Quantum information utilizes the properties of quantum mechanics in realizing applications of information transmission, processes, and storage. To construct any functional quantum computer, one must account for the problem of uncertainty and noise when interacting with the environment. In particular, information transmission will be distorted to various extents before reaching their destinated counterparts thanks to the physical realities of quantum mechanics and information science. Thus, the restoration of quantum information, i.e. quantum error correction, becomes one of the biggest struggles and a pressing issue in the theoretical realization of quantum computing models.
In this talk, we will introduce mathematical formulations of quantum mechanics with an emphasis on quantum information-related constructions. We will then motivate quantum error correction from its classical counterpart. The emphasis of the talk is to introduce existing fundamental theorems and results in the field, illustrating key representations of quantum channels and citing Knill-Laflamme subspace conditions and Shor’s nine-qubit code. We will also discuss a generalized version of the subspace condition to operator systems and the construction of noiseless subspaces.
Students with some linear algebra background should be able to follow the talk.