Quantum parameter estimation with optimal control

A pivotal task in quantum metrology, and quantum parameter estimation in general, is to design schemes that achieve the highest precision with the given resources. Standard models of quantum metrology usually assume that the dynamics is fixed and that the highest precision is achieved by preparing the optimal probe states and performing optimal measurements. However, in many practical experimental settings, additional controls are usually available to alter the dynamics. Here we propose to use optimal control methods for further improvement of the precision limit of quantum parameter estimation. We show that, by exploring the additional degree of freedom offered by the controls, a higher-precision limit can be achieved. In particular, we show that the precision limit under the controlled schemes can go beyond the constraints put by the coherent time, which is in contrast with the standard scheme where the precision limit is always bounded by the coherent time.

Figure 1

(a) Parallel scheme, (b) sequential scheme, (c) controlled sequential scheme.

Figure 2

Flow chart of the algorithm. The key for this algorithm is to update the controls based on the gradient value.

Figure 3

The QFI (normalized by $T^2$) as a function of $\theta$. The dash-dotted green line, purple squares and circled red dots represent the QFI with and without controls, respectively. The dashed black line represents the value of the QFI for unitary evolution. The target time $T=5$, and decay rate $\gamma =0.1$. The states in the bracket in the legend represent the corresponding initial state. The true value of ${\omega }_0$ is assumed to be 1.

Figure 4

Transverse dephasing: (a) The evolution of QFI (normalized by $T$) with and without controls. The red dots and black lines represent the QFI with and without controls, respectively (the wiggles in the black line is not numerical error but caused by some trigonometric functions in the QFI in this case). The dashed blue line is the analytical solution for the QFI (and CFI) under controls. The dash-dotted green line is the CFI without controls. Decay rate $\gamma =0.1$, and the measurement for CFI is $\left\{\right|+\rangle \langle +|,|-\rangle \langle -\left|\right\}$. (b) Optimal controls obtained from GRAPE. (c) The enhanced QFI and CFI (normalized by $T^2$) as a function of ${\widehat{\omega }}_0$. The solid red and dash-dotted blue lines represent the QFI and CFI under controls, respectively. The target time $T=20$ and $\gamma =0.2$. The dashed black line represents the value of QFI without controls. The true values of ${\omega }_0$ in all panels are assumed to be 1.

Figure 5

Parallel dephasing: (a) The evolution of QFI with and without controls. The red dots and solid black lines represent the QFI with and without controls, respectively. The solid yellow line represents the maximum QFI with a single $\pi /2$ pulse along the $y$ axis at a proper time. The green stars and dash-dotted green lines represent the CFI with and without controls, respectively. The measurement for CFI is $\left\{\right|+\rangle \langle +|,|-\rangle \langle -\left|\right\}$. (b) QFI as a function of $t_0$. The solid blue and triangular red lines represent the QFI with rotation at $t_0$, for $T=15$ and $T=5$, respectively. The dotted black and dashed green lines represent the QFIs without rotation.

Figure 6

Parallel dephasing: (a) The evolution of normalized QFI (by $T$) with and without controls. The red dots and solid black line represent the normalized QFI with and without controls, respectively. The solid yellow line represents the maximum normalized QFI with single $\pi /2$ pulse along the $y$ axis at a proper time. The green stars and dash-dotted green lines represent the normalized CFI with and without controls, respectively. The measurement for CFI is $\left\{\right|+\rangle \langle +|,|-\rangle \langle -\left|\right\}$. (b) The controls obtained from GRAPE for the dynamics with $T=20$. The initial guessing is random. (c) The normalized QFI (by $T^2$) for different ${\omega }_0$. The controls are obtained from the GRAPE for ${\omega }_0=1$. It can be seen that the QFI of the controlled scheme is higher than the QFI of the uncontrolled scheme as long as $|{\omega }_0-1|$ is not too big, i.e., as long as the estimated value is reasonably good. The true values of ${\omega }_0$ are assumed to be 1 and the decay rate $\gamma =0.1$ in all panels.

Figure 7

Spontaneous emission: (a) Normalized QFI (by $T$) as a function of $T$. The red dots and solid black lines represent the QFI with and without controls, respectively. And the dashed blue and dash-dotted green lines represent the CFI with and without controls, respectively. Here ${\gamma }_{+}=0, {\gamma }_{-}=0.1$. The true value of ${\omega }_0$ is 1. The measurement for CFI is $\left\{\right|+\rangle \langle +|,|-\rangle \langle -\left|\right\}$. (b) The controls obtained from GRAPE for the dynamics with $T=20$. The initial guessing is all zero. (c) The QFI and CFI (normalized by $T^2$) as a function of ${\omega }_0$. The solid blue and dash-dotted green lines represent the QFI and CFI under the controls given by GRAPE with ${\widehat{\omega }}_0=1$, respectively. The dashed black line is the QFI without controls. (d) QFI as a function of $t_0$. The solid blue and red lines represent the QFI with rotation at $t_0$, for $T=16$ and $T=8$, respectively. The dashed and dash-dotted black lines represent the QFIs without rotation.

Figure 8

Schematic for the effects of parallel dephasing and spontaneous emission on the Bloch sphere. The blue spheres and green spheres represent the initial and evolved state space in the noise.

Figure 9

The energy cost $E\left(t\right)$ (in the scale of ${\omega }_0=1$) for performing the controls in the case of transverse dephasing (solid black line), parallel dephasing (dashed blue line), and spontaneous emission (dash-dotted red line) with the target time $T=10$. The decay rates in all cases are 0.1.