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Elements of Quantum Computation and Quantum Communication by Anirban Pathak

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Chapter 4
Quantum gates and
quantum circuits
Except for the NOT gate, all the other familiar classical gates are irre-
versible in the sense that we cannot uniquely reconstruct the input states
from the output states. For example, OR, AND, NAND, NOR,etc. are
irreversible. They all map a two-bit input state into a single bit output
state. Thus one bit is erased during operation of each of these gates, and
according to Landauer’s principle that requires dissipation of a minimum
amount of energy. Now a simple question arises in our curious mind: Is
it possible to avoid this loss of energy? The answer is yes. If we don’t
erase any bit then we can circumvent this energy loss. Thus we need to
map n bit input states to n bit output states. In addition, if a one-to-
one correspondence exists between the input states and the output states
then only we will be able to uniquely reconstruct the input states from
the output states. In such a case the gate is called reversible. For ex-
ample, if we have f (00) = 00,f(01) = 10,f(10) = 01,f(11) = 11 then f
represents a reversible gate. Quantum evolution operators are unitary so
for every operator U we have an inverse operator U
1
= U
. Therefore,
quantum evolution operators are the natural choice for the construction of
energy efficient reversible gates. However, it is not the only choice. We can
have classical reversible gates, too. Usually by reversible gates we refer to
classical reversible gates, and quantum gates are specifically referred to as
quantum gates. Reversible gates and quantum gates are similar but there
exists a fundamental difference that reversible gates cannot accept super-
position states (e.g. α|0 + β|1) as input states, whereas quantum gates
can. Thus all quantum gates are essentially reversible but the converse is
not true.
In brief, simple unitary operations on qubits are called quantum logic
gates. If a gate acts on a single qubit then it is called a single qubit gate.
137
138 Quantum gates and quantum circuits
Similarly, we can define a two qubit gate, three qubit gate and so on. We
know that gates are combined together to form circuits, so in this chapter
we will first describe single qubit gates, two qubit gates and three qubit
gates. Then we will provide a few examples of quantum circuits and briefly
describe the quantitative measures of the quality of the quantum circuits.
We will also describe a few simple tricks that are usually used to improve
the quality of quantum circuits. This chapter is focused on quantum gates
and quantum circuits, but the techniques described here are also valid for
reversible gates and reversible circuits.
4.1 Single qubit gates
A general structure of single qubit gates is shown in Fig. 4.1. Here the
single qubit gate is a unitary operator which transforms a single qubit state
|ψ
in
to another single qubit state |ψ
out
= U|ψ
in
. In this figure and in all
the subsequent figures that depict quantum gates and quantum circuits,
time moves from left to right, and each horizontal line represents a qubit.
The horizontal lines are often referred to as qubit lines. Single qubit gates
are represented by 2 × 2 unitary matrices. Every unitary operator (U),
whichisrepresentedbya2 × 2 matrix, is a valid single qubit gate. In
principle, we can construct an infinite number of 2 × 2 unitary matrices.
Consequently, there are an infinite number of possible single qubit quantum
gates. However, in the conventional classical circuit theory, only two single
bit logic gates are possible, namely the Identity gate and the logical NOT
gate. Among this infinite number of possible single qubit quantum gates,
some have special importance as they are used most frequently, and as
they can be used as elements of a set of gates, which form a universal gate
library. In this section we will briefly introduce these important and useful
single qubit quantum gates, which are nothing but single qubit quantum
state transformations. Since these transformations are linear, they are
completely specified by their effect on the basis vectors. For instance, if we
know that a single qubit quantum gate A maps |0 to |ψ
0
and |1 to |ψ
1
then linearity implies that the gate maps an arbitrary single qubit state
α|0 + β|1 to α|ψ
0
+ β|ψ
1
. Keeping this in mind, we will now describe
the effect of important single qubit gates on the basis vectors |0 and |1 and
will also provide the corresponding 2 × 2 unitary matrices that represent
the gates.
U
|Ψ
in
|Ψ
out
=U|Ψ
in
Figure 4.1: An arbitrary single qubit gate U .

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