Assessing three closed-loop learning algorithms by searching for high-quality quantum control pulses

Designing a high-quality control is crucial for reliable quantum computation. Among the existing approaches, closed-loop leaning control is an effective choice. Its efficiency depends on the learning algorithm employed, thus deserving algorithmic comparisons for its practical applications. Here we assess three representative learning algorithms, including GRadient Ascent Pulse Engineering (GRAPE), improved Nelder-Mead (NMplus), and Differential Evolution (DE), by searching for high-quality control pulses to prepare the Bell state. We first implement each algorithm experimentally in a nuclear magnetic resonance system and then conduct a numerical study considering the impact of some possible significant experimental uncertainties. The experiments report the successful preparation of the high-fidelity target state by the three algorithms, while NMplus converges fastest, and these results coincide with the numerical simulations when potential uncertainties are negligible. However, under certain significant uncertainties, these algorithms possess distinct performance with respect to their resulting precision and efficiency, and DE shows the best robustness. This study provides insight to aid in the practical application of different closed-loop learning algorithms in realistic physical scenarios.

Figure 1

Schematic framework of closed-loop learning control, which entails the following steps: (1) Either randomly chosen or carefully designed initial controls ${\mathit{u}}^{\left(0\right)}$ are applied to the quantum system. (2) The quantum system will evolve under a specific Hamiltonian $H\left({\mathit{u}}^{\left(k\right)}\right)$ on the $k\mathrm{th}$ cycle around the loop. A suitable measurement is performed to specify the performance function $\tilde{f}\left({\mathit{u}}^{\left(k\right)}\right)$. (3) The performance function obtained from the measurement is fed back to the learning algorithm to design new controls through specific updating laws. (4) These procedures are iteratively performed until the performance stopping criterion is met.

Figure 2

Experimental closed-loop learning results for preparing the Bell state. The control pulses are all divided into $M=10$ slices, the total time period is $T=5$ ms, and the total number of the control parameters is $p=40$. The initial controls are all randomly chosen in the range $[-50,50]\mathrm{Hz}$. The maximum iteration number is set as 15 for GRAPE, 300 for NMplus, and 75 for DE, respectively. The algorithm parameters described in the Appendix are set as follows: (i) GRAPE. ${\lambda }^{\left(0\right)}=2\times {10}^4,{\lambda }^{(l+1)}=0.5{\lambda }^{\left(l\right)},l\le 40$; (ii) NMplus. $\alpha =3,\beta =1/3,\gamma =2,\delta =1/3$; (iii) DE. $R=0.6,C_r=0.95,P_n=10$. (a) The entire experimental learning process, and the subplot enlarges the detailed performance approaching convergence. (b) Plots of the initial and final searched control fields $\mathbf{u}=(u_x^1\left[1\right],...,u_x^1\left[10\right],u_y^1\left[1\right],...,u_y^1\left[10\right],u_x^2\left[1\right],...,u_x^2\left[10\right],u_y^2\left[1\right],...,u_y^2\left[10\right])$ in the time sliced forms. (c) The function evaluations when the measured fidelity (partial state tomography) reaches around 0.65, 0.85, and 0.99, respectively. The experimental efficiency factor, defined by the ratio of function evaluations of each algorithm over function evaluations of NMplus, is also given. The final state fidelity with uncertainties (determined by applying the final controls to test the fidelity of the prepared Bell state ten times) obtained by partial and full state tomography are also listed. (d) Corresponding full state tomography results of the searched final state.

Figure 3

Numerical search results with 500 runs of the three algorithms seeking optimal pulses to prepare the Bell state. (a) State infidelity ($1-F$) vs function evaluations. The thick lines are the statistical average of the data over 500 runs in each case. (b), (c) Function evaluations (abbreviation Feval) utilized when searching for the pulses to prepare the Bell state in the presence of pulse imperfections and measurement errors, respectively. Each algorithm stops when the state infidelity reaches 0.001 within ${10}^5$ function evaluations. Thus we exclude the failed runs and give the corresponding success rates (the rate of runs reaching infidelity 0.001 over 500 runs, abbreviation Sr) at the bottom. The red squares show the mean function evaluations, and the blue bars are the corresponding variance.