Quantum Computingvia Entanglement in Geometric Algebra Approach

Abstract

The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.

Share and Cite:

Soiguine, A. (2024) Quantum Computingvia Entanglement in Geometric Algebra Approach. Journal of Applied Mathematics and Physics, 12, 445-457. doi: 10.4236/jamp.2024.122029.

1. Introduction. What Is Entanglement in Conventional Approach and in the Geometric Algebra Approach

In conventional quantum mechanics entanglement means that multiple objects share a single quantum state [1] [2] . Entanglement is the amount by which multiple objects share a quantum state. They remain indeterminate until they are disentangled by a measurement.

The simplest quantum mechanical state, qubit, reads:

${C}^{2}\ni \left(\begin{array}{c}{z}_{1}\\ {z}_{2}\end{array}\right)={z}_{1}\left(\begin{array}{c}1\\ 0\end{array}\right)+{z}_{2}\left(\begin{array}{c}0\\ 1\end{array}\right)={z}_{1}|0〉+{z}_{2}|1〉$

${z}_{1}={x}_{1}+i{y}_{1}$ , ${z}_{2}={x}_{2}+i{y}_{2}$ , ${‖{x}_{1}+i{y}_{1}‖}^{2}+{‖{x}_{2}+i{y}_{2}‖}^{2}=1$

It has just two observable “things” after measurement, say “up” for $|0〉$ and “down” for $|1〉$ , with probabilities ${z}_{1}^{2}$ and ${z}_{2}^{2}$ . The qubit dimension two is the number of different observable things available after making a measurement on the particle.

In the case of two particles the vector space ${C}^{2}$ is generalized to density matrix defined on tensor product ${C}^{2}\otimes {C}^{2}$ . The appropriateness of tensor products is that the tensor product itself captures all the ways that basic things can “interact” with each other. Quantum entanglement means that aspects of one particle of an entangled pair depend on aspects of the other particle.

To understand the quantum entanglement, it is important to, first, consider quantum superposition. Quantum superposition is the idea that particles exist in multiple states at once. When a measurement is performed, it is as if the particle selects one of the states in the superposition. The quantum entanglement emerges from the reality of quantum superposition.

Consider the decay of a particle called the pi meson. When this particle decays, it produces an electron and a positron that have opposite spin and are moving away from each other. If the electron spin is measured to be up, then the measured spin of the positron could only be down, and vice versa. Because of quantum mechanics, the spin of each particle is both part up and part down until it is measured. Only when the measurement occurs does the quantum state of the spin “collapse” into either up or down—instantaneously collapsing the other particle into the opposite spin.

That’s crucial words: “instantaneously collapsing”. Common wisdom considers that as the physically real transformation of the other particle’s unknown state into the known one.

The scheme suggested in the geometric algebra approach is based on the manipulation and transfer of quantum states as operators acting on observables. Wave functions act in that context on static ${G}_{3}^{+}$ elements through measurements, creating “particles.” [3]

Normalized wave functions as elements of ${G}_{3}^{+}$ are naturally mapped onto unit sphere ${\mathbb{S}}^{3}$ (see Figure 1), [4] [5] . Two-state system is then just a couple of points on ${\mathbb{S}}^{3}$ , say

${\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}={\alpha }_{1}+{I}_{{S}_{1}}{\beta }_{1}={\alpha }_{1}+{\beta }_{1}{b}_{1}^{1}{B}_{1}+{\beta }_{1}{b}_{1}^{2}{B}_{2}+{\beta }_{1}{b}_{1}^{3}{B}_{3}$

${\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}={\alpha }_{2}+{I}_{{S}_{2}}{\beta }_{2}={\alpha }_{2}+{\beta }_{2}{b}_{2}^{1}{B}_{1}+{\beta }_{2}{b}_{2}^{2}{B}_{2}+{\beta }_{2}{b}_{2}^{3}{B}_{3}$

where

${\left({\alpha }_{1}\right)}^{2}+{\left({\beta }_{1}\right)}^{2}\left({\left({b}_{1}^{1}\right)}^{2}+{\left({b}_{1}^{2}\right)}^{2}+{\left({b}_{1}^{3}\right)}^{2}\right)={\left({\alpha }_{1}\right)}^{2}+{\left({\beta }_{1}\right)}^{2}=1$

${\left({\alpha }_{2}\right)}^{2}+{\left({\beta }_{2}\right)}^{2}\left({\left({b}_{2}^{1}\right)}^{2}+{\left({b}_{2}^{2}\right)}^{2}+{\left({b}_{2}^{3}\right)}^{2}\right)={\left({\alpha }_{2}\right)}^{2}+{\left({\beta }_{2}\right)}^{2}=1$

in some bivector basis ${B}_{1}{B}_{2}{B}_{3}=1$ , with multiplication rules ${B}_{1}{B}_{2}=-{B}_{3}$ , ${B}_{1}{B}_{3}={B}_{2}$ , ${B}_{2}{B}_{3}=-{B}_{1}$ .

Figure 1. Geometric representation of a ${G}_{3}^{+}$ element.

