Chapter 9 - Quantum Error Correcting Codes

Introduction

If one were to encode the information contained in some set of quantum states in such a way as to enable the information to be stored redundantly, then it would be possible to use the redundancy to detect and correct errors. Quantum error correcting codes aim to encode quantum information into states in just such a redundant fashion. It is worth noting that classical error correcting codes and coding theory has been around a long time. However, quantum error correction requires extra care when measuring to detect and correct errors. In this chapter, simple examples of quantum error correcting codes are given which correct errors in quantum computation. A later chapter will discuss more general concerns.

Bit-flip Errors: A Classical Code

A classical bit-flip error would turn a 0 into a 1 and a 1 into a 0. A classical error correcting code which protects against such bit-flip errors is the following code. Rather than use the state 0, the state is encoded redundantly; the state 000 is used. This is called an encoded zero state or a logical zero state. Likewise, 111 is used as an encoded 1, or logical 1. Now suppose one bit is flipped when the encoded state 111 is sent, and let this be the first bit. Then having sent 111 and finding 011, one would suppose that it was likely the case that the first bit was flipped. If one of the bits is flipped, the encoded state could be fixed by flipping the one that is different so that it agrees with the others. Let us assume that each error is independent and has probability $p\,\!$. The probability that the bit is not flipped is then $1-p\,\!$. Since the probability is $p\,\!$ that one bit flip occurs, then the probability that two are flipped is $3(1-p)p^{2}\,\!$ (assuming we don't know which one). The probability that three are flipped is $p^{3}\,\!$. So the code will help us if $p>3(1-p)p^{2}+p^{3}\,\!$ which happens when $p<1/2\,\!$.

This example will now be used to find a simple bit-flip code for a quantum system.