How does Steane code use the classical Hamming code for error correction?

Question

I know what the Hamming code is. But I don't understand how could this idea be applied in quantum computation since it's required to cover the case of superposition and entanglement. How could you decide parity in that case? And how is the measurement performed in this coding so that we get the information on where is the error while being able to not change the codeword? Is it possible to know where is the error while not changing the state that was sent? Basically, how is it performed?

Is there a place to read about it which explains it in a concise way and explains it well?

Answer 1

Seems like you are interested in the theory of applying results from classical linear coding theory to quantum information theory. When you want to understand how to generalize the Hamming code to a quantum code - the so-called Steane code - you should definitely get familiar to the "stabilizer formalism" introduced by Daniel Gottesman in his Ph.D. thesis which you can find here.

An easier to read and understand book about this topic is "Quantum Computation and Quantum Information" by Michael Nielsen and Isaac Chuang. In particular, in chapter 10 they explain how to construct the Steane code from the Hamming code. I highly recommend this book when you are only getting started with this topic.