Page 85 - Principles and Applications of NanoMEMS Physics

72 Chapter 2

state and could, accordingly, represent propagation via a wire, in time, in
space, or in any other fashion evolution may be intended to take place. The
gate has control qubits and target qubits. A control qubit, such as x , has its
line of propagation (wire) tapped at a dot. A target qubit, such as y , has its
line of propagation (wire) XOR’ed with a control bit. The gate’s purpose is
to effect a transformation on the target qubit based on the values of the
control qubit, in particular, if the control qubit is set to one, then the target
qubit is inverted. The realization of classical logic gates, which are
inherently irreversible, by totally reversible quantum gates may be effected
with the use of the Toffoli gate, see Figure 2-12(b). The Toffoli gate is an
irreversible gate that takes three inputs, namely, two control qubits and one
target qubit. By applying the Toffoli gate twice to its three input qubits, they
are repoduced, thus the irreversible gate is made reversible [111].

x x ' x' x
x = =' ' x x x
x x
y y ' y' y
y = y=' ' y y ⊕ ⊕ x x
y y z z z z ⊕ ⊕ xy xy
x x x y y y x’ x’ x’ y’ y’ y’
x x x y y y z z z x’ x’ x’ y’ y’ y’ z’ z’ z’
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1
1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 0
0 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 0 0 0
1 1 1 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
1 1 1 0 0 0 1 1 1 1 1 1 0 0 0 1 1 1
1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(a) (b)
x x x = =' ' x y y

y y y = =' ' y x x

x x x y y y x’ x’ x’ y’ y’ y’
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 1 1 1 1 1 1 0 0 0
1 1 1 0 0 0 0 0 0 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1
(c)
Figure 2-12. Truth tables and graphical representations of some quantum gates. (a) Control-
NOT gate. (b) Control-control-NOT (Toffoli) gate. (c) Bit swapping.