24 - Quantum error correction

Summary

This chapter deals with the subject of quantum error correction and the related codes (QECC), which can be applied to noisy quantum channels and quantum memories with the purpose of preserving or protecting the information integrity. I first describe the basics of quantum repetition codes, as applicable to bit-flip and phase-flip quantum channels. Then I consider the 9-qubit Shor code, which has the capability of diagnosing and correcting any combination of bit-flip and phase-flip errors, up to one error of each type. Furthermore, it is shown that the Shor code is, in fact, capable of fully restoring qubit integrity under a continuum of bit or phase errors, a property that has no counterpart in the classical world of error-correction codes. But the exploration of QECC does not stop here! We shall discover the elegant Calderbank–Shor–Steane (CSS) codes, which have the capability of correcting any number of errors t, both bit-flip and phase-flip. As an application of the CSS code, I then describe the 7-qubit Hadamard–Steane code, which can correct up to one error on single qubits. A corresponding quantum circuit, based on an original generator-matrix example, is presented.

Quantum repetition code

In Chapter 11, we saw that the simplest form of error-correction code (ECC) is the repetition code, based on the principle of majority logic. The background assumption is that in a given message sequence, or bit string, the probability of a bit error is sufficiently small for the majority of bits to be correctly transmitted through the channel.