Two Shaky Pillars of Quantum Computing

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Journal of Applied Mathematics and Physics, 2023, 11, 448-456
https://www.scirp.org/journal/jamp
ISSN Online: 2327-4379
ISSN Print: 2327-4352
DOI:
10.4236/jamp.2023.112027 Feb. 13, 2023 448
Journal of Applied Mathematics and Physics
Two Shaky Pillars of Quantum Computing
Alexander Soiguine
SOiGUINE Quantum Computing, Aliso Viejo, USA
Abstract
Superposition and entanglement are two theoretical pillars quantum compu-
ting rests upon. In the g-
qubit theory quantum wave functions are identified
by points on the surface of three-dimensional sphere
3

. That gives differ-
ent, more physically feasible explanation of what superposition and entan-
glement are. The core of quantum computing scheme should be in manipula-
tion and transferring of wave functions on
3

as operators acting on obser-
vables and formulated in terms of geometrical algebra. In this way quantum
computer will be a kind of analog computer keeping and processing informa-
tion by sets of objects possessing infinite number of degrees of freedom, con-
trary to the two value bits or two-dimensional Hilbert space elements, qubits.
Keywords
Geometric Algebra, Wave Functions, Observables, Measurements
1. Introduction: Conventional Entanglement
Complementarity principle in physics says that a complete knowledge of phe-
nomena on atomic dimensions requires a description of both wave and particle
properties. The principle was announced in 1928 by the Danish physicist Niels
Bohr. His statement was that depending on the experimental arrangement, the
behavior of such phenomena as light and electrons is sometimes wavelike and
sometimes particle-like and that it is impossible to observe both the wave and
particle aspects simultaneously.
In the following it will be shown that actual weirdness of all conventional
quantum mechanics comes from logical inconsistence of what is meant in basic
quantum mechanical definitions and has nothing to do with the phenomena
scale and the attached artificial complementarity principle [1] [2] [3] [4].
It will be explained below that theory should speak not about complementari-
ty but about proper separation of measurement process arrangement into oper-
How to cite this paper:
Soiguine, A.
(20
23) Two Shaky Pillars of Quan
tum
Computing
.
Journal of Applied Mathema
t-
ics and Phy
sics
,
11
, 448-456.
https://doi.org/10.4236/jamp.2023.112027
Received:
January 4, 2023
Accepted:
February 10, 2023
Published:
February 13, 2023
Copyright © 20
23 by author(s) and
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Open Access

A. Soiguine
DOI:
10.4236/jamp.2023.112027 449
Journal of Applied Mathematics and Physics
ator, three-sphere
3

element, acting on observable, and operand, measured
observable.
It will be shown that quantum mechanics is not of something deeper but
should be replaced by something conceptually different.
In the suggested alternative it is said that theory should speak not about com-
plementarity but about proper dividing of the measurement process into opera-
tor, wave function, which is the three-sphere
3

element acting on observable,
and operand, the measured observable.
A vector in quantum mechanics
is the mathematical gadget used to describe
the
state
of a quantum system, its status, what it’s capable of doing. A state as-
signed to elementary particles there is given by a unit vector in a vector space,
really a Hilbert space
n
C
, particularly
2
C
, encoding information about the
state. The dimension
n
is the number of different observable things after making
a measurement on the particle.
The simplest quantum mechanical state, qubit, reads:
1
2
1 2 12
2
10
01
01
z
C z z zz
z
  
=+=+
  
 


It has just two observable “things” after measurement, say “up” for
0
and
“down” for
1
, with probabilities
2
1
z
and
2
2
z
.
In the case of two particles vector space
2
C
is generalized to density matrix
defined on tensor product
22
CC⊗
and in the case of
N
particles we get
22 2
CC C⊗ ⊗⊗
,
N
-fold tensor product.
The appropriateness of tensor products is that the tensor product itself cap-
tures all ways that basic things can “interact” with each other.
2. Wave Functions in the g-Qubit Theory
Wave function will be a unit value element of even subalgebra of three-dimensional
geometric algebra. Such elements will execute twisting of observables. Even sub-
algebra
3
G
+
is subalgebra of elements of the form
3S
MI
αβ
= +
, where
α
and
β
are (real)1 scalars and
S
I
is some unit bivector arbitrary placed in
three-dimensional space.
Wave functions as elements of
3
G
+
are naturally mapped onto unit sphere
3

