Chapter 14 Quantum error correction

About more classical error correction and the Hamming codes, how they generalise to quantum codes called CSS codes, including the Steane [[7,1,3]]-code. Also about computing in the presence of errors: logical states, logical operators, and error families — all via the formalism of stabilisers — and the transversal gates that we can implement to act on them.

We have seen a way of dealing with the computational errors introduced by the physical problem of decoherence, namely the Shor [[9,1,3]] code, but this is just the start of the story. There is a vast body of work on classical error correction, so it’s sensible to ask if we can adapt this to help us in the world of quantum computation. As we shall see, we can actually use quite a lot of the theory of classical error-correction codes, and in doing so we will start to really make use of the stabiliser formalism introduced all the way back in Chapter 7. But note that this still isn’t the end of the story: our goal is so-called fault-tolerant computation, which we come to in Chapter 15.