  The quantum Fisher information (QFI), as a function of quantum states,
measures the amount of information that a quantum state carries about an
unknown parameter. The (entanglement-assisted) QFI of a quantum channel is
defined to be the maximum QFI of the output state assuming an entangled input
state over a single probe and an ancilla. In quantum metrology, people are
interested in calculating the QFI of $N$ identical copies of a quantum channel
when $N \rightarrow \infty$, which is called the asymptotic QFI. Over the
years, researchers found various types of upper bounds of the asymptotic QFI,
but they were proven achievable only in several specific situations. It was
known that the asymptotic QFI of an arbitrary quantum channel grows either
linearly or quadratically with $N$. Here we show that a simple criterion can
determine whether the scaling is linear or quadratic. In both cases, the
asymptotic QFI and a quantum error correction protocol to achieve it are
computable via a semidefinite program. When the scaling is quadratic, the
Heisenberg limit, a feature of noiseless quantum channels, is recovered. When
the scaling is linear, we show the asymptotic QFI is still in general larger
than $N$ times the single-channel QFI and furthermore, sequential estimation
strategies provide no advantage over parallel ones.
