Quantum error correction

Suppose for simplicity that our system consists of a single qubit. We start with errors we are familiar with from the classical setting – bit-flips. Such an error converts the original state, say α|0〉+β |1〉, into α|1〉+β |0〉. We can correct these kind of errors using classical error correcting codes. For example, we may use a repetition code, i.e., encoding |0〉 as |000〉 and |1〉 as |111〉. Thus, we encode α|0〉+ β |1〉 as α|000〉+ β |111〉. Now suppose the second bit gets flipped. Then the new state becomes α|010〉+β |101〉. We assume that when any bit gets flipped, it is flipped in all superpositions. In order to locate the error, we attach an “error locating register” initially set to zero. This register is supposed to store the location of the bit in error. The overall state prior to any computation is thus (α|010〉+β |101〉)⊗|0〉.