NISQ: Error Correction, Mitigation, and Noise Simulation.

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 addresses the differences and connections between QEC and QEM, by examining different application scenarios. We demonstrate that QEM protocols, which aim to recover the output density matrix, from a quantum circuit do not always preserve important quantum resources, such as entanglement with another party. We then discuss the implications of noise invertibility on the task of error mitigation, and give an explicit construction called quasi-inverse for non-invertible noise, which is trace-preserving while the Moore-Penrose pseudoinverse may not be. We also study the consequences of erroneously characterizing the noise channels, and derive conditions when a QEM protocol can reduce the noise.