  The Heisenberg limit (HL) and the standard quantum limit (SQL) are two
quantum metrological limits, which describe the scalings of estimation
precision $\Delta \hat\theta$ of an unknown parameter $\theta$ with respect to
$n$, the number of one-parameter quantum channels applied. It was known that
the HL ($\Delta \hat\theta \propto 1/n$) is achievable using quantum error
correction (QEC) strategies when the ``Hamiltonian-not-in-Kraus-span'' (HNKS)
condition is satisfied; and when HNKS is violated, the SQL ($\Delta \hat\theta
\propto 1/n^{1/2}$) is optimal and can be achieved with $n$ repeated
measurements. However, it is unknown whether such limits are still achievable
using restricted quantum devices where the required QEC operations are not
available -- e.g., finite-size devices where only unitary controls are
available or where noiseless ancilla is not available. In this work, we
identify various new noisy metrological limits for estimating one-parameter
qubit channels in different settings with restricted controls. The HL is proven
to be unattainable in these cases, indicating the necessity of QEC in achieving
the HL. Furthermore, we find a necessary and sufficient condition for qubit
channels to attain the SQL, called the
``rotation-generators-not-in-Kraus-span'' (RGNKS) condition. When RGNKS is
satisfied, the SQL is achievable using only unitary controls and a single
measurement. When RGNKS is violated, the estimation precision (in most cases)
has a constant floor when repeated measurements are not allowed. Demonstration
of this separation in metrological powers is within reach of current quantum
technologies.