Then it follows that two wave functions of an arbitrary two-function system are, in any case, connected by the Clifford translation1:

${\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}=\left({\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}\right){\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}\equiv Cl\left({S}_{2},{\phi }_{2},{S}_{1},{\phi }_{1}\right){\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}$

The product of exponents ${\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}$ is trivial in the case ${S}_{1}={S}_{2}\equiv S$ (the case of an unspecified imaginary unit in conventional quantum mechanics,) namely: ${\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}={\text{e}}^{{I}_{S}\left({\phi }_{2}-{\phi }_{1}\right)}$ . In the general case we have more complicated result, see (1.2.2) in [6] :

$\begin{array}{c}Cl\left({S}_{2},{\phi }_{2},{S}_{1},{\phi }_{1}\right)\equiv {\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}\\ =\mathrm{cos}{\phi }_{1}\mathrm{cos}{\phi }_{2}+\left({s}_{1}\cdot {s}_{2}\right)\mathrm{sin}{\phi }_{1}\mathrm{sin}{\phi }_{2}+{I}_{3}{s}_{2}\mathrm{cos}{\phi }_{1}\mathrm{sin}{\phi }_{2}\\ \text{\hspace{0.17em}}\text{ }\text{ }+{I}_{3}{s}_{1}\mathrm{cos}{\phi }_{2}\mathrm{sin}{\phi }_{1}+{I}_{3}\left({s}_{2}×{s}_{1}\right)\mathrm{sin}{\phi }_{1}\mathrm{sin}{\phi }_{2}\end{array}$ (1.1)

where ${s}_{1}$ and ${s}_{2}$ are vectors dual to planes ${S}_{1}$ and ${S}_{2}$ matching orientation of ${I}_{3}$ .

Take a general definition:

${g}_{1}{g}_{2}={\alpha }_{1}{\alpha }_{2}-\left({s}_{1}\cdot {s}_{2}\right){\beta }_{1}{\beta }_{2}+{I}_{{S}_{1}}{\alpha }_{2}{\beta }_{1}+{I}_{{S}_{2}}{\alpha }_{1}{\beta }_{2}-{I}_{3}\left({s}_{1}×{s}_{2}\right){\beta }_{1}{\beta }_{2}$

The bivector part of the product ${g}_{1}{g}_{2}$ belongs to the plane orthogonal to the associated vector as seen in Figure 2.

To get the module $|{g}_{1}{g}_{2}|$ and angle of rotation in the plane of the bivector part of ${g}_{1}{g}_{2}$ we need the scalar part:

${\alpha }_{1}{\alpha }_{2}-\left({s}_{1}\cdot {s}_{2}\right){\beta }_{1}{\beta }_{2}={\alpha }_{1}{\alpha }_{2}-{\beta }_{1}{\beta }_{2}\left({b}_{1}^{1}{b}_{2}^{1}+{b}_{1}^{2}{b}_{2}^{2}+{b}_{1}^{3}{b}_{2}^{3}\right)$

and representation of ${I}_{{S}_{1}}{\alpha }_{2}{\beta }_{1}+{I}_{{S}_{2}}{\alpha }_{1}{\beta }_{2}-{I}_{3}\left({s}_{1}×{s}_{2}\right){\beta }_{1}{\beta }_{2}$ as linear combination of the basis bivectors $\left\{{B}_{1},{B}_{2},{B}_{3}\right\}$ :

Figure 2. Constructing the bivector plane of the product.

$\begin{array}{l}{I}_{{S}_{1}}{\alpha }_{2}{\beta }_{1}+{I}_{{S}_{2}}{\alpha }_{1}{\beta }_{2}-{I}_{3}\left({s}_{1}×{s}_{2}\right){\beta }_{1}{\beta }_{2}\\ ={\alpha }_{2}{\beta }_{1}\left({b}_{1}^{1}{B}_{1}+{b}_{1}^{2}{B}_{2}+{b}_{1}^{3}{B}_{3}\right)+{\alpha }_{1}{\beta }_{2}\left({b}_{2}^{1}{B}_{1}+{b}_{2}^{2}{B}_{2}+{b}_{2}^{3}{B}_{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\beta }_{1}{\beta }_{2}\left[\left({b}_{1}^{2}{b}_{2}^{3}-{b}_{1}^{3}{b}_{2}^{2}\right){B}_{1}+\left({b}_{1}^{3}{b}_{2}^{1}-{b}_{1}^{1}{b}_{2}^{3}\right){B}_{2}+\left({b}_{1}^{1}{b}_{2}^{2}-{b}_{1}^{2}{b}_{2}^{1}\right){B}_{3}\right]\\ ={B}_{1}\left({\alpha }_{2}{\beta }_{1}{b}_{1}^{1}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{1}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{2}{b}_{2}^{3}-{b}_{1}^{3}{b}_{2}^{2}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{B}_{2}\left({\alpha }_{2}{\beta }_{1}{b}_{1}^{2}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{2}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{3}{b}_{2}^{1}-{b}_{1}^{1}{b}_{2}^{3}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{B}_{3}\left({\alpha }_{2}{\beta }_{1}{b}_{1}^{3}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{3}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{1}{b}_{2}^{2}-{b}_{1}^{2}{b}_{2}^{1}\right)\right)\end{array}$

We get:

$\begin{array}{c}{g}_{1}{g}_{2}={\alpha }_{1}{\alpha }_{2}-{\beta }_{1}{\beta }_{2}\left({b}_{1}^{1}{b}_{2}^{1}+{b}_{1}^{2}{b}_{2}^{2}+{b}_{1}^{3}{b}_{2}^{3}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\beta \left({B}_{1}\frac{{\alpha }_{2}{\beta }_{1}{b}_{1}^{1}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{1}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{2}{b}_{2}^{3}-{b}_{1}^{3}{b}_{2}^{2}\right)}{\beta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{B}_{2}\frac{{\alpha }_{2}{\beta }_{1}{b}_{1}^{2}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{2}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{3}{b}_{2}^{1}-{b}_{1}^{1}{b}_{2}^{3}\right)}{\beta }\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{B}_{3}\frac{{\alpha }_{2}{\beta }_{1}{b}_{1}^{3}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{3}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{1}{b}_{2}^{2}-{b}_{1}^{2}{b}_{2}^{1}\right)}{\beta }\right)\end{array}$

where

$\beta =\sqrt{{\alpha }_{1}^{2}{\beta }_{2}^{2}+{\alpha }_{2}^{2}{\beta }_{1}^{2}+{\beta }_{1}^{2}{\beta }_{2}^{2}{|{s}_{1}×{s}_{2}|}^{2}+2{\alpha }_{1}{\alpha }_{2}{\beta }_{1}{\beta }_{2}\left({s}_{1}\cdot {s}_{2}\right)}$