[5] [6] [7].
If in some bivector basis
{ }
123
,,BBB
, with, for example, right-hand screw
multiplication rules
123
1
BBB =
,
12 3
BB B= −
,
13 2
BB B=
,
23 1
BB B
= −
, the
twisting plane bivector is
12 3
123S
I bB bB bB=++
,
then
12 3
123S
I bB b B b B
α βαβ β β
+=+ + +
{}
123
,, ,
S
I bbb
α β αβ β β
+⇒
1In the current formalism scalars can only be real numbers. “Complex” scalars make no sense an
y-
more.

A. Soiguine
DOI:
10.4236/jamp.2023.112027 450 Journal of Applied
Mathematics and Physics
and
() ( )
() ( )( )
()
() ( )
222
22 22
123 1bbb
αβ αβ
+ + + =+=
,
since wave function is normalized and bivector
S
I
is a unit value one.
Wave function can always be conveniently written as exponent, see [7], Sec. 2.
5,
e
S
I
S
I
ϕ
αβ
+=
,
cos
αϕ
=
,
sin
βϕ
=
The product of two exponents is again an exponent, because generally
12 1 2
gg g g
=
and
12 1 2
ee e e 1
SS S S
II I I
αβ α β
= =
.
Multiplication of an exponent by another exponent is often called
Clifford
translation.
Using the term
translation
follows from the fact that Clifford trans-
lation does not change distances between the exponents it acts upon if we iden-
tify exponents as points on unit sphere
3

:
{}
1 12 2 3 3
123
cos sin cos sin sin sin
cos , sin , sin , sin
S
I b Bb B b B
bb b
α αα α α α
αααα
+=+ + +
⇔
( ) ( ) ( )
( )
2
22 2
123
cos sin sin sin 1bbb
αααα
++ + =
This result follows again from
12 1 2
gg g g=
:
( )
12 12 12
ee
SS
II
gg gg gg
αα
− = −=−
Clifford translation of a wave function
2
2
e
S
I
ϕ
by
1
1
e
S
I
ϕ
is displacement of
the wave function, point on
3

, along big circle that is intersection of
3

by
1
S
by parameter
1
ϕ
.
3. The Meaning of Schrodinger Equation
Let us take some vector
( ) ()( ) ( )
()
12 3
31 2 3
Ht I tB tB tB
χχ χ
= ++
and execute
infinitesimal Clifford translation of a wave function
( )
( )
eSt
It
ϕ
using bivector
( )
3
IH t−
and Clifford parameter
( )
0
Ht t∆
at some instant of time
0
t
:
( )
( ) ( )
( )
( )
0
30 0
00
ee
St
Ht
I Ht t It
Ht
ϕ
−∆
With denoting
( )
( ) ()
0
30
0
H
Ht
I It
Ht ≡
we get:
( )
( ) () ()
( )
( )
00
00
00
e ee
St t St
H
I tt I t
I t Ht t
ϕϕ
+∆
+∆ −∆
≈
and
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
( ) ( )
( )
( )
00
00
00
00
0
0
0
00
0
00
ee
lim
1 ee
lim
e
St t St
St St
St
I tt I t
t
It It
H
t
It
H
t
I t Ht t
t
I t Ht
ϕϕ
ϕϕ
ϕ
+∆
+∆
∆→
∆→
−
∆
− ∆−
=∆
= −

A. Soiguine
DOI:
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Journal of Applied Mathematics and Physics
That gives the Schrodinger equation:
()
()
( ) ()
()
( )
ee
St St
It It
H
I t Ht
t
ϕϕ
∂
−=
∂
That means that the Schrodinger equation defines infinitesimal changes of
wave functions under Clifford translations along big circles of
3

