We are now ready to see how computing with qubits can be done. In this
book, we will mainly use the circuit model for computation which was first
introduced by Deutsch [25]. We will represent by quantum “wires,” the qubits
upon which manipulations. The length of the wire is to be interpreted as the
time axis. Manipulations on qubits can be done using basic unitary operators
that are the equivalents of logic “gates.” An algorithm, or a complete set
of steps for achieving a processing task, is a combination of wires and gates
representing a quantum circuit. This circuit must be thought of as a time
sequence of events with every wire a way of representing qubit states, and
with gates representing processing of those states.
|i
n
i
G
1
|o
n
i
|i
n−1
i
•
G
2
|o
n−1
i
.
.
.
G
3
.
.
.
|i
1
i
G
4
|o
1
i
FIGURE 7.1: Illustrating a quantum circuit with n qubits.
This notation is based on one by Richard Feynman, with the convention
that time flows from left to right.
Sometimes, an n-qubit state is represented by a wire with a /
n
decoration
on it, that is referred to as a register.
|ii
n
/
n
Circuit
/
n
|oi
n
In practice the circuit is effectively a unitary operator acting on the input
qubits. A few major differences between classical circuits and quantum ones
are:
• Quantum circuits never contain loops or feedbacks: they are acyclic
• Quantum wires are never fanned out: since arbitrary quantum states
cannot be cloned
121
122 Introduction to Quantum Physics and Information Processing
• Though the action of a circuit can be analyzed using classical states, the
effect on superpositions is what gives it true quantum power.
The fact that quantum evolution is unitary results in quantum gates (and
circuits) being reversible. This means that any manipulation of quantum infor-
mation can be undone, unless an irreversible process such as measurement or
decoherence happens on the system. This, and the peculiar features of qubits
discussed in Chapter 4, makes for startling differences in the way we must
think about quantum algorithms.
Mathematically a gate can be represented as a matrix. Classical reversible
gates can have only ones and zeros as elements: reversibility implies that they
can only perform a permutation of the inputs. For example, a reversible XOR
gate is given by y
0
output in the truth table of what has sometimes been called
a Feynman gate:
x y x
0
y
0
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
=⇒
00 01 10 11
00 1 0 0 0
01 0 1 0 0
10 0 0 0 1
11 0 0 1 0
(7.1)
Such a gate is also implementable as a quantum gate, but the most generic
quantum gate is represented by a complex matrix
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