Arithmetic Using Simulated Qubits - Brain Theory from a Circuits and Systems Perspective - page 115

Fig. 7.5 Symbol for double
controlled toggle or DCT

Fig. 7.6 Reversible modulo-
two addition of i + j

The idea of controlled toggles can be extended to any number of controlling
lines, creating what may be termed a DCT. The symbol for DCT appears in Fig. 7.5 .
The double controlled NOT computes the AND of i and j; if this combination is
true, then k R ¼ NOT(k L ). The i and j are transmitted without change so that i R ¼ i L

and j R ¼ j L , to ensure reversibility. The DCT is also known as the Toffoli gate [ 2 ].

It is a special case of a MCT which depends on the AND of multiple controls. MCT
are useful for detecting when a group of inputs all equal one.

Simple Applications

As a simple application, imagine many registers as in the above figure, one of which
contains a hidden code. As a simple example, q 2 ¼ 1 could be hidden. The

idea is, this code points to secret information. Registers can be analyzed concur-
rently using a DCN. First initialize q 0 !

1, q 1 ¼

0 in each register. Then apply the DCN to
qubits q 1 , q 2 where the code is hidden. That word that contains the code q 2 ¼

1,

q 1 ¼ 1, q 0 ¼ 0 on the left of the wiring diagram will convert to q 2 ¼ 1, q 1 ¼ 1,

q 0 ¼ 1 on the right of the wiring diagram. In other words, the flag in the form of

q 0 ¼ 1 is raised to identify the hidden code. Thus the secret information the hidden

code points to has been located and may be observed.
On a much larger scale the application of MCN suggests a solution to a needle in
the haystack problem.
As an example of logically reversible adding, consider modulo-two addition of
1 + 1 as in Fig. 7.6 . Modulo-two addition means that the sum must show either 0 or
1, but not 2. Therefore 1 + 1 ¼ 0, with the implication that there is a carry of 1 and a

sum equal to 10, or binary 2.
This circuit computes the modulo-two sum, not the carry. Assume three toggle
devices in states i, j, k. Assume that k on the left is initialized to k L ¼ 0. The

modulo-two sum is computed in k R on the right. For example, if a word had on the
left i L ¼

0 , the correct
answer. The modulo-two sum is an exclusive OR. Using wiring diagrams with more
complexity, reversible addition may be extended any number of bits.
It is important to note that data in the above registers does not actually move out
of the register, as would be necessary in microprocessors. This processing is

1, j L ¼ 1, and k L ¼ 0, then on the right is k R ¼ i j ¼