  Twist-untwist protocols for quantum metrology consist of a serial application
of: 1. unitary nonlinear dynamics (e.g., spin squeezing or Kerr nonlinearity),
2. parameterized dynamics $U(\phi)$ (e.g., a collective rotation or phase space
displacement), 3. time reversed application of step 1. Such protocols are known
to produce states that allow Heisenberg scaling for experimentally accessible
estimators of $\phi$ even when the nonlinearities are applied for times much
shorter than required to produce Schr\"{o}dinger cat states. In this work, we
prove that twist-untwist protocols provide the lowest estimation error among
quantum metrology protocols that utilize two calls to a weakly nonlinear
evolution and a readout involving only measurement of a spin operator
$\vec{n}\cdot \vec{J}$, asymptotically in the number of particles. We consider
the following physical settings: all-to-all interactions generated by one-axis
twisting $J_{z}^{2}$ (e.g., interacting Bose gases), constant finite range
spin-spin interactions of distinguishable or bosonic atoms (e.g., trapped ions
or Rydberg atoms, or lattice bosons). In these settings, we further show that
the optimal twist-untwist protocols asymptotically achieve $85\%$ and $92\%$ of
the respective quantum Cram\'{e}r-Rao bounds. We show that the error of a
twist-untwist protocol can be decreased by a factor of $L$ without an increase
in the noise of the spin measurement if the twist-untwist protocol can be
noiselessly iterated as an $L$ layer quantum alternating operator ansatz.
