The application of the hypercomplex number in a three dimensional space in quantum computing

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The application of the hypercomplex number in a three dimensional space in quantum computing

Abdelkarim Assoul

To cite this version:

Abdelkarim Assoul. The application of the hypercomplex number in a three dimensional space in quantum computing. 2018. �hal-01930431�

The application of the hypercomplex number in a three dimensional space in quantum computing

Assoul Abdelkarim

Independent researcher, teacher in secondary schools Annaba-Algeria E-mail : assak_maths@yahoo.fr

Site : https://www.researchgate.net/profile/Assoul_Abdelkarim Tel: +213792556774

27/10/2017 Annaba-Algeria

Abstract

In a publication about the hypercomplex number in a three-dimensional space, we defined a hypercomplex number in a three-dimensional space as follows: s = x + yi + zj with x, y, z real numbers and i, j purely imaginary numbers such as: i² = j² = -1 and ij = ji = 0.

The goal of the article is the use of the hypercomplex number in a three-dimensional space to facilitate the computation of the amplitude and the probability of the superposition of all qubit states that enters into the quantum computing improvement as well as quantum physics.

Keyword: hypercomplex number, superposition, qubit 1. Introduction

The conventional computer calculates by manipulating the bits, each bit carries either a 1 or a 0 whereas the quantum computer calculates by handling the qubits [1, 2], each qubit carries either a 1 or a 0 or a superposition [3] of a 1 and a 0, this way the quantum computer will be more powerful. Many articles were written around the qubit and the quantum computer[1] where the complex numbers were used in the calculation of the amplitude noted z = x + iy and the probability noted |z|² = a² + b².

Our study exactly is the replacement of complex numbers by hypercomplex numbers in a three dimensional Hilbertian space [4] for calculating the amplitude [5] and probability [5,6]

for finding new results.

2. The states of superposition

2.1. The states of superposition of a qubit

In the general case, a qubit has infinity of state superposition (*) denoted by Dirac [7] as follows in fact: = α + β whither and are the basic states and α, βtwo hypercomplex numbers in a three-dimensional Hilbert space such as: |α|²+|β|² = 1

We take: |α|² = p and |β|² = 1- p, p is a real number such as: 0 < p < 1 (1) If , are the image vectors respectively of the numbers α and β in an

orthonormal coordinate system (O, I, J, K) then Mα, Mβ appertain respectively to the spheres S1(0, ), S2(0, ). (2)

The ideal case is to take p = , then Mα, Mβ S (0, ). (3) That is to say Mα, Mβ appertain to the sphere of equation: x²+y²+z² =

If we apply (1), with a simple calculation, we find an example among an infinity of examples, so we search α and β two hypercomplex numbers in a hilbertian space of dimension three such that: = α + β and |α|²+|β|² = 1, as in the following table[1] :

Etats

Amplitude : s = x+yi+zj Probabilité :| s|²= x²+y²+z²

α = Β =

|α|²+|β|² = + = 1

(*)The states of superposition is a purely mathematical consequence of quantum theory while in quantum physics, there are several interpretations namely: The Copenhagen interpretation, Everett's theory David Deutsch, the interpretation of De Broglie-Bohm and Schrödinger cat.

2.2. Superposition states of two qubits

Two qubits are brought together in a superposition of states:

= α + β + λ + μ whither α, β, λ, μ are hypercomplex numbers in a hilbertian space of dimension three such that: |α|²+|β|²+|λ|²+|μ|² = 1

We take in this case: |α|² = p, |β|² = – p, |λ|² = q and |μ|² = – q, (4) p and q two real numbers with: 0 < p < , 0 < q <

Then: |α|²+|β|²+|λ|²+|μ|² = p + – p + q + – q = 1

If , , , are vector images respectively of the numbers α, β, λ, μ then:

Mα є S1 (0, ), Mβ є S2 (0, ), Mλ є S3 (0, ), Mμ є S4 (0, ) (5)

The ideal case is to take p = q = then Mα, Mβ, Mλ, Mμ S (0, ) (6) That is to say Mα, Mβ, Mλ, Mμ appertain to the sphere of equation: x²+y²+z² =

If we apply (4), with a simple calculation, we find an example among an infinity of examples, We search α, β, λ, μ four hypercomplex numbers in a Hilbert space of dimension three such that: = α + β + λ + μ and |α|²+|β|²+|λ|²+|μ|² = 1, as in the following table[1]:

Etats Amplitude : s = x+yi+zj Probabilité :| s|²= x²+y²+z²

α = | α |² =

Β = | β |² =

λ = | λ |² =

μ = | μ |² =

|α|²+|β|²+|λ|²+|μ|² = + + + = 1

2.3. Superposition states of three qubits

Three qubits are brought together in a superposition of states:

= α1 + α2 + α3 + α4 + α5 + α6 + α7 + α8 whither αi, i = are hypercomplex numbers in a hilbertian space of dimension three and

