Pareto-Front Engineering of Dynamical Sweet Spots in Superconducting Qubits

TL;DR

The paper tackles decoherence management in superconducting qubits by developing a fully parameterized, multi-objective framework that uses general periodic flux modulation to engineer dynamical sweet spots (DSSs). By applying Floquet theory and a bi-objective Pareto-front optimization of the energy-relaxation rate $\gamma_1$ and dephasing rate $\gamma_z$, it maps the trade-offs between $T_1$ and $T_\phi$, revealing DSS and double-DSS operating regions and showing that $T_1$ cannot be made arbitrarily large under periodic modulation. The optimized DSS operating points yield substantial coherence gains, with $T_\phi$ enhanced by roughly 3–5× while maintaining $T_1$ in the hundreds of microseconds, and enable high-fidelity gate operations (single- and two-qubit) via additional GRAPE-optimized control pulses. The work provides a general PF engineering framework that improves gate performance under open-system dynamics and is applicable to fluxonium and, more broadly, to other superconducting qubits. Its findings—quantitative PFs, fundamental limits on $T_1$, and robust double-DSS regions—offer practical guidelines for designing robust quantum control in noisy solid-state devices.

Abstract

Operating superconducting qubits at dynamical sweet spots (DSSs) suppresses decoherence from low-frequency flux noise. A key open question is how long coherence can be extended under this strategy and what fundamental limits constrain it. Here we introduce a fully parameterized, multi-objective periodic-flux modulation framework that simultaneously optimizes energy relaxation $T_1$ and pure dephasing $T_φ$, thereby quantifying the tradeoff between them. For fluxonium qubits with realistic noise spectra, our method enhances $T_φ$ by a factor of 3-5 compared with existing DSS strategies while maintaining $T_1$ in the hundred-microsecond range. We further prove that, although DSSs eliminate first-order sensitivity to low-frequency noise, relaxation rate cannot be reduced arbitrarily close to zero, establishing an upper bound on achievable $T_1$. At the optimized working points, we identify double-DSS regions that are insensitive to both DC and AC flux, providing robust operating bands for experiments. As applications, we design single- and two-qubit control protocols at these operating points and numerically demonstrate high-fidelity gate operations. These results establish a general and useful framework for Pareto-front engineering of DSSs that substantially improves coherence and gate performance in superconducting qubits.

Figures (7)

• Figure 1: (a) Aggregated PF with Fourier truncation $n=4$, obtained by aggregating the run-level PFs from ENS-MOEA/D, HypE, SPEA2, and tDEA followed by a final non-dominated sort. Shaded lobes $C1-C3$ indicate sub-regions preferentially explored by different algorithms. Working Point 1 marks the onset of the DSS region. The bracketed segment in $C_2$, labeled “Double-DSS,” indicates the region where candidate double-DSS points occur. (b) Time-domain modulation for Working Point 2: $P(t)=\sum_n p_n e^{in\omega_d t}$, defining $\phi_{\mathrm{ext}}(t)=\phi_{\mathrm{dc}}+\phi_{\mathrm{ac}}P(t)$.
• Figure 2: (a). Control setup for implementing a single-qubit gate in the fluxonium system. (b). The circuit diagram of the device. The system is periodically modulated to stabilize a DSS, while an additional control pulse is applied to steer the system's evolution. (c). Main plot: Closed-system gate fidelity versus GRAPE iteration number for the $X$ gate under periodic flux modulation. Inset: Optimized time-domain control pulse $\Omega_d(t)$ represented using 500 discretized time intervals and constrained to 9 frequency components. The total gate duration is 10 ns, and the pulse amplitude is limited to the range $[-100, 100]$ MHz $\times\,2\pi$.
• Figure 3: (a). Schematic of the two-qubit gate implementation between two fluxonium qubits. (b). The circuit diagram of the device. Each qubit is periodically modulated to reach a DSS and is independently driven by an optimized control pulse $f_{L/R}(t)$ acting along the $\sigma_y$ direction. The interaction between the two qubits is realized through a fixed capacitive coupling term $J \sigma_z \sigma_z$ with $J=48$MHz. The form $\sigma_z \sigma_z$ is caused by $R_y(\pi/2)$ between $\sigma_x$ and $\sigma_z$. (c). Main plot: Optimization results for the $\sqrt{i\mathrm{SWAP}}$ gate using the GRAPE algorithm. The main plot shows the gate fidelity versus iteration number during closed-system optimization. Subfigure: The pulse waveforms $f_L(t)$ and $f_R(t)$ in time-domain, each represented using 500 discretized time steps and constrained to a maximum of 31 frequency components. The total evolution time is 28 ns, with pulse amplitudes limited to the range $[-100, 100]$ MHz $\times\,2\pi$.
• Figure S1: Pareto fronts obtained using four different multi-objective selection strategies (tDEA, SPEA2, ENS-MOEA/D, and HypE) under Fourier truncation length $n=4$. Each PF represents the trade-off between $T_1$ and $T_\phi$ optimized via the periodic function $P(t)$. The positions of three representative dynamical sweet spots (DSS 1–3), also listed in Table \ref{['Params_table']}, are marked in each subfigure. All optimization runs were performed for 2000 iterations.
• Figure S2: Comparison of Pareto fronts obtained under different Fourier truncation lengths $n = 1$ to $5$ for the periodic modulation function $P(t)$. All four selection strategies were applied for each $n$, with a maximum of 2000 optimization iterations. As $n$ increases, the PFs become broader and more expressive, and higher values of $T_1$ are observed. However, the case $n = 5$ exhibits lower maximum $T_\phi$ than the $n = 4$ case (main text Fig. 1).

Theorems & Definitions (1)

• Theorem 1