Chopped random-basis quantum optimization

In this work, we describe in detail the chopped random basis (CRAB) optimal control technique recently introduced to optimize time-dependent density matrix renormalization group simulations [P. Doria, T. Calarco, and S. Montangero, Phys. Rev. Lett. 106, 190501 (2011)]. Here, we study the efficiency of this control technique in optimizing different quantum processes and we show that in the considered cases we obtain results equivalent to those obtained via different optimal control methods while using less resources. We propose the CRAB optimization as a general and versatile optimal control technique.

Figure 1

Infidelity $f_1$ of the final state as a function of number of calls to the optimization algorithm $N_f$ for two capacitively Josephson charge qubits with principal harmonics [dark grey (blue) line] and randomized frequencies [light grey (green) line], for the goal state $|{\psi }_G^1\rangle$ and $N_c=2$ for 30 different random instances.

Figure 2

Optimized infidelity $f$ as a function of the number of optimization parameters $N_c$ with principal harmonics [dark grey (blue), full symbols] and randomized frequencies [light grey (green), empty symbols] for different target states $|{\psi }_G^1\rangle$ (circle), $|{\psi }_G^2\rangle$ (squares), $|{\psi }_G^3\rangle$ (diamonds).

Figure 3

Infidelity as a function of the size in the LMG model. Squares represent the data before the optimization; circles the data after the optimization with CRAB.

Figure 4

Infidelity as a function of the number of control parameters for different sizes in the LMG model. The total evolution time is $T=2T_{\mathrm{QSL}}=2\pi /\Delta$. Inset: infidelity as a function of the number of parameters for a single size $N=32$: comparison between data optimized using as cost function the infidelity (empty circles) and the final energy (full circles). Green squares represent the results with randomized frequencies.

Figure 5

Infidelity as a function of the size in the transfer state problem. Squares represent the data before the optimization; circles the data after the optimization with CRAB.

Figure 6

Infidelity as a function of the number of control parameters for different sizes in the in the state transfer model. The total evolution time $T=2T_{\mathrm{QSL}}$. Inset: infidelity as a function of the number of parameters divided by the size.

Figure 7

Final entanglement entropy as a function of the total evolution time $T$ for $N=10,32,\mathrm{and}100$ of one of two equal bipartitions of the system. The time is measured in units of $J^{-1}$.

Figure 8

Entanglement entropy saturation value as a function of the size (red circles) and the function $A{\log }_2(N/2+1)$ (dashed line). A fit gives $A=0.947$.

Figure 9

Instantaneous excitation probabilities $P_i$ of the $i$-th excited level ($P_1$, continuous; $P_2$, dotted; $P_3$, dot-dashed; $P_4$, dot-dot-dashed; $P_{25}$, dash-dash-dotted line) and total excitation probability $P_{\mathrm{tot}}={\sum }_{i=1}^N{P}_i$ (dashed purple line) in the LMG model with $N=50$ for an evolution induced with a driving field linear in time, $\Gamma \left(t\right)\propto -t/T$, $T/T_{\mathrm{QSL}}=2$, $N=50$. The thick (red) dashed line signals the crossing of the critical point. The time is measured in units of $J^{-1}$.

Figure 10

Optimal instantaneous excitation probabilities $P_i$ in the LMG model for $T/T_{\mathrm{QSL}}=2$, $N=50$. Codes are the same as described in the legend of Fig. 9. The time is measured in units of $J^{-1}$.