  Error-correcting codes were invented to correct errors on noisy communication
channels. Quantum error correction (QEC), however, may have a wider range of
uses, including information transmission, quantum simulation/computation, and
fault-tolerance. These invite us to rethink QEC, in particular, about the role
that quantum physics plays in terms of encoding and decoding. The fact that
many quantum algorithms, especially near-term hybrid quantum-classical
algorithms, only use limited types of local measurements on quantum states,
leads to various new techniques called Quantum Error Mitigation (QEM). This
work examines the task of QEM from several perspectives. Using some intuitions
built upon classical and quantum communication scenarios, we clarify some
fundamental distinctions between QEC and QEM. We then discuss the implications
of noise invertibility for QEM, and give an explicit construction called
Drazin-inverse for non-invertible noise, which is trace preserving while the
commonly-used Moore-Penrose pseudoinverse may not be. Finally, we study the
consequences of having an imperfect knowledge about the noise, and derive
conditions when noise can be reduced using QEM.