.
4. Superposition of Two Basic Wave Functions
Corresponding to
0
and
1
The quantum mechanical qubit state,
12
01zz
ψ
= +
, is linear combination
of two basis states
0
and
1
. In more details:
1
32
i
i
αβ
ψββ
+

=
+

There exist infinite number of options to select triple
{ }
123
,,BBB
. Thus, the
procedure of recovering a g-qubit associated with
12
01zz
ψ
= +
is the fol-
lowing one:
It is necessary [6] [7] firstly, to define bivector
1
i
B
in three dimensions iden-
tifying the torsion plane. Secondly, choose another bivector
2
i
B
orthogonal to
1
i
B
. The third bivector
3
i
B
, orthogonal to both
1
i
B
and
1
i
B
, is then defined
by the first two by orientation (handedness, right screw in the used case):
123
33 3 3ii i
IBIB IB I=
.
Wave functions in the suggested theory are operators acting through mea-
surements on observables:
( )
( )
SS
IC I
αβαβ
++
For any wave function
1
ii
B
αβ
+
,
1, 2 , 3i=
, corresponding to
0
(assuming
22
1
i
αβ
+=
) we get:
( ) ( )
( )
1 1 1 11
22
ii i ii i i i
B B B BB
α β α β αβ
− +=+ =
For the wave functions
( ) ( ) ( ) ( )
2 mod 3 2 mod 3 1 mod3 1 mod 3i i ii
BB
ββ
+ + ++
+
,,
1, 2 , 3i=
, cor-
responding to
1
(with the agreement
3 mod 3 3=
) the value of observable
1
i
B
is (with same assumption
( ) ( )
22
2 mod 3 1 mod 3
1
ii
ββ
++
+=
):
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
( ) ( )
( )
1
11
2 mod 3 2 mo d 3 1 mod3 1 mod3 2 mod3 2 mod 3 1 mod3 1 mod 3
22
2 mod 3 1 mod 3
i
i i ii i i ii
ii
ii
B BB B B
BB
ββ ββ
ββ
+ + ++ + + ++
++
−− +
=−+ =−
Let us take an arbitrary bivector observable expanded in basis
{ }
{ }
123
123
,, , ,
ii i
BB B B B B≡
:
11 2 2 3 3
C CB C B C B=++
The result of measurement by wave function corresponding to
0
is:
( ) ( )
( ) ( )
( ) ( )
11 11
22 22
11 2 1 3 1 2 3 1 2 1 3
11 2 3 2 2 3 3
22
cos 2 sin 2 sin 2 cos 2 ,
BC B
CB C C B C C B
CB C C B C C B
αβ αβ
α β αβ α β αβ
ϕϕ ϕ ϕ
−+
  
= + −− + −+
  
=+− ++
(4.1)

A. Soiguine
DOI:
10.4236/jamp.2023.112027 452 Journal of Applied
Mathematics and Physics
using parametrization
cos
αϕ
=
,
1sin
βϕ
=
.
The result of measurement by wave function corresponding to
1
is:
() ( )
()( )
() ( )
22 33 22 33
22 22
11 2 2 3 323 2 223 3 2 3 3
11 2 3 2 2 3 3
22
cos 2 sin 2 sin ,2 cos 2
B BC B B
CB C C B C C B
CB C C B C C B
ββ ββ
β β ββ ββ β β
θθ θθ
−− +
  
=−+ −+ + − −
  
=−+ + + −
(4.2)
with
2
cos
βθ
=
,
3sin
βθ
=
.
This is a deeper result compared with conventional quantum mechanics
where
1
00
i
αβ
+

