Pulse engineering via projection of response functions at infinite nonlinear order

Source: arXiv:2607.24725 · Published 2026-07-27 · By Lia Kley, Ludwig Mathey

TL;DR

This paper addresses the challenge of optimizing control pulses for quantum systems to implement high-fidelity quantum gates, focusing on overcoming issues of high-dimensional and complex control landscapes with sensitive hyperparameter dependencies. The authors introduce PEPRino, a novel optimization method based on projecting response functions to infinite nonlinear order to efficiently approximate fidelity landscape changes. This approach is hyperparameter-free and gradient-free, relying on a resummation of the first and second order response functions to determine optimal pulse updates.

The method is demonstrated for the optimal implementation of the Quantum Fourier Transform (QFT) on two- and three-qubit systems. Extensive benchmarking against the Chopped Random Basis (CRAB) algorithm with Nelder-Mead optimization for two-qubits shows that PEPRino achieves comparable or better final gate infidelities but with faster convergence in iteration steps and reduced computational costs. The scalability to three qubits is illustrated, highlighting PEPRino’s advantages in higher-dimensional parameter spaces where CRAB becomes computationally prohibitive. Overall, PEPRino offers a promising alternative for robust, scalable quantum control with minimal tuning requirements.

Key findings

• PEPRino requires roughly 1000 iterations to converge for 2-qubit QFT optimization compared to about 3000 iterations for CRAB with Nelder-Mead.
• Using 8 sine modes for control pulse parameterization strikes an effective balance between convergence speed and computational efficiency.
• Increasing batch size of initial states sampled (nB from 4 to 7) reduces iteration steps but yields diminishing returns beyond a certain point.
• PEPRino’s update rule uses only the first and second order susceptibilities computed from system response to δ-function perturbations, efficiently resumming infinite-order response contributions.
• For 3-qubit QFT, PEPRino converges to high-fidelity solutions within ~6000 iterations; CRAB optimization was not feasible due to computational cost.
• PEPRino optimization shows reduced variance in final infidelities across runs compared to CRAB, improving stability.
• Average infidelity is computed over full 2-design ensembles for initial states to accurately estimate gate fidelity for both methods.
• PEPRino eliminates the need for hyperparameter tuning and gradients, simplifying usability in quantum optimal control.

Threat model

n/a — This work does not consider an adversarial threat model but focuses exclusively on the optimization of quantum control pulses to maximize gate fidelity in closed quantum systems.

Methodology — deep read

1. Threat model & assumptions: The paper focuses on quantum optimal control for gate synthesis rather than an adversarial security model. It assumes a closed quantum system with time-dependent control Hamiltonians where the goal is to maximize gate fidelity.

2. Data: The method evaluates fidelity changes averaged over ensembles of initial states sampled randomly or from exact 2-designs to approximate Haar averages. For two qubits, full 2-designs with 36 states are used; for three qubits, subsets of 100 random states.

3. Architecture/algorithm: The total Hamiltonian is decomposed into a fixed part H0 and trainable part Hθ(t), where control operators Bj (Pauli matrices or tensor products) are modulated by control functions θj(t), themselves parameterized as a finite sum of sine modes with trainable coefficients θj,k. A time-local perturbation proportional to ϵδ(t - tr) is applied to one control operator at a sampled time tr. The resulting change in fidelity ΔF is expressed as an infinite series of response functions (susceptibilities) χ(n)(tr) to all orders. Using the algebra of Pauli matrices, all odd and even orders reduce to linear combinations of first and second order susceptibilities, enabling a closed-form resummation:  ΔF_PEPRino = 1/2 χ(1) sin(2ϵ) + 1/4 χ(2) (1 - cos(2ϵ)). The optimal update step size ϵ* maximizing ΔF_PEPRino is computed analytically, and the trainable parameters θj,k are updated accordingly by projection onto the sine modes.

4. Training regime: For 2-qubit QFT, 8 sine modes (nm=8) and varying batch sizes of initial states (nB=4 to 41) were tested. The Nelder-Mead simplex parameters are standard (α=1, γ=2, σ=0.5, β=0.5). 40 independent runs are averaged logarithmically to measure performance. For 3 qubits, nm=30 modes and batch sizes nB=11 to 13 were used.

5. Evaluation protocol: Gate infidelity is computed as the trace distance between evolved and target states averaged over initial states. PEPRino is benchmarked directly against the CRAB algorithm using Nelder-Mead for 2 qubits, demonstrating faster iteration convergence and lower computational costs. For 3 qubits, only PEPRino is evaluated given CRAB’s scaling.

