M. Harutyunyan and S. Guérin of Université Bourgogne Europe have developed a new quantum control strategy that restores two-level qubit behavior within more complex, multi-level systems. The researchers reformulated the Derivative Removal by Adiabatic Gate technique using the iterative quantum Kolmogorov-Arnold-Moser algorithm, enabling the incorporation of existing two-level control pulses.

Results demonstrate that “two-level robustness is restored after only a few iterations,” even with relatively short pulse durations, and the resulting RDK pulses are near optimal in terms of energy compared to other methods. This approach offers versatility for superconducting transmons and other multi-level qubits by effectively reproducing simplified dynamics.

Iterative Quantum KAM Algorithm for Multirobust Control

The iterative application of the quantum Kolmogorov-Arnold-Moser (KAM) algorithm improves control pulse fidelity within multi-level quantum systems. Researchers at Université Bourgogne Europe, led by S. Guérin, demonstrated this enhancement while refining the Derivative Removal by Adiabatic Gate (DRAG) technique, a method traditionally used to suppress unwanted transitions in quantum control.

This reformulation allows the incorporation of control pulses initially designed for simpler, two-level qubits into more complex architectures, offering a way to use existing control strategies. By enforcing a specific match between the multi-level system’s evolution and that of an ideal two-level qubit, the algorithm minimizes leakage outside the computational qubit space and simultaneously addresses robustness against variations in control parameters.

Spectral analysis, conducted with a typical transmon, indicates that high-frequency components, often degraded in real-world hardware, do not significantly impede performance. The algorithm’s compatibility extends to various control pulse types, including α-RIO and αδ-RIO pulses used for complete population transfer, suggesting broad applicability across different quantum gate implementations.

DRAG Technique Reformulation for Qubit Leakage Suppression

This approach bypasses the need for recursive, multi-step DRAG pulses previously used to address multiple leakage pathways in systems like qutrits, quantum systems with three levels. The technique’s efficacy stems from its ability to enforce a block-diagonal structure in the effective Hamiltonian, separating the computational qubit subspace from leakage states.

The researchers assessed RDK pulse performance across varying durations, iteration counts and robustness requirements, demonstrating the technique’s adaptability to different quantum control scenarios. The total Hamiltonian is modeled as H(t) = H₀ + ε V(t), where the small parameter ε identifies the perturbation size, encompassing all orders of control signals.

RDK Pulse Performance Across Iterations and Robustness Types

Spectral analysis, conducted with a typical transmon, indicates that high-frequency components, often degraded in real-world hardware, do not significantly impede performance. The iterative DRAG-KAM algorithm behaves as a superasymptotic series, reaching a performance plateau after a limited number of iterations, with the required iteration count increasing alongside pulse duration.

In fact, seeding GRAPE with RDK pulses yields the best nominal infidelity and converges eightfold faster, as demonstrated by comparative analyses of control fields Ω and Δ. The researchers found similar values of J, indicating a broader robustness profile in detuning, although shallower, than with other methods.

Restoring Two-Level Robustness with Asymptotic Approximation

Analysis reveals that the DRAG-KAM algorithm effectively recovers the robustness of two-level RIO controls after only three iterations, as demonstrated by profiles tracking infidelity versus detuning. This recovery occurs even with pulses designed for both alpha and delta variations, where initial detuning corrections increase error, but are subsequently mitigated by the iterative process. The team applied a time reparametrization technique before implementing the DRAG-KAM algorithm to ensure optimal performance, and the resulting robustness profiles closely mirror those achieved with simpler, single-parameter controls.

Further investigation into the alpha-delta RIO pulse, robust to changes in both alpha and delta, demonstrates that the technique consistently restores two-level robustness within three iterations. Detailed analysis of the delta profile confirms this recovery, showing that the iterative process effectively replicates the robustness originally defined within the two-level qubit subspace. The supplementary material details the step-by-step derivation of this iterative DRAG-KAM algorithm for a three-level system, providing a comprehensive guide to its implementation.

RDK pulses demonstrate energy efficiency comparable to optimal RIO pulses within the computational qubit subspace, a key metric for practical quantum computing applications. Such pulses are well suited for superconducting transmons as well as other forms of qubits embedded in a multi-level structure.

KAM Technique: Reducing Hamiltonian Coupling Terms

The iterative quantum Kolmogorov-Arnold-Moser (KAM) algorithm provides a systematic method for minimizing unwanted interactions within multi-level quantum systems, a process achieved through a series of carefully constructed transformations. The technique focuses on reducing the influence of higher-energy states by eliminating the terms that couple them to the computational qubit subspace, all while preserving the intended control within that subspace. A key element of this method is the decomposition of the Hamiltonian into an unperturbed component and a perturbation, enabling the construction of an effective Hamiltonian that simplifies the system’s dynamics.

This process involves iterative unitary transformations, each designed to lower the order of non-diagonal terms, ultimately leading to a block-diagonalized Hamiltonian. The resulting transformation, denoted as T₁, incorporates a dynamical term that further minimizes off-diagonal elements, effectively suppressing population leakage to unwanted states and maintaining the fidelity of the original gate operations. Subsequent transformations, T₂ and beyond, refine this process, pushing unwanted terms to even higher orders of magnitude. This allows for the construction of robust control protocols compatible with existing two-level strategies.

Formulating the Perturbation with H_0 and V_1

This approach decomposes the total Hamiltonian into an unperturbed component, H₀ and a perturbation, εV₁, allowing for a controlled reduction of coupling between the computational qubit subspace and higher energy levels. Specifically, the method relies on iteratively constructing a transformed Hamiltonian where the perturbation’s order decreases with each iteration, moving from ε to ε², then ε⁴, and so on, achieving a superexponential decrease of order $ε^{2^N}$ after N iterations.

The process begins by defining equations for D₁ and W₁, operators derived from expanding the Hamiltonian in powers of ε and enforcing the cancellation of first-order perturbation terms, which are then used to refine the Hamiltonian. The equation for D₁ is expressed as ∑ν,j,j′, while W₁ is defined as / (λν⁰ − λν′⁰).

This formulation allows for the introduction of free parameters within the definition of W₁, accelerating the convergence of the iterative process. The resulting transformed Hamiltonian, H₂, retains the same general form as the original, enabling the procedure to be repeated with updated H₀,j and Wⱼ operators.

Applying KAM Transformations for Superexponential Convergence

This approach moves beyond simply minimizing error. It actively reshapes the Hamiltonian through successive transformations, accelerating convergence rates beyond those typically observed in quantum control. Each step in the process reduces perturbation from order ε to $ε^{2}$ , then $ε^{4}$ , and so on, establishing a pathway to highly accurate control even with limited computational resources.

The technique offers a way to construct a sequence of transformations that systematically lower the order of non-diagonal terms of the Hamiltonian. This combined approach generates control protocols that systematically reduce the influence of higher-energy states by eliminating the terms coupling them to the qubit subspace while simultaneously ensuring that the original controls within the qubit subspace remain unaltered.

👉 More information
🗞 Multirobust Control Pulses Designed by Iterative Quantum Kolmogorov-Arnold-Moser Algorithm
✍️ Meri Harutyunyan and Stéphane Guérin
🧠 ArXiv: https://arxiv.org/abs/2611.07802v1

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