Traversing Quantum Control Robustness Landscapes: A New Paradigm for Quantum Gate Engineering

Huiqi Xue; Xiu-Hao Deng

REVIEW 4 major objections 5 minor 82 references

Traversing Quantum Control Robustness Landscapes: A New Paradigm for Quantum Gate Engineering

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read This paper claims that walking along a constant-robustness path in a robustness landscape lets one robust seed pulse generate robust pulses for an entire continuous family of parametric quantum gates.

desk verdict RIPV is a genuinely new way to propagate robustness across parametric gate families, but the multi-noise demonstration breaks down exactly where its own regularity condition is violated, so the strongest claims are not yet supported. read the letter →

arxiv 2412.19473 v2 pith:ZQQSNPX3 submitted 2024-12-27 quant-ph

Huiqi Xue , Xiu-Hao Deng This is my paper · ORCID

classification quant-ph

keywords quantumcontrolrobustnesslandscapelevel-settraversalrobustparametricgatesnoisesusceptibilitygradient-orthogonalvariationRIPValgorithmmulti-objective

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's goal is to make robustness, not just fidelity, the quantity that is directly engineered. It defines a Quantum Control Robustness Landscape (QCRL) that maps control parameters to a gate's resistance to noise, and it claims that level sets of this landscape can be traversed: a pulse can be varied continuously so that the gate it implements changes while its robustness stays fixed. The Robustness-Invariant Pulse Variation (RIPV) algorithm does this by moving in the subspace orthogonal to the gradients of all robustness constraints. Starting from a single robust seed pulse, the algorithm generates robust pulses for a continuous family of parametric gates, reported numerically as $R_x(\theta)$ and $R_{XY}(\theta)$ with fidelity at least $0.999$ under quasi-static noise at $2\%$ to $10\%$ of the pulse maximum amplitude. The payoff the paper aims at is that gate families, not just individual gates, can be made robust in one pass rather than by re-optimizing every member.

What carries the argument

The central object is the QCRL level set, the set of all control-parameter vectors $A$ that give the same robustness $R(A)$, together with the variation subspace $V = (\mathrm{span}\{\partial R_i/\partial A\}_i)^\perp$ tangent to it. The mechanism is the gradient-orthogonal variation step: choose a pre-variation $dA_{\mathrm{pre}} = \partial \theta/\partial A$, remove its component parallel to every $\partial R_i/\partial A$, and rescale the remaining component so that $d\theta = \Delta\theta_{\mathrm{ideal}}$. Because $dA$ is orthogonal to all robustness gradients, every robustness value is invariant to first order, while the gate-rotation angle changes by a controlled amount; the repeated steps form a continuous sequence of pulses that can be interpolated to any intermediate $\theta$. When multiple controls or multiple noise sources are present, undesired rotation angles are promoted to the same constraint structure and kept at zero.

What would settle it

Take a parametric gate family whose QCRL level set is known to have two disconnected components containing robust implementations of $\theta_L$ and $\theta_R$, and run RIPV; if the algorithm cannot cross the gap, the claim that one seed suffices for the whole family fails in that case. A simpler numerical check is to reduce the parameter count to exactly $m+1$ for $m$ robustness constraints: the variation subspace then has dimension one, and searching for a point where $\partial \theta/\partial A$ becomes parallel to the constraint gradients would reveal whether irregular points block the walk.

Watch

Extended reading notes

Core claim

The central discovery is that equally robust controls are not isolated points but form a level set in parameter space, and this level set can be walked to change the gate while preserving robustness. The paper defines two families of robustness metrics: integral robustness, the expectation of fidelity between the noiseless and the noisy propagator over all noise paths, and asymptotic robustness, derived from $n$-th order noise susceptibilities $S^n$ in a Magnus expansion. RIPV's step is the variation condition $dA \perp \partial R_i/\partial A$ for every robustness constraint $R_i$, implemented by taking the gate-parameter gradient $\partial \theta/\partial A$ as a pre-variation, projecting it with Gram-Schmidt into the variation subspace, and normalizing so that $\theta$ advances by a fixed $\Delta\theta$. In the simplest case a second-order robust $R_x(2\pi)$ seed becomes a continuous family of $R_x(\theta)$ pulses, and the same construction is extended to two control terms, three noise directions, and the two-qubit $R_{XY}(\theta)$ gate. The paper's claim is that this transfer of robustness from one gate to a whole parametric family is a new capability for quantum gate engineering.

Load-bearing premise

The load-bearing premise is that the level set of the robustness landscape containing the starting pulse is connected enough that a continuous path exists along which $\theta$ can be advanced from $\theta_L$ to $\theta_R$ without hitting an irregular point where $\partial \theta/\partial A$ lies in the span of the robustness gradients, a point the paper explicitly leaves open in Section VI.

Editorial extensions

If this is right

A single robust seed pulse replaces per-gate optimization: the same run produces robust pulses for every $R_x(\theta)$ in the chosen interval, and the continuous parameter sequence can be interpolated for arbitrary intermediate angles.
Robustness to several independent quasi-static noise sources is preserved simultaneously by adding one constraint per source, as long as the control has at least one term that does not commute with each noise direction.
The two-qubit $R_{XY}(\theta)$ family inherits the single-qubit pulse series, so a robust single-qubit family doubles as a robust parametric two-qubit gate family.
The underlying gradient-orthogonal variation procedure works for any objective and any set of invariants, so fidelity, gate time, leakage, or energy consumption can each be adjusted while holding the others fixed.