Thus, the module $|{g}_{1}{g}_{2}|$ is:

$\begin{array}{c}|{g}_{1}{g}_{2}|=\sqrt{{\left({\alpha }_{1}{\alpha }_{2}-{\beta }_{1}{\beta }_{2}\left({s}_{1}\cdot {s}_{2}\right)\right)}^{2}+{\alpha }_{1}^{2}{\beta }_{2}^{2}+{\alpha }_{2}^{2}{\beta }_{1}^{2}+{\beta }_{1}^{2}{\beta }_{2}^{2}{|{s}_{1}×{s}_{2}|}^{2}+2{\alpha }_{1}{\alpha }_{2}{\beta }_{1}{\beta }_{2}\left({s}_{1}\cdot {s}_{2}\right)}\\ =\sqrt{{\alpha }_{1}^{2}{\alpha }_{2}^{2}+{\beta }_{1}^{2}{\beta }_{2}^{2}{|{s}_{1}\cdot {s}_{2}|}^{2}-2{\alpha }_{1}{\alpha }_{2}{\beta }_{1}{\beta }_{2}\left({s}_{1}\cdot {s}_{2}\right)+{\alpha }_{1}^{2}{\beta }_{2}^{2}+{\alpha }_{2}^{2}{\beta }_{1}^{2}+{\beta }_{1}^{2}{\beta }_{2}^{2}{|{s}_{1}×{s}_{2}|}^{2}+2{\alpha }_{1}{\alpha }_{2}{\beta }_{1}{\beta }_{2}\left({s}_{1}\cdot {s}_{2}\right)}\\ =\sqrt{{\alpha }_{1}^{2}{\alpha }_{2}^{2}+{\beta }_{1}^{2}{\beta }_{2}^{2}{\mathrm{cos}}^{2}\left({s}_{1},{s}_{2}\right)+{\alpha }_{1}^{2}{\beta }_{2}^{2}+{\alpha }_{2}^{2}{\beta }_{1}^{2}+{\beta }_{1}^{2}{\beta }_{2}^{2}{\mathrm{sin}}^{2}\left({s}_{1},{s}_{2}\right)}\\ =\sqrt{{\alpha }_{1}^{2}{\alpha }_{2}^{2}+{\beta }_{1}^{2}{\beta }_{2}^{2}+{\alpha }_{1}^{2}{\beta }_{2}^{2}+{\alpha }_{2}^{2}{\beta }_{1}^{2}}=\sqrt{{\alpha }_{1}^{2}\left({g}_{2}^{2}\right)+{\beta }_{1}^{2}\left({g}_{2}^{2}\right)}\\ =\sqrt{\left({g}_{2}^{2}\right)\left({g}_{1}^{2}\right)}=|{g}_{1}||{g}_{2}|\end{array}$

If ${I}_{{S}_{1}}={I}_{{S}_{2}}$ we also have usual formulas valid for complex numbers when formal imaginary unit i is replaced by ${I}_{{S}_{1}}$ (or ${I}_{{S}_{2}}$ .) Particularly, angles are added in multiplication, subtracted in division, and de Moivre’s formula takes place:

If $g=|g|\left(\mathrm{cos}\phi +\mathrm{sin}\phi {I}_{S}\right)$ then ${g}^{n}={|g|}^{n}\left(\mathrm{cos}n\phi +\mathrm{sin}n\phi {I}_{S}\right)$ for positive integer n.

In general case, when ${I}_{{S}_{1}}\ne {I}_{{S}_{2}}$ , the angle of rotation ${g}_{1}{g}_{2}$ is:

${\mathrm{cos}}^{-1}\left(\frac{{\alpha }_{1}{\alpha }_{2}-{\beta }_{1}{\beta }_{2}\left({s}_{1}\cdot {s}_{2}\right)}{|{g}_{1}||{g}_{2}|}\right)$

and the plane of the rotation normal has directional cosines:

$\begin{array}{l}\left\{\frac{{\alpha }_{2}{\beta }_{1}{b}_{1}^{1}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{1}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{2}{b}_{2}^{3}-{b}_{1}^{3}{b}_{2}^{2}\right)}{\beta },\frac{{\alpha }_{2}{\beta }_{1}{b}_{1}^{2}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{2}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{3}{b}_{2}^{1}-{b}_{1}^{1}{b}_{2}^{3}\right)}{\beta },\\ \frac{{\alpha }_{2}{\beta }_{1}{b}_{1}^{3}+{\alpha }_{1}{\beta }_{2}{b}_{2}^{3}+{\beta }_{1}{\beta }_{2}\left({b}_{1}^{1}{b}_{2}^{2}-{b}_{1}^{2}{b}_{2}^{1}\right)}{\beta }\right\}\end{array}$

The result of Clifford translation (1.1) is an ${G}_{3}^{+}$ element. From knowing Clifford translation connecting any two wave functions as points ${\mathbb{S}}^{3}$ it follows that the result of measurement of any observable C by wave function ${\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}$ , for example ${\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}C{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}\equiv C\left({S}_{1},{\phi }_{1}\right)$ , immediately gives the result of (not made) measurement by ${\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}$ :

$\begin{array}{c}{\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}C{\text{e}}^{-{I}_{{S}_{2}}{\phi }_{2}}={\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}{\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}C{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}{\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}{\text{e}}^{-{I}_{{S}_{2}}{\phi }_{2}}\\ ={\text{e}}^{{I}_{{S}_{2}}{\phi }_{2}}{\text{e}}^{-{I}_{{S}_{1}}{\phi }_{1}}C\left({S}_{1},{\phi }_{1}\right){\text{e}}^{{I}_{{S}_{1}}{\phi }_{1}}{\text{e}}^{-{I}_{{S}_{2}}{\phi }_{2}}\\ =Cl\left({S}_{2},{\phi }_{2},{S}_{1},{\phi }_{1}\right)C\left({S}_{1},{\phi }_{1}\right)\stackrel{¯}{Cl\left({S}_{2},{\phi }_{2},{S}_{1},{\phi }_{1}\right)}\end{array}$

When assuming that observables are also identified by points on ${\mathbb{S}}^{3}$ and thus are connected by the formulas similar to the above one we get that measurements of any amount of observables by arbitrary set of wave functions are simultaneously available.

