  We consider a quantum sensor network of qubit sensors coupled to a field
$f(\vec{x};\vec{\theta})$ analytically parameterized by the vector of
parameters $\vec\theta$. The qubit sensors are fixed at positions
$\vec{x}_1,\dots,\vec{x}_d$. While the functional form of
$f(\vec{x};\vec{\theta})$ is known, the parameters $\vec{\theta}$ are not. We
derive saturable bounds on the precision of measuring an arbitrary analytic
function $q(\vec{\theta})$ of these parameters and construct the optimal
protocols that achieve these bounds. Our results are obtained from a
combination of techniques from quantum information theory and duality theorems
for linear programming. They can be applied to many problems, including optimal
placement of quantum sensors, field interpolation, and the measurement of
functionals of parametrized fields.
