Optimal Control of Families of Quantum Gates

Quantum optimal control (QOC) enables the realization of accurate operations, such as quantum gates, and supports the development of quantum technologies. To date, many QOC frameworks have been developed, but those remain only naturally suited to optimize a single targeted operation at a time. We extend this concept to optimal control with a continuous family of targets, and demonstrate that an optimization based on neural networks can find families of time-dependent Hamiltonians realizing desired classes of quantum gates in minimal time.

Figure 1

Optimal control of a continuous family of target gates $U_{\alpha }^{\mathrm{tgt}}$ indexed by the target parameter $\alpha$ which can be either a scalar or a vector. The time-dependent controls $f_{\alpha }(t)$ which now also depend on $\alpha$ are modeled by a neural network (NN). This NN effectively parametrizes a continuous family of controlled gates $U_{\alpha }$ where each point corresponds to the propagator obtained by evolving the system in time (dashed arrows) under the controls produced by the NN (illustrated for two sets of target parameters ${\alpha }^{(0)}$ and ${\alpha }^{(1)}$). Training of the framework consists in optimizing the weights ${\varphi }_{\mathrm{NN}}$ of the NN such that the deviation $\mathcal{I}$ between the controlled and target families of operations is minimized.

Figure 2

Control of the family of arbitrary 1-qubit rotations $U_{\alpha }^1$ defined in Eq. (3) with the Hamiltonian in Eq. (1). The framework is successfully trained to implement any of the target rotations, resulting in an average infidelity of $\overline{\mathcal{I}}=2\times {10}^{-4}$ (assessed on new targets not seen during training). To visualize the controls produced by the NN, ${\alpha }_1$ and ${\alpha }_2$ are kept fixed to a value of $3\pi /4$ and ${\alpha }_3$ is varied in the range $[0,\pi ]$. The amplitudes of the two control fields $f_{\alpha }^{1y}$ (a) and $f_{\alpha }^{1z}$ (b) produced by the NN are plotted as a function of both the time $t$ and ${\alpha }_3/\pi$.

Figure 3

Infidelities and times for the family $U_{\alpha }^1$ of rotations when both the (target-dependent) control functions and times are learned concurrently. For the sake of visualization, results are plotted for a two-dimensional subset of the three-dimensional family of targets, with fixed parameter ${\alpha }_3=3\pi /4$, and discretized ${\alpha }_1,{\alpha }_2\in [0,\pi ]$ over a grid of $75\times 75$ regularly spaced points. For each of the corresponding target gates a heat map indicates the values of the infidelities ${\mathcal{I}}_{\alpha }$ achieved [panel (a), in logarithmic scale] and the control times $T_{\alpha }$ entailed by the framework [panel (b)].