This is a geometrically clear and unambiguous explanation of strict connectivity of the results of measurements instead of quite absurd “entanglement” in conventional quantum mechanics.

2. Maxwell Equation in Geometric Algebra

Without charges and currents, the Maxwell equation is:

$\left({\partial }_{t}+\nabla \right)F=0$ (2.1)

The circular polarized electromagnetic waves are the only type of waves following from the solution of Maxwell equations in free space done in geometric algebra terms. Indeed, let’s take the electromagnetic field in the form:

$F={F}_{0}\mathrm{exp}\left[{I}_{S}\left(\omega t-k\cdot r\right)\right]$ (2.2)

requiring that it satisfies (2.1)

Element ${F}_{0}$ in (2.2) is a constant element of geometric algebra ${G}_{3}$ and ${I}_{S}$ is unit value bivector of a plane S in three dimensions, generalization of the imaginary unit [7] . The exponent in (2.2) is the unit value element of ${G}_{3}^{+}$ :

${\text{e}}^{{I}_{S}\phi }=\mathrm{cos}\phi +{I}_{S}\mathrm{sin}\phi$ , $\phi =\omega t-k\cdot r$

Solution of (2.1) should be the sum of a vector (electric field $e$ ) and bivector (magnetic field ${I}_{3}h$ ):

$F=e+{I}_{3}h$

with some initial conditions:

$e+{{I}_{3}h|}_{t=0,\stackrel{\to }{r}=0}={F}_{0}={e|}_{t=0,\stackrel{\to }{r}=0}+{{I}_{3}h|}_{t=0,\stackrel{\to }{r}=0}={e}_{0}+{I}_{3}{h}_{0}$

Substitution of (2.2) into Maxwell’s (2.1) will show us what the solution looks like.

The derivative by time gives

$\frac{\partial }{\partial t}F={F}_{0}{\text{e}}^{{I}_{S}\phi }{I}_{S}\frac{\partial }{\partial t}\left(\omega \text{ }t-k\cdot r\right)={F}_{0}{\text{e}}^{{I}_{S}\phi }{I}_{S}\omega =F{I}_{S}\omega$

The geometric algebra product $\nabla F$ is:

$\nabla F={F}_{0}{I}_{S}{\text{e}}^{{I}_{S}\phi }\nabla \left(\omega \text{ }t-k\cdot r\right)=-{F}_{0}{\text{e}}^{{I}_{S}\phi }{I}_{S}k=-F{I}_{S}k$

or

$\nabla F={F}_{0}{\text{e}}^{{I}_{S}\phi }\nabla \left(\omega \text{ }t-k\cdot r\right){I}_{S}=-{F}_{0}{\text{e}}^{{I}_{S}\phi }k{I}_{S}=-Fk{I}_{S}$ ,

depending on do we write ${I}_{S}\left(\omega \text{ }t-k\cdot r\right)$ or $\left(\omega \text{ }t-k\cdot r\right){I}_{S}$ . The result should be the same because $\omega \text{ }t-k\cdot r$ is a scalar.

Commutativity ${I}_{S}k=k{I}_{S}$ is valid only if $k×{I}_{3}{I}_{S}=0$ . The following agreement takes place between orientation of ${I}_{3}$ , orientation of ${I}_{S}$ and direction of vector k. The vector ${I}_{3}{I}_{S}={I}_{S}{I}_{3}$ is orthogonal to the plane of ${I}_{S}$ and its direction is defined by orientations of ${I}_{3}$ and ${I}_{S}$ . Rotation of right/left hand screw defined by orientation of ${I}_{S}$ gives the movement of right/left hand screw. This is the direction of the vector ${I}_{3}{I}_{S}={I}_{S}{I}_{3}$ . That means that the matching between $\stackrel{^}{k}$ and ${I}_{S}$ should be $\stackrel{^}{k}=±{I}_{3}{I}_{S}⇒\stackrel{^}{k}{I}_{S}=\mp {I}_{3}$ (see Figure 3).

Assume first that orientation is ${I}_{3}=\stackrel{^}{k}{I}_{S}$ . Then Maxwell equation becomes:

$F\left({I}_{S}\omega -{I}_{3}|k|\right)=F\left(\omega {I}_{S}-|k|\stackrel{^}{k}{I}_{S}\right)=0$

or

$\left(e+{I}_{3}h\right)\omega =\left(e+{I}_{3}h\right)k$

Figure 3. Two possible orientations of the wave vector.

Left hand side of equation is sum of vector and bivector, while right hand side is scalar $e\cdot k$ plus bivector $e\wedge k$ , plus pseudoscalar ${I}_{3}\left(h\cdot k\right)$ , plus vector ${I}_{3}\left(h\wedge k\right)$ . It follows that both and h lie on the plane of ${I}_{S}$ and then:

$\omega e={I}_{3}hk$ , $\omega {I}_{3}h=ek\to \frac{{\omega }^{2}}{{|k|}^{2}}{I}_{3}hk=\omega e$

Thus, $\omega =|k|$ and we get equation ${I}_{3}h\stackrel{^}{k}=e$ from which particularly follows ${|e|}^{2}={|h|}^{2}$ and $\stackrel{^}{e}\stackrel{^}{k}\stackrel{^}{h}={I}_{3}$ .