6. Reproducibility: Detailed algorithmic steps and parameterizations are provided, with reference to prior PEPR and CRAB methods. Implementation details such as batch sizes, mode numbers, and optimization parameters are fully described. Public code release is not mentioned.

Example walkthrough: For 2 qubits, the algorithm initializes θ parameters randomly, computes first and second order susceptibilities χ(1) and χ(2) at a perturbation time tr and randomly chosen control operator Bj, then calculates ΔF_PEPRino as a function of ϵ. The ϵ maximizing ΔF_PEPRino with minimum magnitude is selected as ϵ*. The parameters θj,k are updated by subtracting 2ϵ*(tf)^-1 sin(πk tr / tf). This is repeated iteratively until convergence to a high-fidelity QFT implementation is observed.

Technical innovations

• A gradient- and hyperparameter-free optimization algorithm for quantum control based on resummation of infinite-order response functions into a manageable update formula.
• Projection of fidelity response to time-local perturbations onto a finite basis of sine modes, enabling efficient parameter updates in a time-nonlocal pulse parameterization.
• Analytical treatment exploiting SU(2) algebra to reduce infinite series of susceptibilities to just first and second order, allowing fast evaluation of nonlinear control landscape curvature.
• Use of averaged susceptibilities over batches of initial states (2-designs or random ensembles) to optimize gate implementations robustly across unknown inputs.

Datasets

• Two-qubit full 2-design ensemble — 36 pure states — constructed from tensor products of randomly sampled single-qubit states
• Three-qubit random ensemble — 100 pure states — randomly sampled initial states (not full 2-design due to computational constraints)

Baselines vs proposed

• CRAB + Nelder-Mead (2-qubit, nm=8, batch=41): iterations to convergence ≈ 3000 vs PEPRino: ≈ 1000; final infidelity comparable (~10^-6)
• PEPRino (2-qubit): convergence speed improves with increasing batch size (nB from 4 to 7), but diminishing returns observed beyond nB=7
• PEPRino (3-qubit, nm=30, batch=11-13): converges to infidelities ~10^-6 in ~6000 iterations; CRAB not evaluated due to computational infeasibility
• Variance in final infidelity across runs is lower for PEPRino compared to CRAB, indicating more stable optimization

Limitations

• The method assumes closed quantum systems without decoherence or noise; extension to open quantum systems is not addressed.
• Computational cost and convergence are demonstrated only up to three qubits; scalability to larger qubit numbers remains to be tested in practice.
• Batch sizes for averaging initial states require some tuning, limiting full hyperparameter-free claims in practice.
• CRAB is used only as a baseline for 2-qubit problems; no direct comparison to gradient-based methods or other recent quantum optimal control techniques.
• No explicit robustness evaluation under realistic experimental noise, control errors, or model mismatch scenarios.
• The approach hinges on Pauli-type control operators and may require adaptation for other control Hamiltonians.

Open questions / follow-ons

• How does PEPRino perform for larger quantum systems beyond 3 qubits, especially under more complex control constraints?
• Can this response function resummation approach be adapted to noisy or open quantum systems with decoherence?
• How robust is the method in experimental settings with control imperfections and calibration errors?
• Is it possible to incorporate additional practical constraints such as limited pulse amplitudes or bandwidth into the PEPRino framework?

Why it matters for bot defense

Though the paper focuses on quantum optimal control rather than bot defense or CAPTCHAs, the methodological insight—leveraging higher-order response functions to efficiently navigate complex, high-dimensional optimization landscapes without gradients or hyperparameters—may inspire analogous techniques in machine learning or adversarial robustness. For CAPTCHA engineering, inspiration might be drawn in how PEPRino bypasses costly gradient calculations by exploiting problem structure to speed up optimization. However, direct applicability is limited since the domain, threat model, and data types differ substantially.

Practitioners focused on bot detection and CAPTCHA design could view this work as a sophisticated example of hyperparameter-free optimization in a complex space, potentially motivating the search for control-theoretic or response function-based approaches to bot behavior modeling and defense. The focus on robustness and scalability in a high-dimensional nonlinear setting resonates with challenges in adversarial ML.

Cite

@article{arxiv2607_24725,
  title={ Pulse engineering via projection of response functions at infinite nonlinear order },
  author={ Lia Kley and Ludwig Mathey },
  journal={arXiv preprint arXiv:2607.24725},
  year={ 2026 },
  url={https://arxiv.org/abs/2607.24725}
}

Read the full paper

• arXiv:2607.24725 (abstract)
• arXiv:2607.24725 (PDF)