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Elements of Quantum Computation and Quantum Communication by Anirban Pathak

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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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