The result for the case ${I}_{3}=\stackrel{^}{k}{I}_{S}$ is that the solution of (2.1) is

$F=\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\left(\omega \text{ }t-k\cdot r\right)\right]$

where ${e}_{0}$ and ${h}_{0}$ are arbitrary mutually orthogonal vectors of equal length, lying on the plane S. Vector k should be normal to that plane, $\stackrel{^}{k}=-{I}_{3}{I}_{S}$ and $|k|=\omega$ .

In the above result, the sense of the ${I}_{S}$ orientation and the direction of k were assumed to agree with ${I}_{3}=\stackrel{^}{k}{I}_{S}$ . Opposite orientation, $-{I}_{3}=\stackrel{^}{k}{I}_{S}$ , that’s k and ${I}_{S}$ compose left hand screw and $\stackrel{^}{k}={I}_{3}{I}_{S}$ , will give solution $F=\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\left(\omega t-k\cdot r\right)\right]$ with $\stackrel{^}{e}\stackrel{^}{k}\stackrel{^}{h}={I}_{3}$ .

Summary:

For a plane S in three dimensions Maxwell equation (2.1) has two solutions

· ${F}_{+}=\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\left(\omega \text{ }t-{k}_{+}\cdot r\right)\right]$ , with ${\stackrel{^}{k}}_{+}={I}_{3}{I}_{S}$ , $\stackrel{^}{e}\stackrel{^}{k}{\stackrel{^}{h}}_{+}={I}_{3}$ , and the triple $\left\{\stackrel{^}{e},\stackrel{^}{k},{\stackrel{^}{h}}_{+}\right\}$ is right hand screw oriented, that’s rotation of $\stackrel{^}{e}$ to $\stackrel{^}{h}$ by π/2 gives the movement of right hand screw in the direction of ${k}_{+}=|k|{I}_{3}{I}_{S}$ .

· ${F}_{-}=\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\left(\omega \text{ }t-{k}_{-}\cdot r\right)\right]$ , with ${\stackrel{^}{k}}_{-}=-{I}_{3}{I}_{S}$ , $\stackrel{^}{e}\stackrel{^}{k}{\stackrel{^}{h}}_{-}=-{I}_{3}$ , and the triple $\left\{\stackrel{^}{e},\stackrel{^}{k},{\stackrel{^}{h}}_{-}\right\}$ is left hand screw oriented, that’s rotation of $\stackrel{^}{e}$ to $\stackrel{^}{h}$ by π/2 gives movement of left hand screw in the direction of ${k}_{-}=-|k|{I}_{3}{I}_{S}$ or, equivalently, movement of right hand screw in the opposite direction, $-{k}_{-}$ .

· ${e}_{0}$ and ${h}_{0}$ , initial values of $e$ and $h$ , are arbitrary mutually orthogonal vectors of equal length, lying on the plane S. Vectors ${k}_{±}=±|{k}_{±}|{I}_{3}{I}_{S}$ are normal to that plane. The length of the “wave vectors” $|{k}_{±}|$ is equal to the angular frequency $\omega$ .

Maxwell Equation (2.1) is a linear one. Then any linear combination of ${F}_{+}$ and ${F}_{-}$ saving the structure of (2.2) will also be a solution.

Let’s write:

$\left\{\begin{array}{c}{F}_{+}=\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\omega \left(t-\left({I}_{3}{I}_{S}\right)\cdot r\right)\right]\\ =\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\omega \text{ }t\right]\mathrm{exp}\left[-{I}_{S}\left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right]\\ {F}_{-}=\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\omega \left(t+\left({I}_{3}{I}_{S}\right)\cdot r\right)\right]\\ =\left({e}_{0}+{I}_{3}{h}_{0}\right)\mathrm{exp}\left[{I}_{S}\omega \text{ }t\right]\mathrm{exp}\left[{I}_{S}\left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right]\end{array}$ (2.3)

Then for arbitrary (real2) scalars $\lambda$ and $\mu$ :

$\lambda {F}_{+}+\mu {F}_{-}=\left({e}_{0}+{I}_{3}{h}_{0}\right){\text{e}}^{{I}_{S}\omega t}\left(\lambda {\text{e}}^{-{I}_{S}\left[\left({I}_{3}{I}_{S}\right)\cdot r\right]}+\mu {\text{e}}^{{I}_{S}\left[\left({I}_{3}{I}_{S}\right)\cdot r\right]}\right)$ (2.4)

is solution of (2.1). The item in the second parenthesis is a weighted linear combination of two states with the same phase in the same plane but opposite sense of orientation. The states are strictly coupled, entangled if you prefer, because the bivector plane should be the same for both, does not matter what happens with that plane.

Arbitrary linear combination (2.4) can be rewritten as:

$\lambda {\text{e}}^{{I}_{Plane}^{+}{\phi }^{+}}+\mu {\text{e}}^{{I}_{Plane}^{-}{\phi }^{-}}$ (2.5)

where

${\phi }^{±}={\mathrm{cos}}^{-1}\left(\frac{1}{\sqrt{2}}\mathrm{cos}\omega \left(t\mp \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right)\right)$ ,

$\begin{array}{c}{I}_{Plane}^{±}={I}_{S}\frac{\mathrm{sin}\omega \left(t\mp \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right)}{\sqrt{1+{\mathrm{sin}}^{2}\omega \left(t\mp \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right)}}+{I}_{{B}_{0}}\frac{\mathrm{cos}\omega \left(t\mp \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right)}{\sqrt{1+{\mathrm{sin}}^{2}\omega \left(t\mp \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right)}}\\ \text{\hspace{0.17em}}+{I}_{{E}_{0}}\frac{\mathrm{sin}\omega \left(t\mp \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right)}{\sqrt{1+{\mathrm{sin}}^{2}\omega \left(t\mp \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\right)}}\end{array}$

The triple of unit value basis orthonormal bivectors $\left\{{I}_{S},{I}_{{B}_{0}},{I}_{{E}_{0}}\right\}$ is comprised of the ${I}_{S}$ bivector, dual to the propagation direction vector; ${I}_{{B}_{0}}$ is dual to an initial vector of the magnetic field; ${I}_{{E}_{0}}$ is dual to the initial vector of the electric field. The expression (2.5) is a linear combination of two geometric algebra states, g-qubits.

