We consider a quantum sensor network of qubit sensors coupled to a field f (x; theta) analytically parameterized by the vector of parameters theta. The qubit sensors are fixed at positions x(1), ..., x(d). While the functional form of f (x; theta) is known, the parameters theta are not. We derive saturable bounds on the precision of measuring an arbitrary analytic function q(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.