Reading between the lines

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

Inference: if level-set connectivity holds, a practical workflow could be to store a small library of robust seed pulses and generate any parameterized gate family on demand, avoiding repeated robust optimization on the device.
Inference: a testable extension is to run RIPV while sweeping a physical parameter other than rotation angle, such as qubit detuning or coupling strength, treating it as the varied gate parameter and checking whether the same invariance holds.
Inference: the observed degradation of fidelity plateaus near $\theta = \pi/4$ and other turning points is attributed by the paper to the linear approximation; the editorially inferred test is to use smaller $\Delta\theta$ or a correction step there, and to distinguish numerical drift from a genuine topological obstruction.
Inference: if accessibility fails for some gate families, the framework would still be useful for calibrating within a connected patch, suggesting that characterizing which parametric families lie on connected level sets is the first question worth attacking.

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.

Desk Editor's Note

read the letter

Here's my read of arXiv:2412.19473. The core idea is genuinely new: instead of optimizing robustness gate-by-gate, define a robustness landscape and walk along its level sets, using gradient-orthogonal steps to keep susceptibility fixed while the gate parameter sweeps. That is distinct from D-MORPH and from Sauvage–Mintert, and the single-noise results make the case well. Starting from a second-order robust pulse from their earlier work, first-order RIPV keeps S1 within 1.414–1.425 across the whole Rx(θ) family, with fidelity above 0.999 under 5% noise; the second-order version maintains S2 around 10.5–10.7. The two-qubit RXY(θ) reduction is clean and useful.

The soft spots are in the multi-noise demonstration, and they are not minor. Their Figure 5(c,d) shows exactly what you'd see if the variation direction degenerates: S1_x rises sharply near θ ≈ π/4 and 5π/4, the undesired rotations ϑy and ϑz deviate from zero, and Δθ deviates from Δθ_ideal. The authors call this 'linear approximation error,' but the pattern matches their own definition of an irregular point in Section IVD3, where ∂θ/∂A lies in the span of the constraint gradients and the normalization factor blows up. They assert in that section that irregular points never occur in their experiments, yet the broad features of Figure 5 contradict that assertion. Also, noiseless fidelity degrades near θ = 0, so the 'robust' pulses are not even maintaining the zero-noise gate. They propose fixes—smaller steps, a correcting step—and that is credible, but the central 'preserving robustness' claim is not actually demonstrated for the multi-noise case.

Two other things. The abstract says the gates 'exceed the quantum error correction threshold,' but no threshold is defined or computed; the text only shows infidelity curves, and a single fidelity number does not establish a QEC threshold. That should be fixed. Also, there is no code or data, which makes reproducing the traversal and the breakdown hard. Section VI honestly lists accessibility and connectivity as open, which is good, but it means the guarantee is conditional. The use of starting pulses from ref. [45] (same group) is not a problem; those pulses are published and the dependence is explicit.

This paper deserves a serious referee. The idea is worth engaging, the formalism is clear, and the single-noise evidence is solid. The multi-noise issue is addressable, not fatal to the concept. I would send it to review with the expectation of major revision: tone down the abstract, either repair or honestly delimit the multi-noise claims, and release code and data. The audience that gets value from this is people working on quantum control, parametric gate calibration, and robust gate design—and anyone interested in level-set traversal as a multi-objective tool.

Referee Report

4 major / 5 minor

Summary. The manuscript introduces the Quantum Control Robustness Landscape (QCRL), a map from control parameters to a noise-robustness measure defined either as an integral over noise paths or as Magnus/susceptibility-based asymptotic robustness, and proposes the Robustness-Invariant Pulse Variation (RIPV) algorithm, which uses gradient-orthogonal variation to move along a level set of robustness constraints while changing a gate parameter θ. Starting from robust seed pulses taken from ref. [45], the authors numerically generate families of R_x(θ) pulses under single σ_z noise (first- and second-order robustness), R_x(θ) pulses under simultaneous σ_x, σ_y, σ_z noise with two control terms, and an R_XY(θ) two-qubit gate obtained from the single-qubit pulses. They report fidelities at or above 0.999 under 2–10% quasi-static noise and claim these gates exceed quantum error correction thresholds. Section VI explicitly leaves accessibility and connectivity of QCRL level sets as open questions.

Significance. The QCRL/GOV/RIPV framework is a genuinely useful conceptual proposal: it decouples robustness from individual gate synthesis, provides a general multi-constraint variation principle, and the QEED visualization offers intuitive access to error accumulation. The single-noise numerical results are concrete evidence for the core idea: S_1 stays within a narrow band and fidelity plateaus remain above 0.999 across the full R_x(θ) family. However, the manuscript as written overclaims. The multi-noise demonstration itself shows robustness drift and undesired rotations in the very regime where the central claim is being tested, and the abstract's 'exceed the quantum error correction threshold' statement is not established by the simulations. If the multi-noise example is repaired with the correction step of Section IVD2, or the claims are restricted to the demonstrated θ ranges, the framework would be a solid contribution.

major comments (4)