Linear combination of the two equally weighted basic solutions of the Maxwell equation ${F}_{+}$ and ${F}_{-}$ , $\lambda {F}_{+}+\mu {F}_{-}$ with $\lambda =\mu =1$ reads:

$\begin{array}{c}{\lambda {F}_{+}+\mu {F}_{-}|}_{\lambda =\mu =1}=2\mathrm{cos}\omega \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\left(\frac{1}{\sqrt{2}}\mathrm{cos}\omega \text{ }t+{I}_{S}\frac{1}{\sqrt{2}}\mathrm{sin}\omega \text{ }t\\ \text{\hspace{0.17em}}+{I}_{{B}_{0}}\frac{1}{\sqrt{2}}\mathrm{cos}\omega \text{ }t+{I}_{{E}_{0}}\frac{1}{\sqrt{2}}\mathrm{sin}\omega \text{ }t\right)\end{array}$ (2.6)

where $\mathrm{cos}\phi =\frac{1}{\sqrt{2}}\mathrm{cos}\omega \text{ }t$ and $\mathrm{sin}\phi =\frac{1}{\sqrt{2}}\sqrt{1+{\left(\mathrm{sin}\omega \text{ }t\right)}^{2}}$ . It can be written in

standard exponential form $\mathrm{cos}\phi +\mathrm{sin}\phi {I}_{B}={\text{e}}^{{I}_{B}\phi }$ .3

I will call such g-qubits spreons because they spread over the whole three-dimensional space for all values of time and instantly change under Clifford translations over the whole three-dimensional space for all values of time, along with the results of measurement of any observable.4

3. Measurement of Observables by Spreons

Measurement of any observable ${C}_{0}+{C}_{1}{B}_{1}+{C}_{2}{B}_{2}+{C}_{3}{B}_{3}$ (actually Hopf fibration) by a state $\alpha +{\beta }_{1}{B}_{1}+{\beta }_{2}{B}_{2}+{\beta }_{3}{B}_{3}$ in the current formalism is, see, for example, [6] :

$\begin{array}{l}{C}_{0}+{C}_{1}{B}_{1}+{C}_{2}{B}_{2}+{C}_{3}{B}_{3}\stackrel{\alpha +{\beta }_{1}{B}_{1}+{\beta }_{2}{B}_{2}+{\beta }_{3}{B}_{3}}{\to }\\ {C}_{0}+\left({C}_{1}\left[\left({\alpha }^{2}+{\beta }_{1}^{2}\right)-\left({\beta }_{2}^{2}+{\beta }_{3}^{2}\right)\right]+2{C}_{2}\left({\beta }_{1}{\beta }_{2}-\alpha {\beta }_{3}\right)+2{C}_{3}\left(\alpha {\beta }_{2}+{\beta }_{1}{\beta }_{3}\right)\right){B}_{1}\\ +\left(2{C}_{1}\left(\alpha {\beta }_{3}+{\beta }_{1}{\beta }_{2}\right)+{C}_{2}\left[\left({\alpha }^{2}+{\beta }_{2}^{2}\right)-\left({\beta }_{1}^{2}+{\beta }_{3}^{2}\right)\right]+2{C}_{3}\left({\beta }_{2}{\beta }_{3}-\alpha {\beta }_{1}\right)\right){B}_{2}\\ +\left(2{C}_{1}\left({\beta }_{1}{\beta }_{3}-\alpha {\beta }_{2}\right)+2{C}_{2}\left(\alpha {\beta }_{1}+{\beta }_{2}{\beta }_{3}\right)+{C}_{3}\left[\left({\alpha }^{2}+{\beta }_{3}^{2}\right)-\left({\beta }_{1}^{2}+{\beta }_{2}^{2}\right)\right]\right){B}_{3}\end{array}$

In the case of spreon (2.6):

${B}_{1}={I}_{S}$ , ${B}_{2}={I}_{{B}_{0}}$ , ${B}_{3}={I}_{{E}_{0}}$ ,

$\begin{array}{c}\alpha =2\mathrm{cos}\omega \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\frac{1}{\sqrt{2}}\left(\mathrm{cos}\gamma \mathrm{cos}\omega \text{ }t-{\gamma }_{1}\mathrm{sin}\gamma \mathrm{cos}\omega \text{ }t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{2}\mathrm{sin}\gamma \mathrm{cos}\omega \text{ }t-{\gamma }_{3}\mathrm{sin}\gamma \mathrm{sin}\omega \text{ }t\right)\end{array}$

$\begin{array}{c}{\beta }_{1}=2\mathrm{cos}\omega \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\frac{1}{\sqrt{2}}\left(\mathrm{cos}\gamma \mathrm{sin}\omega \text{ }t+{\gamma }_{1}\mathrm{sin}\gamma \mathrm{cos}\omega \text{ }t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\gamma }_{2}\mathrm{sin}\gamma \mathrm{sin}\omega \text{ }t+{\gamma }_{3}\mathrm{sin}\gamma \mathrm{cos}\omega \text{ }t\right)\end{array}$

