  We investigate different quantum parameter estimation scenarios in the
presence of noise, and identify optimal probe states. For frequency estimation
of local Hamiltonians with dephasing noise, we determine optimal probe states
for up to 70 qubits, and determine their key properties. We find that the
so-called one-axis twisted spin-squeezed states are only almost optimal, and
that optimal states need not to be spin-squeezed. For different kinds of noise
models, we investigate whether optimal states in the noiseless case remain
superior to product states also in the presence of noise. For certain spatially
and temporally correlated noise, we find that product states no longer allow
one to reach the standard quantum limit in precision, while certain entangled
states do. Our conclusions are based on numerical evidence using efficient
numerical algorithms which we developed in order to treat permutational
invariant systems.
