Adaptive Hybrid Optimal Quantum Control for Imprecisely Characterized Systems

Optimal quantum control theory carries a huge promise for quantum technology. Its experimental application, however, is often hindered by imprecise knowledge of the input variables, the quantum system’s parameters. We show how to overcome this by adaptive hybrid optimal control, using a protocol named Ad-HOC. This protocol combines open- and closed-loop optimal control by first performing a gradient search towards a near-optimal control pulse and then an experimental fidelity estimation with a gradient-free method. For typical settings in solid-state quantum information processing, adaptive hybrid optimal control enhances gate fidelities by an order of magnitude, making optimal control theory applicable and useful.

Figure 1

(a) Sketch of quantum control experiment. The unit generating the control pulses, typically an arbitrary waveform generator (AWG), at room temperature generates the control pulses that are sent through the control transfer chain (sketched as the chain of cylinders) to finally reach the quantum system, often cooled to less than a Kelvin. Error sources are in the parameters modeling the “chip,” the electronics, and the calibration of the control signals. (b) Rapid degradation of a 99.99% fidelity CZ gate between two qubits coupled via a bus assuming only an error on the $g_1$.

Figure 2

Ad-HOC protocol: The physical system and surrounding control and measurement apparatus are designed taking control problems into consideration. For instance the AWG has to have a sufficient bandwidth for the desired control task. The system is then characterized with the best possible precision. Using the resulting parameters the control pulses are created. These are then fine tuned to the system using closed-loop OCT. The pulses are then ready to be used in the experiment and can be recalibrated at a later time to account for drift.

Figure 3

Convergence during optimization of random gates. 100 pulses were optimized each for a different realization of the random TLS. The target fidelity of $1–10^{-5}$ is reached rapidly as indicated by the median. The shaded area includes 68% of all runs centered around the median.

Figure 4

Debugging procedure for parameter errors, control transfer chain errors and control dc offset errors. The error of the initial pulse was minimized using a gradient search down to machine precision. The pulses are then calibrated to a specific realization of the system. (a) Histograms for 300 system realizations. The red histograms show the fidelity of the initial uncalibrated numerical pulse. The blue histograms show the improvement in average gate fidelity after running Ad-HOC. (b) Gate errors as function of the calibration algorithms iteration number. (c) Histogram of the number evaluations of $\overline{\mathcal{F}}$ needed to calibrate the pulse, i.e., to take the red histograms to the blue ones.

Figure 5

(a) Convergence speed of a single optimization comparing the cases when a depolarizing channel adds noise and when the optimization is noiseless. (b) Difference in fidelity of the worst point in the simplex between subsequent iterations in a noisy optimization. As long as, on average, this difference is greater than the noise level, the optimization continues.