[§V B2, Fig. 5(c,d)] The multi-noise demonstration does not, as presented, support the claim that RIPV preserves robustness while varying θ. Near θ≈π/4 and θ≈ 5π/4, the first-order susceptibility S_1^x rises sharply, the undesired rotations ϑ_y and ϑ_z visibly deviate from zero, the achieved Δθ deviates from Δθ_ideal, and the noiseless fidelity degrades as θ approaches 0. The authors attribute this to linear approximation error, but this is the regime in which the central claim is being tested. The pattern is consistent with the variation direction becoming nearly degenerate, i.e., the irregular-point condition of §IVD3, and the manuscript provides no diagnostic or control for that possibility. Please either implement the correction step described in §IVD2 and show that S_1^j and ϑ_y,ϑ_z remain bounded over [0,2π], or explicitly restrict the 'preserving robustness' claim to the θ range where the demonstrated quantities actually stay constant.
[§V B1 and Eq. (14)] There is an internal inconsistency in the interpretation of the QEED results. The text states that the error curves in Fig. 4(a) 'closed at the origin (black dot), indicating that S_1 was kept near zero,' but a few sentences later it reports 1.414 < S_1 < 1.425 throughout the variation. Given the definition S_1 = ||M_1(T)|| in Eq. (14), a nonzero S_1 of order 1.4 is incompatible with the final error-curve point returning to the origin. Either the QEEDs do not actually close at the origin, in which case the 'near zero' statement is incorrect, or 'first-order robustness' in this paper means a small but nonzero constant S_1, in which case the QEED interpretation and the 'closed at the origin' wording must be corrected. This matters because the single-noise demonstration is the strongest positive evidence for the framework.
[§VI and §IVD3] The algorithm's termination requires the existence of a continuous path in the common level set of the constraints along which θ changes monotonically from θ_L to θ_R while avoiding irregular points where ∂θ/∂A lies in the span of the constraint gradients. Section §VI explicitly leaves accessibility and connectivity of QCRL level sets open. This is not a flaw in itself, but it does mean that the abstract's language about generating robust pulses for 'any arbitrary gate' and the introduction's claim of 'a once impossible task – optimizing a gate family' go beyond what is established. Please either state the existence condition as a conjecture with the numerical evidence clearly delimited, or add a sufficient regularity/controllability condition under which the variation path exists.
[Abstract and §VII] The claim that the simulated gates 'exceed the quantum error correction threshold even with substantial noise' is not substantiated. The numerical evidence is gate fidelity as a function of quasi-static noise amplitude; a QEC threshold is a property of a code, decoder, and full error model, and cannot be inferred from a single-gate infidelity curve. I recommend replacing this with a precise quantitative statement, such as 'gate infidelity remains below 10⁻³ for noise strengths up to X% of the pulse amplitude,' and reserving any QEC-threshold remark for an actual end-to-end threshold calculation.

minor comments (5)

[Title/Abstract and §IV] The algorithm is called 'Robustness-Invariant Pulse Variation' in the abstract and Section IV, but the chapter heading in Section IV reads 'Robustness-Invariant Pulse Variance'; please make the naming consistent.
[§IVB4] There is a dangling phrase: 'The selection of this pre-variation dApre is very important to roughly determine the direction along which the pulse A should be adjusted, since.' The word 'since' appears to be a leftover and should be removed or completed.
[§IIIB2] The construction of the path measure is admittedly informal and the manuscript does note this, but for a journal readership it would help to state explicitly that Eq. (3) is a definition of the integral on parametrized noise paths rather than a theorem from measure theory; the current Remark 2 can be used for this.
[Algorithm 1] The pseudocode termination condition `while θnow < θR` only covers the increasing-θ case; the decreasing case used in §VB1b and §VB2 should either be stated separately or written symmetrically.
[§V B2] The text says the fidelity plateau on σ_x noise 'shrank fast' as θ approaches 0, but the quantitative S_1^x values in Fig. 5(c) are not given in the text; providing the numerical values would make the extent of the robustness drift concrete and reproducible.

Circularity Check

1 steps flagged · score 2.0 of 10

One load-bearing self-citation supplies the robust starting pulses, but the RIPV variation and numerical demonstrations are not equation-level circular.

self citation load bearing [Section V.B.1 (Single noise source), initialization of the Rx(θ) pulse family]
"The beginning pulse parameters used in these simulations are all from ref. [45]."

The robust Rx(θ) family is generated by RIPV starting from pulses whose parameters and robustness data are taken exclusively from ref. [45], a paper by Y.-J. Hai, J. Li, J. Zeng, D. Yu, and X.-H. Deng, with Deng as the corresponding author of the present work. Since the RIPV step is designed to keep the robustness constraints Ri unchanged, any robustness in the output family is inherited from the starting pulse; if the cited prior result were not independently established, the central numerical claim would reduce to that self-citation. The paper does independently simulate the output fidelity and QEEDs, so the citation is not the only evidence, but the initial condition of every demonstration is an unverified-in-this-paper self-cited input.

full rationale

I walked the derivation chain: robustness is defined independently (Definition 1 and the Magnus-expansion susceptibilities in Section III.C) as a functional of the noiseless and noisy propagators, not in terms of the target gate parameter θ. The RIPV variation condition (Eq. 20) requires dA ⊥ ∂Ri/∂A, and the step normalization in Section IV.B.5 sets dθ = Δθideal; these are constructive constraints rather than fitted predictions. The numerical demonstrations then verify infidelity versus noise for the generated pulses, so the claim that the family is robust is supported by simulation in this paper, not merely by the starting-pulse citation. The remaining issues are limitations rather than circularity: Section VI explicitly leaves accessibility and connectivity of QCRL level sets open, and Figure 5(c,d) shows that near θ≈π/4 the first-order susceptibility S1_x, the undesired rotations, and the achieved Δθ deviate from their ideal values, which the authors attribute to linear-approximation error. Those are correctness concerns, not cases where the derivation reduces to its inputs. The only circularity-adjacent element is the load-bearing use of the authors' own prior work [45] to supply all beginning pulses and the orthogonal-control condition, giving a modest score of 2 rather than 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on unproved connectivity and accessibility of QCRL level sets, on the validity of small-noise Magnus expansion, and on externally supplied robust starting pulses. No new physical particles, forces, or dimensions are introduced; QCRL is a mathematical map rather than an invented physical entity.

