Minimal-Energy Optimal Control of Tunable Two-Qubit Gates in Superconducting Platforms Using Continuous Dynamical Decoupling

Abstract

We present a unified scheme for generating high-fidelity entangling gates in superconducting platforms by continuous dynamical decoupling (CDD) combined with variational minimal-energy optimal control. During the CDD stage, we suppress residual couplings, calibration drifting, and quasistatic noise, resulting in a stable effective Hamiltonian that preserves the designed ZZ interaction intended for producing tunable couplers. In this stable $\mathrm{SU}(4)$ manifold, we calculate smooth low-energy single-quibt control functions using a variational geodesic optimization process that directly minimizes gate infidelity. We illustrate the methodology by applying it to CZ, CX, and generic engangling gates, achieving virtually unit fidelity and robustness under restricted single-qubit action, with experimentally realistic control fields. These results establish CDD-enhanced variational geometric optimal control as a practical and noise-resilient scheme for designing superconducting entangling gates.

Figures (4)

• Figure 1: At the top is shown the integrand of the energetic cost functional, Eq. (\ref{['Efunctional']}). The area under each curve corresponds to the actual cost of implementing the gate. In the legends, we show the numerical value of the integral for each curve, which is in units of $\hbar^2/\tau$. At the bottom is the gate fidelity as a function of time. Again, the curves for the Monte Carlo and Variational methods are practically the same.
• Figure 2: Optimal control Hamiltonian for the CZ gate obtained with each of the methods: (a) Monte Carlo, (b) Krotov, and (c) Variational. The indices $j=1,2,3$ correspond to the operators $\sigma_j \otimes I$, which operate on the first qubit, and $j=4,5,6$ correspond to the operators $I \otimes \sigma_{j-3},$ as explained just following Eq. (\ref{['Hc1']}). Notice that the solutions in (a) and (c) differ only by a multiplicative factor of $-1$ in the $\sigma_x$ and $\sigma_y$ components.
• Figure 3: Analogous to Fig. \ref{['CZ_graphs']}, but for the general entangling gate $U_\mathrm{R}$ given by Eq. (\ref{['generic_U']}), which is locally equivalent to the CX (and CZ) gate, meaning they only differ by local operations. At the top is shown the integrand of the energetic cost functional, Eq. (\ref{['Efunctional']}). In the legends, we show the numerical value of the integral for each curve, which is in units of $\hbar^2/\tau$. At the bottom is the gate fidelity as a function of time.
• Figure 4: Energetic cost and gate fidelity for $U_\mathrm{R}$ gate with each of the three components separately suppressed. In all cases, the three methods can find an optimal control Hamiltonian that results in $U_\mathrm{R}$ with unitary fidelity at $t=\tau$. Similar to the previous results, the Variational and Monte Carlo methods reached equivalent solutions.