Chapter 6

Quantum error correction

In Chapter 4 we introduced diﬀerent quantum gates and quantum circuits

and in Chapter 5 we described several quantum algorithms. So far the

analysis of the algorithms and the circuits has been made with the as-

sumption that the required unitary operations can be implemented exactly

without any error. This is an abstraction. In an actual physical system

that implements a quantum algorithm or a quantum circuit there are sev-

eral sources of errors. The situation is similar in classical computation

and to circumvent this problem error correcting codes are used in classical

digital computing. Since qubits are more delicate and more susceptible

to errors we need to introduce quantum error correcting codes. It is very

important, as without such codes it will not be possible to build a scalable

quantum computer. Such quantum error correcting codes are introduced

in this chapter. However, we will not limit ourselves to the discussion of

error correcting codes alone. We will also discuss more general aspects

of errors and error correction, like decoherence, decoherence free subspace

and fault-tolerance.

6.1 Quantum error correction

In Chapter 2 we brieﬂy discussed the possibilities of appearance of errors

in a classical channel. We also noted the diﬃculties associated with the

error correction in analog computing. However, we have not discussed the

following important issues:

(a) How can classical errors be corrected?

(b) What are the diﬀerences between classical and quantum errors?

(c) Can we use classical error correcting schemes to correct quantum

errors?

(d) If not, then how to correct quantum errors?

In the following sections we will discuss these issues. But before we

207

208 Quantum error correction

start the technical discussion we would prefer to recall the answer to a

more fundamental question: Why don’t we use analog classical computers?

The answer is obvious: the possibility of inﬁnitely many states makes it

impossible to correct errors. Apparently, the same logic is applicable to

quantum states. This is because the qubit is a two level quantum system,

which can have continuum of possible states speciﬁed by

|ψ = α|0+ β|1,

where α and β are arbitrary complex numbers which satisfy |α|

2

+|β|

2

=1.

In the early days of quantum information theory, Landauer [83] and others

were convinced by this logic and it was believed that quantum error cor-

rection would not be possible [84]. Later on clever techniques of quantum

error correction were introduced [24], but to understand them we need to

understand: (a) What is an error model and (b) How are classical errors

corrected?

6.2 Basic idea of an error model

When we wish to protect our information against possible errors we ﬁrst

need to deﬁne the particular form of error from which we wish to protect our

information. Each type of error leads to an error model. We will further

clarify this point soon. We do not want a bit/qubit to change its value

when it is either stored or it is moving from one place to another place.

But because of error model the set of bits/qubits get changed (evolved).

A particular error model describes a certain type of evolution and is often

referred to as a channel. For example, we are interested in identity channel

which implies an error free channel. The simplest example of classical error

model is a bit ﬂip channel. In this simplest model the state of a bit ﬂips

with probability p and remains unchanged with probability (1 −p).Asthe

probability of bit ﬂip is the same for both 0 and 1, this channel is often

referred to as a symmetric bit ﬂip channel. Such a channel is shown in Fig.

6.1. We can now easily add some more complexities to this simple error

model and generate new error models. The following examples elaborate

how to obtain new error models by modifying the symmetric bit ﬂip channel

described above.

Example 6.1: Consider that the probability of bit ﬂip for 0 is p and that

for 1 is q,wherep = q. This asymmetric bit ﬂip channel provides us with

a new and a little more complex error model.

Example 6.2: Consider that the bit ﬂipping does not occur independently

in diﬀerent bits. That is, bit ﬂip errors are correlated. This provides

another example of an error model, which is a bit more general in nature.

The above examples of classical error models are provided to clear the

concept of an error model. Once the error model is known then the task at

hand is to protect the information from that particular kind of error. This

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