  Nonlinear interactions are recognized as potential resources for quantum
metrology, facilitating parameter estimation precisions that scale as the
exponential Heisenberg limit of $2^{-N}$. We explore such nonlinearity and
propose an associated quantum measurement scenario based on the nonlinear
interaction of $N$-probe entanglement generating form. This scenario provides
an enhanced precision scaling of $D^{-N} /(N-1)!$ with $D > 2$ a tunable
parameter. In addition, it can be readily implemented in a variety of
experimental platforms and applied to measurements of a wide range of
quantities, including local gravitational acceleration $g$, magnetic field, and
its higher-order gradients.