free parameters (3)

Starting pulse parameters A0 from ref. [45] = Not listed in this paper; supplied in ref. [45] supplementary material
All generated family pulses inherit robustness from this starting configuration, and no independent values or code are provided here.
Step size Delta theta_ideal = 0.001 rad (single noise), 5e-4 rad (multi-noise)
Chosen by hand; controls linearization error, number of iterations, and the drift observed near theta = pi/4 in the multi-noise example.
Pulse parametrization dimension and basis = 9 or 18 parameters with sin-envelope Fourier parametrization (Eq. 21)
The landscape dimension is a modeling choice; the authors state that more parameters help the variation subspace but provide no systematic selection criterion.

assumptions (4)

domain assumption The QCRL level set containing the starting pulse is connected enough to allow continuous variation of theta over [theta_L, theta_R] with no irregular points.
Invoked implicitly in RIPV step 2 (Section IVA) and the Variation Condition (Eq. 20); the authors explicitly state in Section VI that accessibility and connectivity of QCRL level sets are unknown.
domain assumption Noise is correctly modeled by a stochastic Hamiltonian Hn with small quasi-static perturbations, and the Magnus expansion applies.
Assumed in Definition 1 and used throughout Examples 1-3 and the numerical susceptibility computations; the small-noise limit is central to the asymptotic robustness metric.
domain assumption Tangent-space linear steps remain on the level set, so a correction step is unnecessary.
Section IVD2 states that the correcting step is not implemented because 'linear approximation is good enough'; the multi-noise numerics show S1_x drifting, so this assumption is load-bearing.
standard math Standard results of Magnus expansion, Gram-Schmidt orthogonalization, and matrix logarithm extraction are valid for the time-ordered evolution used here.
Used in Examples 1-2 and Section IVC1; these are standard mathematical tools, not introduced by the paper.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Traversing Quantum Control Robustness Landscapes: A New Paradigm for Quantum Gate Engineering." pith.science (2026). https://pith.science/paper/ZQQSNPX3

@misc{pith2026241219473,
  author       = {Pith},
  title        = {Pith review of: Traversing Quantum Control Robustness Landscapes: A New Paradigm for Quantum Gate Engineering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQQSNPX3}},
  note         = {Machine review of arXiv:2412.19473}
}

read the original abstract

The optimization of robust quantum control is often tailored to specific tasks and suffers from inefficiencies due to the complexity of cost functions. Our recent findings indicate a highly effective methodology for the engineering of quantum gates by initiating the process with a robust control configuration of any arbitrary gate. We first introduce the Quantum Control Robustness Landscape (QCRL), a conceptual framework that maps control parameters to noise susceptibility. This framework facilitates a systematic investigation of equally robust controls for diverse quantum operations. By navigating through the level sets of the QCRL, our Robustness-Invariant Pulse Variation (RIPV) algorithm allows for the variation of control pulses while preserving robustness. Numerical simulations demonstrate that our single- and two-qubit gates exceed the quantum error correction threshold even with substantial noise. This methodology opens up a new paradigm for quantum gate engineering capable of effectively suppressing generic noise.

Figures

Figures reproduced from arXiv: 2412.19473 by the authors.

**Figure 2.** FIG. 2: Comparison of different approaches to obtain [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3: Probability of two vectors being orthogonal in [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4: Robust [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

**Figure 5.** Figure 5: FIG. 5: Robust [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: FIG. 6: Infidelity-noise graph when using single control [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: FIG. 7: Decoupling two objectives [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

82 extracted references · 69 canonical work pages

[45]

Zhang, T

J. Zhang, T. H. Kyaw, S. Filipp, L.-C. Kwek, E. Sjöqvist, and D. Tong, Geometric and holonomic quantum compu- tation, Physics Reports1027, 1 (2023)

work page 2023
[1]

Each Ωk(A) is a function oft, denoted byΩk(t; A): Ωk : A 7→ Ωk(A), s.t

A vector of maps⃗Ω = {Ωk}k from control parame- ters A to control pulsesΩk(t; A), each correspond- ing to one of the control termsHc,k. Each Ωk(A) is a function oft, denoted byΩk(t; A): Ωk : A 7→ Ωk(A), s.t. Ωk(A)(t) = Ωk(t; A)
[2]

Hc[⃗Ω](t) = X k Ωk(t)Hc,k

A mapHc from pulses ⃗Ω to a control Hamiltonian Hc[⃗Ω], where Hc[⃗Ω] is a time-dependent operator: Hc : ⃗Ω 7→ Hc[⃗Ω], s.t. Hc[⃗Ω](t) = X k Ωk(t)Hc,k
[3]

Usc[Hc](t) = U Hs+Hc(τ )(t)

A map Usc from control Hamiltonian Hc to the noiseless propagator Usc[Hc], where Usc[Hc] is a time-independent operator: Usc : Hc 7→ Usc[Hc], s.t. Usc[Hc](t) = U Hs+Hc(τ )(t)

work page
[4]

Finally, we obtain the robustness map from control pa- rameters Ato some metric of robustnessRby composing all the maps, R : A 7→ ⃗Ω(A) 7→ Hc[⃗Ω] 7→ Usc[Hc] 7→ R[Usc(A)]

A map from the propagator Usc(t) to any metric of robustness R (integral robustness, asymptotic robustness, etc.): R : Usc 7→ R[Usc] . Finally, we obtain the robustness map from control pa- rameters Ato some metric of robustnessRby composing all the maps, R : A 7→ ⃗Ω(A) 7→ Hc[⃗Ω] 7→ Usc[Hc] 7→ R[Usc(A)] . (17) If we write out the time dependence explicitl...

