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Fig. 4.15
“Wiring” diagram
example
is usually understood from the context of a
discussion of state vectors and not written. So
a
The direct product symbol
j
>
j
b
>
means
a
j
>
j
b
>
.
Subject to the normalization conditions,
j
ψ >
may be filled with fractions of
differing values such that: 0
1. Note again that two qubits
resulted in four states. Generally n qubits have 2
n
states.
a
1
, a
2
, b
1
, b
2
Classical Simulated Qubits Versus Quantum Qubits
The electron, for instance, can exist partly with spin up and partly with spin down.
In contrast, an intermediate multivibrator frequency is not a combination of low
frequency f
0
and high frequency f
1
. The probability mix is only “simulated.” This is
a major difference.
Other lesser differences are: There is only one phase variable in the above version of
a simulated qubit whereas a physical qubit has two phase variables, one for
0
>
j
and
one for
1
>
. The above simulated qubit has only one phase
variable. Clever design can bring in another phase variable for simulated qubits, as in
another chapter, but it is unnecessary for most neural situations.
If the waveforms from multiple multivibrators are synchronized (all start at the
same moment), and are harmonically related, probabilities take on discrete values,
unlike quantum probabilities, since the available frequencies are assumed to go up
as integers. Thus, if f
0
¼
j
. That is, for
α
and
β
1 Hz and f
x
is an integer up to a few hundred, probability
from the above equations takes on discrete values.
Another difference is that in classical simulation entanglement is impossible.
Entanglement
An important feature of physical qubits is that they can be entangled. Consider two
qubits
a
in a quantummechanical system. The entangle-
ment process is diagrammed in Fig.
4.15
. The following theoretical procedure may be
applied. Step (1) Qubit
a
j
>
,
b
j
>
initialized to
0
j
>
0
j
>
maybepreparedtobe
a
>¼ η½
11
,
η ¼
1/
2. Step (2) A
√
controlled NOTmay be applied such that if
a
j
>
is
0
j
>
, nothing happens; but if
a
j
>
is
j
1
>
, then
b
j
>
is flipped (inverted) from
0
j
>
to
1
j
>
, for that fraction of
a
j
>
that is
j
1
>
. The result will be two entangled qubits with an equation of the form
0
ψ >¼ η
0
>
0
> þ
1
>
j
1
>
(4.13)
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