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We introduce a strategy to develop optimally designed fields for continuous dynamical decoupling. Using our methodology, we obtain the optimal continuous field configuration to maximize the fidelity of a general one-qubit quantum gate. To achieve this, considering dephasing-noise perturbations, we employ an auxiliary qubit instead of the boson bath to implement a purification scheme, which results in unitary dynamics. Employing the sub-Riemannian geometry framework for the two-qubit unitary group, we derive and numerically solve the geodesic equations, obtaining the optimal time-dependent control Hamiltonian. Also, due to the extended time required to find solutions to the geodesic equations, we train a neural network on a subset of geodesic solutions, enabling us to promptly generate the time-dependent control Hamiltonian for any desired gate, which is crucial in circuit optimization., Comment: 14 pages, 10 figures

We consider the case of a $\sqrt{\mathrm{SWAP}}$ quantum gate and its optimized entangling action, via continuous dynamical decoupling, in the presence of dephasing noise. We illustrate the procedure in the specific case where only the two-qubit operation is controlled and no single-qubit operations are included in the description. To compare the optimized dynamics in the presence of noise with the ideal case, we use the standard fidelity measure. Then we discuss the importance of using optimized gates in the quantum operational-probabilistic theory. Because of their importance for the explicit construction of the completely positive maps representing the operations, we derive optimized Kraus operators in this specific case, focusing on the entanglement operation. We then show how to visualize the time evolution of each Kraus operator as a curve in a three-dimensional Euclidean space. Finally, we connect this formalism with the operational framework of quantum mechanics by describing a possible set of measurements that could be performed to obtain the Kraus operators.