=

and
32
0
1i
ββ

=
+

Conclusion:
• Measurement of observable
11 2 2 3 3
C CB C B C B=++
by any wave function
corresponding to
0
is bivector with the
1
B
component equal to un-
changed value
1
C
. The
2
B
and
3
B
components of the result of measure-
ment are equal to
2
B
and
3
B
components of
C
rotated by angle
2
ϕ
de-
fined by
cos
αϕ
=
,
1
sin
βϕ
=
where plane of rotation is
1
B
.
• Measurement of observable
11 2 2 3 3
C CB C B C B=++
by any wave function
corresponding to
1
is bivector with the
1
B
component equal to flipped
value
1
C−
. The
2
B
and
3
B
components of the result of measurement are
equal to
2
B
and
3
B
components of
C
rotated by angle 2
θ
defined by
2
cos
βθ
=
,
3
sin
βθ
=
where plane of rotation is
1
B
but direction of rota-
tion is opposite to the case of
0
.
If we denote by
0
so
and
1
so
arbitrary wave functions represented in
3
G
+
by
11
B
αβ
+
and
()
2233 3213
B B BB
β β ββ
+=+
they only differ by factor
3
B
in
1
so
, thus for the measurement by them we have:
33
11 0 0
so Cso B so Cso B=
That simply means that the measurement on the left side is received from
00
so Cso
by its flipping in plane
3
B
.
Probabilities of the results of measurements are measures of wave functions
on
3

surface giving considered results
.
Suppose we are interested in the probability of the result of measurement in
which the observable component
11
CB
does not change. This is relative meas-
ure of wave functions
22 1
11
22 22
11
B
β
α
αβ αβ αβ


++

++

in the measure-
ments:
22 22
11
1 11 1
22 22 22 22
11 11
BC B
ββ
αα
αβ αβ
αβ αβ αβ αβ
  
  
+− ++
  
++ ++
  
(4.3)
That measure is equal to
22
1
αβ
+
, that is equal to
2
1
z
in the down mapping
from
3
G
+
to Hilbert space of
12
01zz+
. Thus, we have clear explanation of

A. Soiguine
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Journal of Applied Mathematics and Physics
common quantum mechanics wisdom on “probability of finding system in state
0
”.
Similar calculations explain correspondence of
22
32
ββ
+
to
2
2
z
in the qubit
12
01
zz+
when the component
11
CB
in measurement just got flipped.
Let us consider superposition of
11
B
αβ
+
and
22 33
BB
ββ
+
with some coef-
ficients
1
p
and
2
p
,
( )
( )
1 11 2 2 2 3 3
p Bp B B
αβ β β
++ +
,
and measuring by it of
11 2 2 3 3
C CB C B C B=++
.
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( ) ( )
( ) ( )
1 11 2 22 3 3 1 11 2 2 2 33
1 11 1 11 2 2 2 3 3 2 2 2 3 3
2 2 2 33 1 11 1 11 2 2 2 3 3
1 11 1 11 2 2 2 3 3 2 2 2 3 3
1 11 1 11 1
p Bp B BCp Bp B B
p B Cp B p B B Cp B B
p B B Cp B p B Cp B B
p B Cp B p B B Cp B B
p B Cp B p
αβ β β αβ β β
αβ αβ β β β β
β β αβ αβ β β
αβ αβ β β β β
αβ αβ α
  
− +− − + + +
  
= − + +− − +
+− − + + − +
= − + +− − +
+− +
( )
( )
( ) ( ) ( )
( )
11 2 2 2 3 3
2 22 33 2 22 33 2 2 2 33 1 11
Bp B B
p B BCp B Bp B Bp B
β ββ
ββ ββ ββ αβ
−+
+−− + −− +
It follows from this formula that the result of measurement by wave function
( )
( )
1 11 2 2 2 3 3
p Bp B B
αβ β β
++ +
makes the
11
CB
component unchanged and
two other components rotated around the normal to
1
B
, see (4.1) and (4.3),
with probability
2
1
p
(item
( ) ( )
1 11 1 11
p B Cp B
αβ αβ
−+
). Then it just flips the
11
CB
component and two other components rotated around the normal to
1
B
,
but in opposite direction see (4.2) with probability
2
2
p
(item
() ( )
2 22 33 2 22 33
p B B Cp B B
ββ ββ
−− +
).
Other two items are correspondingly the first above item subjected to Clifford
(parallel) translation on
3