work page
[5]

Wefirst discretize thetime interval[0, T]into N pieces, namely t0 = 0, t1 = T N ,

Piecewise constant parametrization. Wefirst discretize thetime interval[0, T]into N pieces, namely t0 = 0, t1 = T N , . . . , tk = kT N , . . . , tN = T . The pulse is then defined by the parameter vectorA = (A0, . . . , AN ) as Ω(t) = ( 0 t /∈ [0, T] Ak t ∈ [tk, tk+1]. The parametrization is straightforward to implement, but applying constraints could be...

work page
[6]

There are many functional bases to choose

Functional basis parametrization. There are many functional bases to choose. Two com- mon methods are the Taylor expansion and Fourier ex- pansion, where the pulse parameters are the coefficients of the expansion terms. Besides, wavelets are among the most useful functional bases for constructing pulses, for they are finitely supported and have various me...

work page
[7]

After optimization, we obtain a pulse A0 and its corresponding gate parameterθ0

(Initialization.) Starting from an arbitrary pulse Ainit (called the initial pulse), use any optimiza- tion algorithm to optimize the robustness function R(A). After optimization, we obtain a pulse A0 and its corresponding gate parameterθ0. Note that we only record but do not designate the valueθ0

work page

Show all 82 references

[8]

But we choose ∆A smartly so that R stays the same butθ changes by∆θ

(Variation step.) Starting fromA0 (called thebe- ginning pulse), we add a small vector ∆A each time. But we choose ∆A smartly so that R stays the same butθ changes by∆θ. That is, we require, in each step, R(A + ∆A) = R(A) (18) θ(A + ∆A) = θ(A) ± ∆θ . (19) The sign before ∆θ is...
[9]

We get controlsA0, A1,

(Termination condition.) Repeat the variation step for M iterations. We get controlsA0, A1, . . . ,AM with θ0, θ1, . . . , θM. Terminatewhen [θ0, θM ]covers the desired range [θL, θR]. In the desired range, we obtain N + 1 pulses where N = θR−θL ∆θ . To aid discussions, it is ...
[10]

constraints,

(Relabel and output.) We relabel theN pulses as- sociated with θ ∈ [θL, θR]by 0, 1, . . . , N. Finally, 12 we obtainN + 1pulses A0, A1, . . . ,AN, along with gate parameters θL = θ0, θ1, . . . , θN = θR. Here we implicitly assumed only one noise source, hence onlyonerobustness...
[11]

constraint

Variation vs. optimization Before diving into GOV, we emphasize that GOV is not an optimization algorithm. Traversing a level set, such as when implementing a robust parametric gate, is fundamentally different from an optimization algorithm (referred to as “OPT” in this sectio...
[12]

pre- variation

Variation condition Let’s reformulate Equation 18 into a more useful condi- tion. Recallthatincalculus, wehavethissimpleequation dy = dy dx dx that states the infinitesimal change ofy as a function of x is equal to the infinitesimal change ofx multiplied by the derivativedy dx...
[13]

Orthogonalization In short, the Gram-Schmidt process orthogonalizes a vector v to a group of vectorsu1, u2, . . . ,uN. The pro- cess is mathematically equivalent to decomposingv as v = v⊥ + v∥ , 13 where v⊥ (or v∥) is orthogonal to (or inside)span{ui}i = span{u1, . . . ,uN }. ...
[14]

official variation

Choice of pre-variation The “official variation”dA is an orthogonal projection of dApre into the variation subspaceV. Therefore, the selection of this pre-variationdApre is very important to roughly determine the direction along which the pulseA should be adjusted, since. This...
[15]

official variation

Normalization Now that the direction of variation is determined as dApre ⊥ , the step size of variation is to be derived. Given our objective to implementU (θ) for every value ofθ, it is imperative to ensure that θ is uniformly distributed along the interval [θL, θR], with the...
[16]

The general proce- dure is as follows

Summary of GOV To summarize, the goal of GOV is to maintain con- straints {Ri}i unchanged, for which we need to achieve the variation condition dA ⊥ ∂Ri ∂A. The general proce- dure is as follows
[17]

Choose an arbitrary pre-variation vectordApre, ei- ther random or task-specific
[18]

Apply the Gram-Schmidt process to getdApre ⊥ ∈ V = span{ ∂Ri ∂A }i ⊥
[19]

Choose an appropriateα according to the task, and vary A by dA = α dApre ⊥ . For our RIPV algorithm, where we traverse the rota- tionangle θ whilemaintainingseveralfixedcriteria {Ri}i, we need to further specify the pre-variationdApre and normalization factor α
[20]

Calculate the pre-variationdApre = ∂θ ∂A. 14
[21]

Apply the Gram-Schmidt process to achieve dApre ⊥ ∈ V = span{ ∂Ri ∂A }i ⊥
[22]

The GOV algorithm extends beyond robust quantum control aimed at countering quasi-static noise

Normalize the vector and adjust the step size to obtain dA = ∆θideal ⟨dApre ⊥ , ∂θ ∂A ⟩ dApre ⊥ . The GOV algorithm extends beyond robust quantum control aimed at countering quasi-static noise. It allows for adjusting the rotation angle while preserving robust- ness against an...
[23]

Extraction of rotation angle For the simplest control schemeHc(t) = Ω( t)σ where σ is a constant operator, the rotation angle is simply the integral θ(A) = R T 0 Ω(t; A) dt. For example, if we expect to implement Rx(θ) with σx control, then this integration gives the exact rot...
[24]

These could be carried out by hand and hard-coded into the program, but we adopt a more general solution, Automatic Differentiation, here- inafter referred to as autodiff

