Dressing the chopped-random-basis optimization: A bandwidth-limited access to the trap-free landscape

In quantum optimal control theory the success of an optimization algorithm is highly influenced by how the figure of merit to be optimized behaves as a function of the control field, i.e., by the control landscape. Constraints on the control field introduce local minima in the landscape—false traps—which might prevent an efficient solution of the optimal control problem. Rabitz et al. [Science 303, 1998 (2004)] showed that local minima occur only rarely for unconstrained optimization. Here, we extend this result to the case of bandwidth-limited control pulses showing that in this case one can eliminate the false traps arising from the constraint. Based on this theoretical understanding, we modify the chopped-random-basis (CRAB) optimal control algorithm and show that this development exploits the advantages of both (unconstrained) gradient algorithms and of truncated basis methods, allowing one to always follow the gradient of the unconstrained landscape by bandwidth-limited control functions. We study the effects of additional constraints and show that for reasonable constraints the convergence properties are still maintained. Finally, we numerically show that this approach saturates the theoretical bound on the minimal bandwidth of the control needed to optimally drive the system.

Figure 1

Schematic view on the control landscape $J\left(f\right)=F(\left|\psi \right(T)\rangle )$. While $F$ has a strict global maximum point at $|\zeta \rangle$ (left panel), this state can be reached by different control functions $f_1$ and $f_2$ corresponding to multiple global maximum points in the landscape $J$ (right panel).

Figure 2

Success probability $p_C$ as a function of the number $N_C$ of basis functions in the CRAB expansion. As we increase the size of the function space false traps are removed. The symbols are the mean values over 10 different random pairs of final and initial states $|\xi \rangle$ and $|\zeta \rangle$ with 10 different starting points and frequencies each. The error bars show the standard deviation over the different final and initial states. The total time was set to $T=6\pi ,10\pi ,16\pi$ and the allowed bandwidth to $T{\omega }_{\max }/\left(2\pi \right)=8,20,40$ for $N=2,3,4$ qubits, respectively. An optimization was counted as success when the residual error $\varepsilon$ was smaller than $\eta ={10}^{-3}$. The black lines indicate the empirical “$2\times 2^N-2$” rule for the required number of coefficients $N_C$.

Figure 3

Success probability $p_d$ as a function of the number $N_C$ of basis functions in a single call of CRAB within the DCRAB superiterations. Independently of $N_C$ no false traps are encountered and $p_d=1$. The total time, allowed bandwidth and error threshold are as in Fig. 2.

Figure 4

Number of function evaluations $n_f$ for DCRAB (red) and original CRAB (black) as a function of the number $N_C$ of basis functions involved in a single call of the respective algorithm. The error bars show the logarithmic standard deviation. Optimization was stopped when the error $\varepsilon$ crossed the threshold $\eta ={10}^{-3}$. The total time was $T=16\pi$, the allowed bandwidth $T{\omega }_{\max }/\left(2\pi \right)=40$ and the system size $N=4$ qubits. Similar results are obtained for other choices of the parameters.

Figure 5

Infidelity $\varepsilon =1-F$ as a function of the limited bandwidth ${\omega }_{\max }$ for $|\xi \rangle =|0000\rangle$ and $|\zeta \rangle =|1111\rangle$ in the system of $N=4$ qubits and total time $T=16\pi$ with fixed random coefficients ${\alpha }_i, {\beta }_i$ in the Hamiltonian [Eq. (24)]. The gray diamonds show the infidelity $\varepsilon$ for CRAB ($N_C=40$), while the green circles show it for DCRAB ($N_C=max\{2{\omega }_{\max }T/2\pi ,40\}$). The black dashed line indicates the error threshold of $\eta ={10}^{-3}$. The error bars report the logarithmic standard deviation.

Figure 6

Infidelity $\varepsilon =1-F$ as a function of the limited pulse height $f_{\max }$ for the transfer from $|\xi \rangle =|00\rangle$ to $|\zeta \rangle =|11\rangle$ in the two-qubit system with constrained DCRAB. The black line indicates the error threshold of $\eta ={10}^{-3}$. The orange triangles are obtained with Lagrange multipliers [see Eq. (25)], while the brown circles are obtained with a cutoff at $f_{\max }$ [see Eq. (26)]. The error bars indicate the standard deviation over 10 different starting points of the optimization. The orange triangles have also error bars in $f_{\max }$ since the Lagrange multiplier is not a hard wall and instead we plot the maximum absolute value the optimal pulse takes in each realization.