of Ref. [22], Eq. (11), and the Taylor argument in Section III. The numerical fidelity curves in Figure 7 are independent evaluations of that fidelity expression using the solved pulse coefficients, not fits to the curves. The power optimization is a standard Rayleigh-quotient problem on analytically constructed matrices, and the reported power values in Figure 8 are computed, not assumed. The only notable weakness is the b_n = 0 restriction in Appendix D, which is explicitly labeled an ansatz and justified only heuristically; this is a model assumption that could affect the global power-optimality claim, but it is not a circular reduction of the central scaling result. There are no load-bearing self-citations and no fitted inputs renamed as predictions.

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

No free parameters are fitted to experimental data. The Fourier coefficients are optimization variables, not fitted constants. The axioms are the standard single-mode MS model, the leading-order fidelity expansion, and the unproven restriction to a_n coefficients in the power optimization.

assumptions (4)

domain assumption The two-ion MS gate is described by the interaction Hamiltonian in Eq. (1), restricted to the center-of-mass mode, with carrier and fast-rotating terms neglected.
Invoked in Section II.A; also assumes Lamb-Dicke regime and Ω ≪ δ. Ref. [35] estimates neglected terms contribute coherent errors around 1e-4.
standard math The gate infidelity is dominated by the leading-order term 1-F ≈ (n̄+1/2)(F²+G²)/2, with higher-order corrections from the geometric-phase deviation being negligible.
Derived in Appendix A via Taylor expansion; valid for small Δt and moderate phonon number n̄.
ad hoc to paper The sine Fourier coefficients b_n can be set to zero without losing power-optimality.
Appendix D provides a heuristic sign-based argument but no proof that the global optimum lies in the a_n-only subspace.
domain assumption Truncating the Fourier series at order N and numerically diagonalizing the matrix P'^{-1/2}A'P'^{-1/2} yields the claimed minimal-power pulses.
Used in Section III.D and Appendix E; assumes the numerical procedure converges and the truncation is sufficient.

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Pith. "Pith review of Power-optimized amplitude modulation for robust trapped-ion entangling gates: a study of gate-timing errors." pith.science (2026). https://pith.science/paper/W5BCYF36

@misc{pith2026241217789,
  author       = {Pith},
  title        = {Pith review of: Power-optimized amplitude modulation for robust trapped-ion entangling gates: a study of gate-timing errors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5BCYF36}},
  note         = {Machine review of arXiv:2412.17789}
}

abstract

Trapped-ion systems are a promising route toward the realization of both near-term and universal quantum computers. However, one of the pressing challenges is improving the fidelity of two-qubit entangling gates. These operations are often implemented by addressing individual ions with laser pulses using the Molmer-Sorensen (MS) protocol. Amplitude modulation (AM) is a well-studied extension of this protocol, where the amplitude of the laser pulses is controlled as a function of time. We present an analytical study of AM, using a Fourier series expansion to maintain the generality of the laser amplitude's functional form. We then apply this general AM method to gate-timing errors by imposing conditions on these Fourier coefficients, producing trade-offs between the laser power and fidelity at a fixed gate time. The conditions derived here are linear and can be used, in principle, to achieve arbitrarily high orders of insensitivity to gate-timing errors. Numerical optimization is then employed to identify the minimum-power pulse satisfying these constraints. Our central result is that the leading order dependence on gate timing errors is improved from $\mathcal{O}(\Delta t^2)$ to $\mathcal{O}(\Delta t^6)$ with the addition of one linear constraint on the Fourier coefficients and to $\mathcal{O}(\Delta t^{10})$ with two linear constraints without a significant increase in the average laser power. The increase approaches zero as more Fourier coefficients are included. In further studies, this protocol can be applied to other error sources and used in conjunction with other error-mitigation techniques to improve two-qubit gates.

Figures

Figures reproduced from arXiv: 2412.17789 by the authors.

**Figure 2.** FIG. 2. Pulse envelopes of the power-optimized pulses, where [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Trajectory ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Trajectories ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. The population dynamics of MS gate (a) compared [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 7.** Figure 7: FIG. 7. Error rate comparison between constant pulse and [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: FIG. 8. The power required to produce an amplitude modu [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

C. D. Bruzewicz, J. Chiaverini, R. McConnell, and J. M. Sage, Trapped-ion quantum computing: Progress and challenges, Applied Physics Reviews 6, 021314 (2019)

2019

[2] [2]

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