  Quantum information science provides powerful technologies beyond the scope
of classical physics. In practice, accurate control of quantum operations is a
challenging task with current quantum devices. The implementation of high
fidelity and multi-qubit quantum operations consumes massive resources and
requires complicated hardware design to fight against noise. An approach to
alleviating this problem is to replace quantum operations with classical
processing. Despite the common practice of this approach, rigorous criteria to
determine whether a given quantum operation is replaceable classically are
still missing. In this work, we define the classically replaceable operations
in four general scenarios. In each scenario, we provide their necessary and
sufficient criteria and point out the corresponding classical processing. For a
practically favorable case of unitary classically replaceable operations, we
show that the replaced classical processing is deterministic. Beyond that, we
regard the irreplaceability of quantum operations by classical processing as a
quantum resource and relate it to the performance of a channel in a non-local
game, as manifested in a robustness measure.
