The Phase of the “1” - Brain Theory from a Circuits and Systems Perspective - page 164

Table 10.1 The four
functions of a single binary
variable

x i g 0 g 1 g 2 g 3

0 0 0 1 1

1 0 1 0 1

Table 10.2 Procedure to
identify constant and
nonconstant functions

x i g 2 Prepared list Tagged list Assumed list structure

011 1 ¼a 1

101 1 ¼a 2

2 2 2

b 1 ¼ð 1 η Þ p þðη Þð q Þ¼ 1 2 η ¼ 0 : (10.5)

2 2

b 2 ¼ð 1 η Þ p ðη Þð q Þ¼ 1 : (10.6)

2

[0 1] 0 which after sampling is 1 or true,
determinedwith certaintywith one sampling (or measurement). So a systemcan know
that the original vector is

1] 0 transforms to 1 Since η ¼ 0.5, η [1 ¼

1) 0 without having to probe and perhaps disturb the
multivibrator. This transform is reversible. For example, [0 1] 0 may be transformed
to be [pq] 0 where p

η (1

¼ 0; q ¼ 1/ η , returning the vector to b ¼ η [1 1] 0 .

Binary Function Classification Involving One Simulated Qubit

Assuming an ordinary binary function of a single bit, possibilities are, the function
is constant: either g(x i )

¼ 0org(x i ) ¼ 1 for i ¼ 1, 2; (note that x 1 ¼ 0, x 2 ¼ 1)

x i 0 , the complement of x i .
This information is presented in a truth table, Table 10.1 .
Two functions, g 0 and g 3 , are constant functions as x i varies; the other two, g 1
and g 2 , are nonconstant.
Recall that simulated qubits have at their core a set of multivibrator waveforms
operating at a certain frequency and phase. A multivibrator function is a sequence
of rotations, polar and azimuthally, of the probability vector that introduces nega-
tive signs into the presampled output possibilities. For example, two simulated
qubits [a 1 a 2 ] 0 ,[b 1 b 2 ] 0 may be prepared to be [1 1] 0 ,[11] 0 meaning that each has
50 % probability of being true and 50 % probability of being false. Overall there are
finite probabilities for the following combinations a 1 b 1 , a 1 b 2 , a 2 b 1 , a 2 b 2 .
A multivibrator function is defined to be one that places a negative sign on one
more of these terms.
Generalizing, for any number of simulated qubits, the locations of the negative
signs are congruent to the 1s of a particular Boolean function. Negative signs are
invisible in the readout method, so with simple sampling fails to identify the
multivibrator function should it be unknown. A special procedure is required to
identify or at least classify the function. For example, refer to the inverter function,
g 2 , in Table 10.2 .

otherwise it is nonconstant: either g(x i ) ¼ x i or g(x i ) ¼