by
( )
( )
1 2 11 2 2 3 3
pp B B B
αβ β β
−+
and the second
item subjected to opposite Clifford translation
()
( )
1 2 2 2 3 3 11
pp B B B
β β αβ
−− +
.
They are neither (4.1) nor (4.2) and their probabilities to make
11
CB
unchanged
or flipped are zero. Thus, they give two other different available measurement
results.
5. Superposition of Two Arbitrary Wave Functions
Any arbitrary
3
G
+
wave function
11 2 2 3 3
BBB
αβ β β
++ +
can be rewritten ei-
ther as 0-type wave function or 1-type wave function:
( )
123
222
11 2 2 33 1 2 3
,,S
BBB I
ββ β
αβ β β α βββ
+++=+ ++
,
where
( )
123
11 2 2 3 3
,, 222
1 23
S
BBB
I
ββ β
ββ β
βββ
++
=++
, 0-type,
or
( )
( )
()
21
112233 32112 33
222
3 1 23
,,S
B B B B B BB
IB
β βα
αβ β β β β β α
β αββ
−−
+++=+−−
= + ++
,
where
( )
21
21 12 3
,, 222
12
S
BBB
I
β βα
ββα
αββ
−−
−−
=++
, 1-type.

A. Soiguine
DOI:
10.4236/jamp.2023.112027 454 Journal of Applied
Mathematics and Physics
All that means that any
3
G
+
wave function
11 2 2 3 3
BBB
αβ β β
++ +
measur-
ing observable
11 2 2 3 3
CB CB C B
++
does not change the observable projection
onto plane of
( )
123
11 2 2 3 3
,, 222
1 23
S
BBB
I
ββ β
ββ β
βββ
++
=++
and just flips the observable pro-
jection onto plane
( )
21
21 12 3
,, 222
12
S
BBB
I
β βα
ββα
αββ
−−
−−
=++
.
Take two arbitrary wave functions and rewrite the first one as 0-type wave
function and the second one as 1-type wave function. Then all the results of Sec.
2 become applicable for their superposition. It will follow that there will be a re-
sult of measurement
()
()
( )
()
123 123
2 222 222
1 123 1 23
,, ,,
SS
p I CI
ββ β ββ β
α βββ α βββ
− ++ + ++
not changing the projection of
C
onto plane of
()
123
,,S
I
ββ β
and keeping proba-
bility
2
1
p
; plus, result of measurement
()
()
()
( )
()
()
21 21
2 222 222
233 12 3 123
,, ,,SS
pB I C I B
β βα β βα
β αββ β αββ
−− −−
− − ++ + ++
just flipping projection of
C
in plane of
( )
21
,,S
I
β βα
−−
and keeping probability
2
2
p
. Two other results represent the first two subjected to Clifford (parallel)
translations on the sphere
3

by
( )
()
( )
()
21
123
222 222
12 1233 12
,,
,, S
S
pp I I
β βα
ββ β
α ββββ αββ
−−
− ++ + ++
and
( )
()
( )
()
21 123
222 222
123 12 123
,, ,,
SS
pp I I
β βα ββ β
β αββα βββ
−−
− ++ + ++
correspondingly.
6. Entanglement in Measurements
Whilst the Schrodinger equation governs infinitesimal transformations of a wave
function by Clifford translations a finite Clifford translation moves a wave func-
tion along a big circle of
3

by any Clifford parameter.
In
3
G
+
multiplication is:
( )( )
1 2 1 2 12
12 1 1 2 2 1 2 21 12 12S S S S SS
gg I I I I I I
α β α β αα α β αβ ββ
=+ +=+ + +
It is not commutative due to the not commutative product of bivectors
12
SS
II
. Indeed, taking vectors to which
1
S
I
and
2
S
I
are dual:
1
13S
s II= −
,
2
23S
s II= −
, we have, see [7], Sec. 1.1:
( )
12
12 31 2SS
II s s I s s=−⋅ − ×
Then:
( ) ( )
12
12 12 1 2 12 21 12 3 1 2 12SS
gg s s I I I s s
αα ββ α β αβ ββ
= −⋅ + + − ×
and
( ) ( )
12
21 12 1 2 12 21 12 3 1 2 12SS
gg s s I I I s s
αα ββ α β αβ ββ
= −⋅ + + + ×