Calculation of derivatives We need to calculate the value of the derivatives∂θ ∂A and ∂Ri ∂A at data points. These could be carried out by hand and hard-coded into the program, but we adopt a more general solution, Automatic Differentiation, here- inafter referred to as autodi...
[25]

We only have to vary one rotation angleθ and maintain one robustness R

Single noise source The simplest RIPV algorithm can deal with one con- trol term Hc,0 and one quasi-static noise source Hn,0, assuming the noise is correctable, i.e.,[Hc,0, Hn,0] ̸= 0. We only have to vary one rotation angleθ and maintain one robustness R. The pseudocode of th...
[26]

We need to discuss two cases, where one of them is trivial and another one requires a modified RIPV algorithm

Multiple noise sources To deal with multiple noise sources, we might need to modify our algorithm. We need to discuss two cases, where one of them is trivial and another one requires a modified RIPV algorithm. Case 1. There is only one control,Hc(t) = Ω( t)Hc,0 satisfying [Hs,...
[27]

Since the rotation angle is no longer a simple inte- gration of the pulse, we have to use the projection method mentioned in section IVC1 instead of sim- ple integral to calculate rotation angles, including θ and ϑj
[28]

orthogonalcontrol

ϑj’s should also remain unchanged. In the orthog- onalization step, we should orthogonalizedApre to both the gradient of robustness against each noise source ∂Ri/∂A and the gradient of undesired ro- tations ∂ϑi/∂A. Consider, for instance, the implementation of Rx(θ) which exhi...
[29]

official

Number of parameters The number of independent control parameters, equiv- alently the dimension of the landscape, matters. It must be sufficiently large to accommodate a viable solution and ensure that the desired solutions are in the same connected component as the beginning ...
[30]

No matter how small a step we choose for variation, we can only slow down the accumulation of error but not eliminate it

Error from linear approximation The RIPV algorithm inevitably introduces error when ittakesfinite-lengthstepsinthetangentspaceofthelevel set instead of the actual level set. No matter how small a step we choose for variation, we can only slow down the accumulation of error but...
[31]

This problem might appear when we orthogonalizedApre to W = span { ∂Ri ∂A }m i=1

Viability of variation direction Even if we used a lot of free parameters, there is still a chance that we cannot find a viable direction fordA. This problem might appear when we orthogonalizedApre to W = span { ∂Ri ∂A }m i=1. If we chose the pre-variation dApre = ∂θ ∂A and fo...
[32]

8-shaped

Interpolation of gate parameters As mentioned in section IIC, we need to interpolate between the pulses after RIPV. After all, we can only generate a discrete sequence of pulses, instead of a gen- uinely continuous function. Since our algorithm is a local gradient-based algori...
[33]

According to Hai et al

Single noise source In this part, we assume the noise is onσz. According to Hai et al. [45], we need only one control on either one of the perpendicular axes, which means control on either σx or σy. We chooseσx as our control term to implement robust Rx(θ). To summarize, we as...
[34]

robust pulses

Multiple noise sources If there are multiple noise sources, as we discussed in section IVC4, the algorithm is slightly different. To sup- press quasi-static noise from all three directions, we need at least 2 orthogonal controls. The rotation angleθj on direction σj where j = ...
[35]

Cheng, X.-H

B. Cheng, X.-H. Deng, X. Gu, Y. He, G. Hu, P. Huang, J. Li, B.-C. Lin, D. Lu, Y. Lu,et al., Noisy intermediate- scale quantum computers, Frontiers of Physics18, 21308 (2023)

2023
[36]

E. T. Campbell, B. M. Terhal, and C. Vuillot, Roads towards fault-tolerant universal quantum computation, Nature 549, 172 (2017)

2017
[37]

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.,Quantumoptimalcontrolin quantum technologies. strategic report on current status, visions and goals for research in europe, EPJ Quantum Tec...

2022
[38]

Chakrabarti and H

R. Chakrabarti and H. Rabitz, Quantum control land- scapes, International Reviews in Physical Chemistry26, 671 (2007)

2007
[39]

Ge, R.-B

X. Ge, R.-B. Wu, and H. Rabitz, The optimization landscape of hybrid quantum–classical algorithms: From quantum control to nisq applications, Annual Reviews in Control 54, 314 (2022)

2022
[40]

A.Rothman, T.-S.Ho,andH.Rabitz,Exploringthelevel sets of quantum control landscapes, Physical Review A 73, 053401 (2006)

2006
[41]

Dominy and H

J. Dominy and H. Rabitz, Exploring families of quantum controls for generating unitary transformations, Journal of Physics A: Mathematical and Theoretical41, 205305 (2008)

2008
[42]

Chen, R.-B

Q.-M. Chen, R.-B. Wu, T.-M. Zhang, and H. Rabitz, Near-time-optimalcontrolforquantumsystems,Physical Review A92, 063415 (2015)

2015
[43]

A. M. Souza, G. A. Alvarez, and D. Suter, Robust dy- namicaldecouplingforquantumcomputingandquantum memory, Physical review letters106, 240501 (2011)

2011
[44]

G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vi- tanov, Correction of arbitrary field errors in population inversion of quantum systems by universal composite pulses, Physical review letters113, 043001 (2014)

2014
[46]

Khodjasteh and L

K. Khodjasteh and L. Viola, Dynamically error-corrected gates for universal quantum computation, Physical re- view letters102, 080501 (2009)

2009
[47]

Zeng, X.-H

J. Zeng, X.-H. Deng, A. Russo, and E. Barnes, Gen- eral solution to inhomogeneous dephasing and smooth pulse dynamical decoupling, New Journal of Physics20, 033011 (2018)

