Tunable, Flexible, and Efficient Optimization of Control Pulses for Practical Qubits

Quantum computation places very stringent demands on gate fidelities, and experimental implementations require both the controls and the resultant dynamics to conform to hardware-specific constraints. Superconducting qubits present the additional requirement that pulses must have simple parameterizations, so they can be further calibrated in the experiment, to compensate for uncertainties in system parameters. Other quantum technologies, such as sensing, require extremely high fidelities. We present a novel, conceptually simple and easy-to-implement gradient-based optimal control technique named gradient optimization of analytic controls (GOAT), which satisfies all the above requirements, unlike previous approaches. To demonstrate GOAT’s capabilities, with emphasis on flexibility and ease of subsequent calibration, we optimize fast coherence-limited pulses for two leading superconducting qubits architectures—flux-tunable transmons and fixed-frequency transmons with tunable couplers.

Figure 1

Optimized flux modulation $\delta (t)$ generating an iswap in the flux-tunable coupler of Eqs. (8)–(10). The infidelity is ${10}^{-12}$ in a fully coherent model and a noise-limited 99.5% with $T_1=T_2=40\mu \mathrm{s}$ and thermalization to 25 mK [55]. (a) Unconstrained Fourier pulse, sum of six frequency components. (b) Constraints applied using sigmoids, in both time (pulse starts and ends smoothly at zero) and amplitude (limited to $0.3{\mathrm{\Phi }}_0$). (c) Output of waveform generator, including carrier and dc bias. See Appendix C of the Supplemental Material [35] for further details.

Figure 2

Simulation of 300 GOAT pulse calibrations, assuming inaccurate characterization of system parameters. Fractional error in the coupling $g$ and anharmonicity $\eta$ is taken to be normal distributed, with a standard deviation of 3%. (a) Initial (dotted lines) and final (solid lines) drive shape for a two tunable-qubits cZ gate parametrized with error functions using only 16 parameters. Red and blue lines depict the two qubit controls, $a_1(t)$ and $a_2(t)$ of Eq. (13). (b) Distribution of the number of fidelity measurements needed to calibrate the pulse using the quasi-Newton algorithm. The low pulse parameter count implies less than 1500 fidelity measurements are needed in most simulated systems. (c) Distribution of precalibration (blue) and postcalibration (red) infidelity. Pulses have been calibrated up to the coherence limit imposed by Markovian processes, ${10}^{-3}$.