A. Soiguine
DOI:
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Journal of Applied Mathematics and Physics
I the case when both elements are of exponent form:
1
1
1
12 3
1 1 1 11 1 11 2 11 3
e
S
I
S
I bB b B b B
ϕα βαβ β β
=+=+ + +
2
2
2
12 3
2 2 2 22 1 22 2 22 3
eS
I
S
I bB bB bB
ϕ
α βαβ β β
=+=+ + +
,
with
() ( )
() ( )( )
()
()()
222
22 22
123
11111 11
1
bbb
αβ αβ
+ ++ =+=
( ) ()
( ) ( ) ()
()
() ( )
222
22 22
123
22222 22
1bb b
αβ αβ
+ ++ = + =
,
as in the case a wave function and Clifford translation, we get:
( )
( )
21
21
1 212 1232 12
31 2 1 3 2 1 1 2
e e cos cos sin sin cos sin
cos sin sin sin
SS
II
s s Is
Is I s s
ϕϕ
ϕ ϕ ϕϕ ϕϕ
ϕϕ ϕϕ
= +⋅ +
+ −×
Then it follows that two wave functions are, in any case, connected by the
Clifford translation:
()
( )
2 21 1 1
2 21 1 1
2 211
e e e e , ,, e
S SSS S
I III I
Cl S S
ϕ ϕϕ ϕ ϕ
ϕϕ
−
= ≡
,
where
( )
()
( )
21
21
2 211
1 212 1232 12
31 2 1 3 2 1 1 2
, ,, e e
cos cos sin sin cos sin
cos sin sin sin
SS
II
Cl S S
s s Is
Is I s s
ϕϕ
ϕϕ
ϕϕ ϕϕ ϕϕ
ϕϕ ϕϕ
−
≡
= +⋅ +
+ +×
.
From knowing Clifford translation connecting any two wave functions as
points on
3

it follows that the result of measurement of any observable
C
by
wave function
1
1
eS
I
ϕ
, for example
( )
11
11
11
ee ,
SS
II
C CS
ϕϕ ϕ
−≡
, immediately gives
the result of (not made) measurement by
2
2
eS
I
ϕ
:
( )
( ) ( ) ( )
2 2 211 112
2 2 211 112
21 1 2
2 1 12
11
2 21 1 11 2 21 1
e e e ee ee e
e e ,e e
, ,, , , ,,
S S SSS SSS
S S SS
I I III III
I I II
CC
CS
Cl S S C S Cl S S
ϕ ϕ ϕϕ ϕ ϕ ϕϕ
ϕϕ ϕϕ
ϕ
ϕϕ ϕ ϕϕ
− −− −
−−
=
=
= −− −−
When assuming observables are also identified by points on
3

and thus are
connected by formulas as the above one we get that the measurements of any
amount of observables by arbitrary set of wave functions are simultaneously
available.
7. Conclusion
The suggested formalism gives different, more physically feasible explanation of
what is superposition and entanglement. Superposition of any two wave func-
tions in the frame of g-qubit theory gives another wave function the result of
measurement by which is more complicated than in conventional quantum me-
chanics. In addition to the two results of measurements coming from composed
items of the wave functions there appear two additional items which are Clifford
(parallel) translations of the first two results in opposite directions on the sphere
3

.
The core of quantum computing scheme should be in manipulation and

A. Soiguine
DOI:
10.4236/jamp.2023.112027 456 Journal of Applied
Mathematics and Physics
transferring of wave functions on
3

as operators acting on observables and
formulated in terms of geometrical algebra.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this pa-
per.
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