Summary

Over the course of this chapter, we have seen the rudiments of classical error correction, where introducing redundant information is key to recovering from errors, provided they occur with a low-enough probability. We introduced simple bit-repetition models that corrected bit-flip errors, thanks to syndrome computation and measurement and their generalization – linear codes.

Then, we dived into quantum error computation and saw that even though the no-cloning theorem prevents us from directly duplicating a state, it is still possible to encode a qubit by entangling it with other qubits, thus introducing redundancy without breaking any quantum law. Moreover, using ancillary qubits, it is possible to compute a syndrome and measure it without interfering with the quantum information that we wish to transmit.

After introducing two simple error correction codes for correcting either a bit-flip or a phase-flip error, we looked at Shor code, which can, by combining the...