The Phase of the “1” - Brain Theory from a Circuits and Systems Perspective - page 165

[a 1 , a 2 ] 0 ¼ η [1, 1] 0 ; this is represented by

Imagine a qubit that is prepared to be η

a Prepared List in the table; in this case, a 1 ¼ 1. A function is applied in

such a way so as to tag the prepared list with negative signs at those inputs for which
the function is true. That is, the 1s in the prepared list are converted to

1, a 2 ¼

1s in those
rows where the truth table for g 2 has a 1. In this case the top element is tagged (refer
to the Tagged List). 1
Processing is now applied to see the effects of the function on the original
prepared qubit which had a 1 ¼ 1. Variables in the Assumed List Structure

are equated to the Tagged List and solved to determine that a 1 ¼1, a 2 ¼ 1, the
solution of which is trivial for a single qubit but more involved for many qubits. The
qubit is assumed to be transformed to a 1 ¼

1, a 2 ¼

1. The overall result of the
above multivibrator function is to change the phase of the simulated qubit from

1, a 2 ¼

η [1,

1] 0 1, 1] 0 . This phase change does not affect the probabilities of what is

sampled and read out; true will occur 50 % of the time, and false 50 % of the time.
It is desired to use an h-transformation. Upon h-transformation, the result

to η [

1] 0 occurs in the multivibrator. When this is sampled it is observed to
be a true. Note that the minus sign in [0

1 ¼ [0

1] 0 does not affect the truth value. Either

+ 1 or 1 internally will give a true after sampling.

Using the above procedure, it is soon discovered that any nonconstant function
will result in the observation of a true, and that any constant function will result in
the observation of a false.
The above demonstration, although elementary, leads to an interesting, if subtle,
consequence for simulated qubits that hold false and true simultaneously. If an
unknown multivibrator function has been applied to a simulated qubit, there is an
easy way to identify that unknown binary function. As explained above, it is not
permitted to place an oscilloscope probe directly the multivibrator in some sort of
direct observation.
To help identify the function, probability processing as above is used. After
applying the h-transformation, the multivibrator vector may end up being a ¼

[1 0] 0 . This means that the sampling readout will be false with certainty, so it may be
concluded that the multivibrator function is constant, either g(a i ) ¼ 0org(a i ) ¼ 1
for i ¼ 1, 2 (a 1 ¼ 0, a 2 ¼ 1). But if after h-transformation the result is a ¼[0 1] 0

then the sampled readout is true with certainty, and it may be concluded that the
function is nonconstant, either g(a i ) a i 0 .

Constancy or non-constancy is a global property, which can be determined, as
above, by probability transformation and then observing a to be 0 or 1 . Normally,
function classification for constancy or non-constancy usually requires evaluations
of the function at least two times, once for an argument of 0 and once for an
argument of 1. But using multivibrator functions, only one observation is necessary
to make this determination.

¼ a i or g(a i ) ¼

1 Note that a function is implemented by operations that change the sign of selected terms in the
Prepared List to become the Tagged List.