$\begin{array}{c}{\beta }_{2}=2\mathrm{cos}\omega \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\frac{1}{\sqrt{2}}\left(\mathrm{cos}\gamma \mathrm{cos}\omega \text{ }t+{\gamma }_{1}\mathrm{sin}\gamma \mathrm{sin}\omega \text{ }t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\gamma }_{2}\mathrm{sin}\gamma \mathrm{cos}\omega \text{ }t-{\gamma }_{3}\mathrm{sin}\gamma \mathrm{sin}\omega \text{ }t\right)\end{array}$

$\begin{array}{c}{\beta }_{3}=2\mathrm{cos}\omega \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\frac{1}{\sqrt{2}}\left(\mathrm{cos}\gamma \mathrm{sin}\omega \text{ }t-{\gamma }_{1}\mathrm{sin}\gamma \mathrm{cos}\omega \text{ }t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\gamma }_{2}\mathrm{sin}\gamma \mathrm{sin}\omega \text{ }t+{\gamma }_{3}\mathrm{sin}\gamma \mathrm{cos}\omega \text{ }t\right),\end{array}$

and we get a ${G}_{3}^{+}$ element spreading through the three-dimensional space for all values of the time parameter t. It is ${G}_{3}^{+}$ element spreading through the three-dimensional space for all values of time parameter t:

$\begin{array}{l}4{\mathrm{cos}}^{2}\omega \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]\left[{C}_{0}+{C}_{3}{I}_{S}+\left({C}_{1}\mathrm{sin}2\omega \text{ }t+{C}_{2}\mathrm{cos}2\omega \text{ }t\right){I}_{{B}_{0}}\\ +\left({C}_{2}\mathrm{sin}2\omega \text{ }t-{C}_{1}\mathrm{cos}2\omega \text{ }t\right){I}_{{E}_{0}}\right]\end{array}$ (3.1)

Geometrically, that means that the measured observable is rotated by π/2 in the ${I}_{{B}_{0}}$ plane, such that the ${C}_{3}{I}_{S}$ component becomes orthogonal to the plane ${I}_{S}$ and remains unchanged. Two other components became orthogonal to ${I}_{{B}_{0}}$ and ${I}_{{E}_{0}}$ and continue rotating in ${I}_{S}$ with angular velocity $2\omega t$ . The factor $4{\mathrm{cos}}^{2}\omega \left[\left({I}_{3}{I}_{S}\right)\cdot r\right]$ defines the dependency of those transformed values through all points of the three-dimensional space.

The current approach transcends common quantum computing schemes since the latter are principally based on qubit entanglement (whatever it is) and thus have tough problems of creating large sets of entangled qubits. In the current scheme, any test observable can be placed anywhere into the continuum of the $\left(t,r\right)$ dependent values of the spreon state. The above formula (3.1) gives the result of measurements simultaneously at all points $\left(t,r\right)$ .

Let us get back to the geometric algebra entanglements explained at the end of Sec.1. Assume we have two observables, ${C}_{0}^{1}+{C}_{1}^{1}{B}_{1}+{C}_{2}^{1}{B}_{2}+{C}_{3}^{1}{B}_{3}$ and ${C}_{0}^{2}+{C}_{1}^{2}{B}_{1}+{C}_{2}^{2}{B}_{2}+{C}_{3}^{2}{B}_{3}$ . Write them in exponential form:

$\begin{array}{c}{C}^{1}\equiv {C}_{0}^{1}+{C}_{1}^{1}{B}_{1}+{C}_{2}^{1}{B}_{2}+{C}_{3}^{1}{B}_{3}\\ =|{C}^{1}|\left(\frac{{C}_{0}^{1}}{|{C}^{1}|}+\frac{|{C}_{B}^{1}|}{|{C}^{1}|}\frac{{C}_{1}^{1}{B}_{1}}{|{C}_{B}^{1}|}+\frac{|{C}_{B}^{1}|}{|{C}^{1}|}\frac{{C}_{2}^{1}{B}_{2}}{|{C}_{B}^{1}|}+\frac{|{C}_{B}^{1}|}{|{C}^{1}|}\frac{{C}_{3}^{1}{B}_{3}}{|{C}_{B}^{1}|}\right)\\ =|{C}^{1}|\left(\mathrm{cos}{\phi }_{1}+\mathrm{sin}{\phi }_{1}{S}_{1}\right)=|{C}^{1}|{\text{e}}^{{S}_{1}{\phi }_{1}}\end{array}$

where $|{C}^{1}|=\sqrt{{\left({C}_{0}^{1}\right)}^{2}+{\left({C}_{1}^{1}\right)}^{2}+{\left({C}_{2}^{1}\right)}^{2}+{\left({C}_{3}^{1}\right)}^{2}}$ , $|{C}_{B}^{1}|=\sqrt{{\left({C}_{1}^{1}\right)}^{2}+{\left({C}_{2}^{1}\right)}^{2}+{\left({C}_{3}^{1}\right)}^{2}}$ ,

$\mathrm{cos}{\phi }_{1}=\frac{{C}_{0}^{1}}{|{C}^{1}|}$ , $\mathrm{sin}{\phi }_{1}=\frac{|{C}_{B}^{1}|}{|{C}^{1}|}$ , ${S}_{1}=\frac{{C}_{1}^{1}{B}_{1}}{|{C}_{B}^{1}|}+\frac{{C}_{2}^{1}{B}_{2}}{|{C}_{B}^{1}|}+\frac{{C}_{3}^{1}{B}_{3}}{|{C}_{B}^{1}|}$

