What is the code distance in quantum information theory?

Question

What is the code distance in quantum information theory? Code distance seems to be a very important concept in fault tolerant quantum computation and topological quantum computation.

Answer 1

Rather than give a more mathematical answer, to which I'll refer you to another answer, let me give you slightly less precise explanation. Basically, the distance is the shortest path in a certain "space of errors" which maps between two orthogonal quantum states that are in the code. The natural space of errors is that of single qubit errors of the form $\sigma_X$, $\sigma_Y$ or $\sigma_z$, in the case where the Hilbert space is that of $n$ qubits. So you can think of distance as the shortest path to get from one state to another by operations on single qubits, applied one at a time sequentially.