2018
[48]

Dong and I

D. Dong and I. R. Petersen,Learning and robust control in quantum technology(Springer, 2023)

2023
[49]

X. Cao, J. Cui, M. H. Yung, and R.-B. Wu, Robust con- trol of single-qubit gates at the quantum speed limit, Physical Review A110, 022603 (2024)

2024
[50]

R. L. Kosut, G. Bhole, and H. Rabitz, Robust quan- tum control: Analysis & synthesis via averaging, arXiv preprint arXiv:2208.14193 (2022)

2022 arXiv
[51]

Ramakrishna, M

V. Ramakrishna, M. V. Salapaka, M. Dahleh, H. Rab- itz, and A. Peirce, Controllability of molecular systems, Physical Review A51, 960 (1995)

1995
[52]

H. Fu, S. G. Schirmer, and A. I. Solomon, Complete con- trollability of finite-level quantum systems, Journal of Physics A: Mathematical and General34, 1679 (2001)

2001
[53]

S. G. Schirmer, H. Fu, and A. I. Solomon, Complete con- trollability of quantum systems, Physical Review A63, 063410 (2001)

2001
[54]

Polack, H

T. Polack, H. Suchowski, and D. J. Tannor, Uncontrol- lable quantum systems: A classification scheme based on lie subalgebras, Physical Review A79, 053403 (2009)

2009
[55]

H. A. Rabitz, M. M. Hsieh, and C. M. Rosenthal, Quan- tum Optimally Controlled Transition Landscapes, Sci- ence 303, 1998 (2004)

2004
[56]

T.-S.Ho, J.Dominy,andH.Rabitz,Landscapeofunitary transformations in controlled quantum dynamics, Physi- cal Review A79, 013422 (2009)

2009
[57]

C. Brif, R. Chakrabarti, and H. Rabitz, Control of quan- tum phenomena: Past, present and future, New Journal of Physics12, 075008 (2010)

2010
[58]

A. N. Pechen and D. J. Tannor, Are there Traps in Quan- tum Control Landscapes?, Physical Review Letters106, 120402 (2011)

2011
[59]

De Fouquieres and S

P. De Fouquieres and S. G. Schirmer, A Closer Look at Quantum Control Landscapes & their Implication for Control Optimization, Infinite Dimensional Analysis, Quantum Probability and Related Topics16, 1350021 (2013)

2013
[60]

Birtea, I

P. Birtea, I. Caşu, and D. Comănescu, Constraint opti- mization and SU(N) quantum control landscapes, Jour- nal of Physics A: Mathematical and Theoretical 55, 115301 (2022)

2022
[61]

Are There Traps in Quantum Control Land- scapes?

H. Rabitz, T.-S. Ho, R. Long, R. Wu, and C. Brif, Com- ment on “Are There Traps in Quantum Control Land- scapes?”, Physical Review Letters108, 198901 (2012)

2012
[62]

R. Wu, J. Dominy, T.-S. Ho, and H. Rabitz, Singularities of Quantum Control Landscapes, Physical Review A86, 013405 (2012), arxiv:0907.2354 [quant-ph]

2012 arXiv
[63]

Russell, H

B. Russell, H. Rabitz, and R.-B. Wu, Control landscapes are almost always trap free: A geometric assessment, Journal of Physics A: Mathematical and Theoretical50, 205302 (2017)

2017
[64]

Pechen, D

A. Pechen, D. Prokhorenko, R. Wu, and H. Rabitz, Control landscapes for two-level open quantum systems, Journal of Physics A: Mathematical and Theoretical41, 045205 (2008)

2008
[65]

R.-B. Wu, H. Ding, D. Dong, and X. Wang, Learning ro- bust and high-precision quantum controls, Physical Re- 24 view A99, 042327 (2019)

2019
[66]

Clarke and F

J. Clarke and F. K. Wilhelm, Superconducting quantum bits, Nature453, 1031 (2008)

2008
[67]

Krantz, M

P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gus- tavsson, and W. D. Oliver, A quantum engineer’s guide to superconducting qubits, Applied physics reviews 6 (2019)

2019
[68]

J. Koch, T. M. Yu, J. Gambetta, A. A. Houck, D. I. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Charge-insensitive qubit design de- rived from the Cooper pair box, Physical Review A76, 042319 (2007)

2007
[69]

Sarovar, T

M. Sarovar, T. Proctor, K. Rudinger, K. Young, E. Nielsen, and R. Blume-Kohout, Detecting crosstalk errors in quantum information processors, Quantum4, 321 (2020)

2020
[70]

Rudinger, C

K. Rudinger, C. W. Hogle, R. K. Naik, A. Hashim, D. Lobser, D. I. Santiago, M. D. Grace, E. Nielsen, T. Proctor, S. Seritan, et al., Experimental characteri- zation of crosstalk errors with simultaneous gate set to- mography, PRX Quantum2, 040338 (2021)

2021
[71]

Yi, Y.-J

K. Yi, Y.-J. Hai, K. Luo, J. Chu, L. Zhang, Y. Zhou, Y. Song, S. Liu, T. Yan, X.-H. Deng,et al., Robust quan- tum gates against correlated noise in integrated quantum chips, Physical Review Letters132, 250604 (2024)

2024
[72]

Bylander, S

J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, Noise spectroscopy through dy- namical decoupling with a superconducting flux qubit, Nature Physics7, 565 (2011)

2011
[73]

K. R. Brown, A. W. Harrow, and I. L. Chuang, Arbitrar- ily accurate composite pulse sequences, Physical Review A 70, 052318 (2004)

2004
[74]

