Optimal Control at the Quantum Speed Limit

Optimal control theory is a promising candidate for a drastic improvement of the performance of quantum information tasks. We explore its ultimate limit in paradigmatic cases, and demonstrate that it coincides with the maximum speed limit allowed by quantum evolution.

Figure 1

Infidelity $\mathcal{I}$ versus number of iterations $n$ of the Krotov algorithm [2] for different values of $T$ (in units of $\hslash /\omega$) for $\Gamma (T)/\omega =500$. The dashed line is the estimated QSL ($T_{\mathrm{QSL}}=1.5688$) while the dot-dashed line is an exponential fit. Inset: Second derivative of the infidelity logarithm with respect to the logarithm of $n$.

Figure 2

Comparison between the time-independent estimate (dashed line) and the second derivative criterion (circles, for $-\Gamma (0)/\omega =5$, 10, 20, 100, 500, 5000 from right to left) for $T_{\mathrm{QSL}}$ for the LZ model.

Figure 3

Left: Infidelity as a function of the number of iterations $n$ for a spin chain of length $N=101$ and different durations $T$ (in units of $\hslash /J$). Right: Minimum infidelity $\mathcal{I}$ for chain length $N$ and the (rescaled) transfer time $(T-b)/N$ [$b\sim 1.53$ is the $y$-axis intercept of the function $T_{\mathrm{QSL}}^{*}(N)$ reported in the inset], showing the expected linear behavior of $T_{\mathrm{QSL}}$ with the size $N$. The green line follows from the estimate $T_{\mathrm{QSL}}^{*}$ of the QSL time obtained by choosing the time $T(N)$ at which the infidelity reaches the value ${\mathcal{I}}^{*}=5\times {10}^{-5}$ for $n={10}^5$.

Figure 4

Left: $\Delta E_2(t)$ for a spin chain of length $N=101$, and different total times $T=68.97$ (green solid line) and $T=56.66\approx T_{\mathrm{QSL}}^{*}$ (blue dashed line) (the times have been rescaled to make them fit on the same axis). The central portion of the plot has been enlarged in the inset. Right: $\frac{\pi (N-1)}{2\Delta {\mathcal{E}}_1}$ (orange solid line), $\frac{\pi (N-1)}{2\Delta {\mathcal{E}}_2}$ (red circles), and $T_{\mathrm{QSL}}^{*}$ (green squares) versus chain length $N$.