The Phase of the “1” - Brain Theory from a Circuits and Systems Perspective - page 169

x 1 0 x 2 Table 10.8 Analysis for g d ¼

Assumed list
structure

Mod1 permutated
tagged list

Mod2 1st entry
sign reversal Tagged list

1 a 1 b 1 1 1

1 a 1 b 2 1 1

1 a 2 b 1 1 1

1 a 2 b 2 1 1

code satisfies the function g as , and that there is only one negatively tagged entry in
the Tagged List. Now, to demonstrate the method, it must be assumed that what
satisfies the multivibrator function is unknown.
Simulated qubits, n in number, can be prepared to provide all 2 n combinations of
true and false. It is assumed that a multivibrator function will tag a given combina-
tion with a minus sign where the function is satisfied, as suggested in the table. The
location of the tag cannot be determined by direct sampling because negative signs
do not affect the probability of a given output. Electrical probing of the waveforms
is assumed to be impossible.
In an attempt to discover which a i b j term has the negative tag, a transformation is
going to be applied prior to sampling and readout. The transformation is assumed
available for a multivibrator function. First the Tagged List is permutated using
an H matrix of H matrices, or H 2 giving the result Mod1 shown in Table 10.8 .
Note that

1
2

1
2

11
1

HH
H

H 2

H ¼ p ¼ p :

;

1 H

This list is modified again with a transformation that reverses only the phase of
the first entry a 1 b 1 as shown in the right of the table under Mod2.
There are now an even number of negative signs in the list under Mod2 in the
table. Having an even number of minus signs is the key; consequently a result can
be identified.
Using the above solving procedure, a transformation is defined by solving the
four equations in the table, a 1 b 1 ¼

1, a 1 b 2 ¼ 1, a 2 b 1 ¼ 1, a 2 b 2 ¼ 1, to give

[1 1] 0 and b 11] 0 . The probabilities can be h-transformed as above to a ¼ ¼ [

show ab 01 . The minus sign will not be observed after sampling, so the output

will be 01 which happens to look like a count of binary 1. Counting from zero,
position 1 is where the original 1 locates in the original Table for this function. So
what satisfies the function has been found.
This method of transformation was inspired by Grover's quantum algorithm [ 4 ].
It works for more than two simulated qubits, but for n

¼

1 iteration is necessary,
about 2 n/2 iterations. To summarize, (1) apply the function to tag a prepared
list, which in reality is filled with various fractions to achieve probability normali-
zation; (2) apply H n to the tagged list in order to reorganize it; (3) reverse the phase
of the first term in the list; and (4) solve for an updated result ab . This last step is
equivalent to applying H n again.

>