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|
In the early days of quantum information theory, Landauer  and others
were convinced by this logic and it was believed that quantum error cor-
rection would not be possible . Later on clever techniques of quantum
error correction were introduced , but to understand them we need to
understand: (a) What is an error model and (b) How are classical errors
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
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