In fact if we take: = p, = - p, 0 < p <

= , = - , <

= q, = - q, 0 < q <

= , = - , < (7) Then:

If , i = are vector images respectively of the numbers αi, i = then the images Mαi , i = appertain respectively to the following spheres:

Mα1 є S1 (0, ), Mα2 є S2 (0, ), Mα₃ є S3 (0, ), Mα4 є S4 (0, ),

Mα5 є S5 (0, ), Mα6 є S6 (0, ), Mα₇ є S7 (0, ), Mα8 є S8 (0, ), (8) The ideal case is to take:

p = q = = = whither Mαi, i = appertain to the sphere of equation: S (0, ) (9) that is to say the sphere of equation: x²+y²+z² =

If we apply (7), with a simple calculation, we find an example among infinity of examples, we search αi, i = eight hypercomplex numbers in a three-dimensional Hilbert space such as:

= α1 + α2 + α3 + α4 + α5 + α6 + α7 +α8 and

As in the following table [1]:

Etats Amplitude : s = x+yi+zj Probabilité :| s|²= x²+y²+z² α₁ = | ₁|² =

α₂ = | _₂|² = α₃ = | _₃|² = α₄ = | ₄|² = α₅ = | ₅|² = α₆ = | ₆|² = α₇ = | _₇|² = α₈ = | _₈|² = 3. Generalization: the states of superposition of n qubits, nєN*

n qubits are in a superposition of states: = such as = 1

Whither αi, i = are hypercomplex numbers in a hilbertian space of dimension three and , i = are all states.

In this general case, we have:

| ₁|² = p₁ and | ₂|² = - p₁ such as: 0 < p₁ < (10)

| ₃|² = p₂ and | ₄|² = - p₂ such as: 0 < p₂ <

| _₅|² = p₃ and | _₆|² = - p₃ such as: 0 < p₃ <

. . .

| _{₂ⁿ -1}|² = p₂ⁿ⁻¹ and | _₂ⁿ |² = - p₂ⁿ⁻¹ such as: 0 < p₂ⁿ⁻¹ <

Then: | ₁|² + | ₃|² + | ₃|² + ………. + | ₂ⁿ -1|² + | ₂ⁿ |² = . = 1

If , i = are vector images respectively of the numbers αi, i = Then the images Mαi, i = appertain respectively to the following spheres:

Mα₁ є S₁ (0, ), Mα₂ є S₂ (0, ), Mα_₃ є S₃ (0, ), Mα₄ є S₄ (0, ) …

Finally we find M ₂ⁿ -1 є S₂ⁿ -1 (0, ), and M ₂ⁿ є S₂ⁿ (0,

) (11)

If , i = are vector images respectively of the numbers αi, i =

So the images Mαi, i = appertain in the ideal case to the sphere S (0, ) (12) That is to say the sphere of equation: x²+y²+z² =

Note. We can take simple examples of ideal cases using spherical coordinates:

z = [8]

With = , = arc cos ( ), =

and =2π -

Remarque

We know that the order of magnitude of the radius of the atom ra is 10^-10 m [9] and the states of superposition of n qubits can be represented by 2ⁿ points of the sphere of center O and radius rs= so if: ra = rs that is to say = 10^-10, we find 2ⁿ = 10²⁰ then n = ≈ 66.43.

This allows us to say that the necessary number of qubit to have a really efficient quantum computer with atoms is : n = 67

Conclusion

Our study shows that the states of superposition of a single qubit can be represented by any two points of the sphere S (0, ),the superposition states of two qubits can be represented by any four points of the sphere S (0, ),the superposition states of three qubits can be represented by any eight points of the sphere S (0, )and the states of superposition of n qubits can be represented by any 2ⁿ points of the sphere S (O, ).

According to my calculation in this article if we take the number of qubits n 67, we will have an ideal quantum computer!

References

[1] Calculateur quantique — Wikipédia

https://fr.wikipedia.org/wiki/Calculateur_quantique [2] Qubit — Wikipédia

https://fr.wikipedia.org/wiki/Qubit

[3] Principe de superposition quantique — Wikipédia

https://fr.wikipedia.org/wiki/Principe_de_superposition_quantique [4] Assoul Abdelkarim, 2016,

https://www.researchgate.net/publication/308969073_Research_in_the_Mathematical_Sci ences_Hypercomplex_number_in_three_dimensional_spaces_--Manuscript_Draft--

_Manuscript_Number_RMSS-D-16- [5]Amplitude de probabilité — Wikipédia

https://fr.wikipedia.org/wiki/Amplitude_de_probabilitè [6] Jerzy Hanckowiak, 2015,

https://www.researchgate.net/publication/273706661_Taming_the_Probability_Amplitude [7] Notation bra-ket — Wikipédia

https://fr.wikipedia.org/wiki/Notation_bra-ket [8]Coordonnées sphériques — Wikipédia

https://fr.wikipedia.org/wiki/Coordonnées_sphériques [9]www.maxicours.com/se/fiche/3/2/358832.html