Error Correction vs. Mitigation in Noisy Quantum Systems

Error-correcting codes were originally developed to protect both classical computation and communication from noise. Quantum error correction (QEC) adapts these ideas to the quantum setting, where the no-cloning theorem and continuous-variable error models raise qualitatively new difficulties and underlie schemes for fault-tolerant quantum computation. In contrast, quantum error mitigation (QEM) offers practical techniques to reduce noise effects in quantum devices without the significant hardware overhead required by QEC. 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.