Shi, J.-T

Z.-C. Shi, J.-T. Ding, Y.-H. Chen, J. Song, Y. Xia, X. Yi, and F. Nori, Supervised learning for robust quantum con- trol in composite-pulse systems, Physical Review Applied 21, 044012 (2024)

2024
[75]

Zhang, Z

Z. Zhang, Z. Miao, Y. Chen, and X.-H. Deng, Smolyak al- gorithmassistedrobustcontrolforquantumsystemswith uncertainties, arXiv preprint arXiv:2410.14286 (2024)

2024 arXiv
[76]

B. Li, T. Calarco, and F. Motzoi, Experimental error suppression in cross-resonance gates via multi-derivative pulse shaping, npj Quantum Information10, 66 (2024)

2024
[77]

Barnes, F

E. Barnes, F. A. Calderon-Vargas, W. Dong, B. Li, J. Zeng, and F. Zhuang, Dynamically corrected gates from geometric space curves, Quantum Science and Tech- nology 7, 023001 (2022)

2022
[78]

W. Dong, F. Zhuang, S. E. Economou, and E. Barnes, Doubly geometric quantum control, PRX Quantum 2, 030333 (2021)

2021
[79]

Y.-J. Hai, J. Li, J. Zeng, D. Yu, and X.-H. Deng, Univer- sal robust quantum gates by geometric correspondence of noisy quantum evolution (2023), arxiv:2210.14521 [quant-ph]

2023 arXiv
[80]

Sauvage and F

F. Sauvage and F. Mintert, Optimal control of families of quantum gates, Physical Review Letters129, 050507 (2022)

2022
[81]

Barnes and J

E. Barnes and J. Zeng, Generating error-resistant quan- tum control pulses from geometrical curves (2022), uS Patent App. 17/765,875

2022
[82]

Foxen, C.Neill, A

B. Foxen, C.Neill, A. Dunsworth, P.Roushan, B. Chiaro, A. Megrant, J. Kelly, Z. Chen, K. Satzinger, R. Barends, F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, S. Boixo, D. Buell, B. Burkett, Y. Chen, R. Collins, E. Farhi, A. Fowler, C. Gidney, M. Giustina, R. Graff, M....

2020

Pith tools

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

[1] [45]

Zhang, T

J. Zhang, T. H. Kyaw, S. Filipp, L.-C. Kwek, E. Sjöqvist, and D. Tong, Geometric and holonomic quantum compu- tation, Physics Reports1027, 1 (2023)

work page 2023

[2] [1]

Each Ωk(A) is a function oft, denoted byΩk(t; A): Ωk : A 7→ Ωk(A), s.t

A vector of maps⃗Ω = {Ωk}k from control parame- ters A to control pulsesΩk(t; A), each correspond- ing to one of the control termsHc,k. Each Ωk(A) is a function oft, denoted byΩk(t; A): Ωk : A 7→ Ωk(A), s.t. Ωk(A)(t) = Ωk(t; A)

[3] [2]

Hc[⃗Ω](t) = X k Ωk(t)Hc,k

A mapHc from pulses ⃗Ω to a control Hamiltonian Hc[⃗Ω], where Hc[⃗Ω] is a time-dependent operator: Hc : ⃗Ω 7→ Hc[⃗Ω], s.t. Hc[⃗Ω](t) = X k Ωk(t)Hc,k

[4] [3]

Usc[Hc](t) = U Hs+Hc(τ )(t)

A map Usc from control Hamiltonian Hc to the noiseless propagator Usc[Hc], where Usc[Hc] is a time-independent operator: Usc : Hc 7→ Usc[Hc], s.t. Usc[Hc](t) = U Hs+Hc(τ )(t)

work page

[5] [4]

Finally, we obtain the robustness map from control pa- rameters Ato some metric of robustnessRby composing all the maps, R : A 7→ ⃗Ω(A) 7→ Hc[⃗Ω] 7→ Usc[Hc] 7→ R[Usc(A)]

A map from the propagator Usc(t) to any metric of robustness R (integral robustness, asymptotic robustness, etc.): R : Usc 7→ R[Usc] . Finally, we obtain the robustness map from control pa- rameters Ato some metric of robustnessRby composing all the maps, R : A 7→ ⃗Ω(A) 7→ Hc[⃗Ω] 7→ Usc[Hc] 7→ R[Usc(A)] . (17) If we write out the time dependence explicitl...

work page

[6] [5]

Wefirst discretize thetime interval[0, T]into N pieces, namely t0 = 0, t1 = T N ,

Piecewise constant parametrization. Wefirst discretize thetime interval[0, T]into N pieces, namely t0 = 0, t1 = T N , . . . , tk = kT N , . . . , tN = T . The pulse is then defined by the parameter vectorA = (A0, . . . , AN ) as Ω(t) = ( 0 t /∈ [0, T] Ak t ∈ [tk, tk+1]. The parametrization is straightforward to implement, but applying constraints could be...

work page

[7] [6]

There are many functional bases to choose

Functional basis parametrization. There are many functional bases to choose. Two com- mon methods are the Taylor expansion and Fourier ex- pansion, where the pulse parameters are the coefficients of the expansion terms. Besides, wavelets are among the most useful functional bases for constructing pulses, for they are finitely supported and have various me...

work page

[8] [7]

After optimization, we obtain a pulse A0 and its corresponding gate parameterθ0

(Initialization.) Starting from an arbitrary pulse Ainit (called the initial pulse), use any optimiza- tion algorithm to optimize the robustness function R(A). After optimization, we obtain a pulse A0 and its corresponding gate parameterθ0. Note that we only record but do not designate the valueθ0

work page