Optimal control for quantum metrology via Pontryagin's principle

Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.

Figure 1

Optimal controls using 8 (a), 16 (b), 32 (c), and 64 (d) controls (time intervals) with $(N,\chi )=(20,4)$. Increasing the number of controls reduces the magnitudes of the switching function and results in a flatter negative c-Hamiltonian.

Figure 2

Optimal controls (blue, left axis), the overlap $|\langle {\psi }_{\text{HL}}|\psi \left(t\right)\rangle |$ (red dashed, right axis), and QFI/${\left(Nt\right)}^2$ (red solid, right axis) for ($N, \chi$) = (20, 0.1) (a) and (20,4) (b). In (a), $\mathrm{\Omega }\left(t\right)$ is nonzero during the whole evolution, and the overlap to $|{\psi }_{\text{HL}}\rangle$ increases to $\lesssim 0.1$ around $t=0.96$. In contrast in (b), the optimal control $\mathrm{\Omega }\left(t\right)$ vanishes after $t\gtrsim 0.18$ around which $|\langle {\mathrm{\Psi }}_{\text{HL}}|\psi \left(t\right)\rangle |$ is approaching one. 64 controls are used in these simulations. (c) The dimensionless controls [Eq. (10)] for $(N,\chi )=(20,2)$, (20,4), (30,1), (30,2), (40,1), and (50,1): they almost collapse to a single curve. (d) The dimensionless controls using $|{\mathrm{\Psi }}_{\text{HL}}\rangle$ as the target state for $(N,\chi ,T)=(20,2,1/4)$, (20,4,1/8), (30,1,1/3), (30,2,1/6), (40,1,0.26), and (50,1,0.21). All overlaps $|\langle {\mathrm{\Psi }}_{\text{HL}}|\psi \left(T\right)\rangle |$ are larger than 0.985.

Figure 3

Optimal control of $N=100, \chi =0.1$ using 100 time intervals. (a) No constraint on the control. (b)–(d) Optimal controls with $\mathrm{\Omega }\left(t\right)<u_{\text{max}}$: (b) $u_{\text{max}}=6$; (c) $u_{\text{max}}=4$; (d) $u_{\text{max}}=2$. When the optimal control takes one of the extreme values, $\mathrm{\Phi }\left(t\right)$ has an opposite sign.

Figure 4

(a) Optimal control for $N=4, \chi =1$ using 64 controls to maximize CFI. The optimal CFI is about 8.19 with $\varphi =\pi /2$. (b) Optimal control for $N=100, \chi =0.1$ using 64 controls to maximize CFI. The optimal CFI is about 2867.5 with $\varphi =0$.