$\begin{array}{c}{C}^{2}\equiv {C}_{0}^{2}+{C}_{1}^{2}{B}_{1}+{C}_{2}^{2}{B}_{2}+{C}_{3}^{2}{B}_{3}\\ =|{C}^{2}|\left(\frac{{C}_{0}^{2}}{|{C}^{2}|}+\frac{|{C}_{B}^{2}|}{|{C}^{2}|}\frac{{C}_{1}^{2}{B}_{1}}{|{C}_{B}^{2}|}+\frac{|{C}_{B}^{2}|}{|{C}^{2}|}\frac{{C}_{2}^{2}{B}_{2}}{|{C}_{B}^{2}|}+\frac{|{C}_{B}^{2}|}{|{C}^{2}|}\frac{{C}_{3}^{2}{B}_{3}}{|{C}_{B}^{2}|}\right)\\ =|{C}^{2}|\left(\mathrm{cos}{\phi }_{2}+\mathrm{sin}{\phi }_{2}{S}_{2}\right)=|{C}^{2}|{\text{e}}^{{S}_{2}{\phi }_{2}}\end{array}$

where $|{C}^{2}|=\sqrt{{\left({C}_{0}^{2}\right)}^{2}+{\left({C}_{1}^{2}\right)}^{2}+{\left({C}_{2}^{2}\right)}^{2}+{\left({C}_{3}^{2}\right)}^{2}}$ , $|{C}_{B}^{2}|=\sqrt{{\left({C}_{1}^{2}\right)}^{2}+{\left({C}_{2}^{2}\right)}^{2}+{\left({C}_{3}^{2}\right)}^{2}}$ ,

$\mathrm{cos}{\phi }_{2}=\frac{{C}_{0}^{2}}{|{C}^{2}|}$ , $\mathrm{sin}{\phi }_{2}=\frac{|{C}_{B}^{2}|}{|{C}^{2}|}$ , ${S}_{2}=\frac{{C}_{1}^{2}{B}_{1}}{|{C}_{B}^{2}|}+\frac{{C}_{2}^{2}{B}_{2}}{|{C}_{B}^{2}|}+\frac{{C}_{3}^{2}{B}_{3}}{|{C}_{B}^{2}|}$

The observable ${C}^{2}$ can be written as:

${C}^{2}=\frac{|{C}^{2}|}{|{C}^{1}|}{C}^{1}{\text{e}}^{-{S}_{1}{\phi }_{1}}{\text{e}}^{{S}_{2}{\phi }_{2}}$ ,

and then its measurement by any g-qubit ${\text{e}}^{S\phi }$ reads:

${\text{e}}^{-S\phi }{C}^{2}{\text{e}}^{S\phi }={\text{e}}^{-S\phi }\frac{|{C}^{2}|}{|{C}^{1}|}{C}^{1}{\text{e}}^{S\phi }{\text{e}}^{-S\phi }{\text{e}}^{-{S}_{1}{\phi }_{1}}{\text{e}}^{{S}_{2}{\phi }_{2}}{\text{e}}^{S\phi }$

that is, up to the factor $\frac{|{C}^{2}|}{|{C}^{1}|}$ , the result of measurement of ${C}^{1}$ , multiplied by the result of measurement of ${\text{e}}^{-{S}_{1}{\phi }_{1}}{\text{e}}^{{S}_{2}{\phi }_{2}}$ .

4. Software Simulation of Analog Computing

The hardware creating superfields requires special implementation of a photonic/laser device that does not exist yet. Instead, we have a very convenient equivalent simulation scheme where the amount of simultaneously available space/time points of observable measured values is only restricted, for example, by the overall available Nvidia GPU number of threads.

Formula (3.1) gives the result of measuring any type of ${G}_{3}^{+}$ element ${C}_{0}+{C}_{1}{B}_{1}+{C}_{2}{B}_{2}+{C}_{3}{B}_{3}$ by the sprefield state.

Let us consider pieces of code for possible implementation.

The CUDA code begins with allocating an initial array of (3.1) on the host at $t=0$ . For example:

size_t memsize = 0;

checkCudaErrors(cudaGetDeviceProperties(&deviceProp, 0));

memsize = deviceProp.totalGlobalMem;

// We can never use all the memory so to keep things simple we aim to

// use around half the total memory

memsize /= 2;

int defaultDim = (int)floor(pow((memsize / (2.0 * sizeof(float))), 1.0 / 3.0));

// above is the size of cube for all considered points

float step = volumeSize.width / defaultDim;

float4* r = (float3*)malloc(memsize * sizeof(float4));

r[0].x = r[0].y = r[0].z = r[0].w = 0.0;

for (int k = 0; k < defaultDim; k++)

{

r[k].x = r[0].x + k * step;

r[k].y = r[0].y + k * step;

r[k].z = r[0].z + k * step;

}

float omega = 12560000.0; // an option from laser beams

float* factor = (float*)malloc(defaultDim * sizeof(float));

for (int k = 0; k < defaultDim; k++)

{

// below from formula (3.1)

factor[k] = 4. * (cosf(omega * r[k].x)) * (cosf(omega * r[k].x));

r[k].x = factor[k] * C3;

r[k].y = factor[k] * C2;

r[k].z = -factor[k] * C1;

r[k].w = factor[k] * C0;

}

Now, copy to the device memory:

Float4* d_r = NULL;

checkCudaErrors(cudaMalloc(&d_r, memsize));

checkCudaErrors(cudaMemcpy(d_r, r, memsize, cudaMemcpyHostToDevice));

Run the CUDA part of calculations and graphics output:

// create VBO

// run the cuda part

runCuda(&cuda_vbo_resource);

// start rendering mainloop

glutMainLoop();

with specially written kernel function

vbo_kernel <<< grid, block >>> (pos, mesh_width, mesh_height, mesh_depth, time);

running inside the runCuda. The vbo_kernel executes in the parallel recalculation of all measured observable values sitting in the cube, changing with time and their graphics output.

5. Conclusions

The geometric algebra lift of conventional quantum mechanics qubits is the game-changing quantum leap forward potentially kicking from the quantum computing market big fishes (IBM, Microsoft, Google, dozens of smaller ones) investing billions in elaborating quantum computing devices. The approach brings into reality a kind of physical field spreading through the whole three-dimensional space and values of the time parameter. The fields can be modified instantly in all points of space and time values. All measured observable values are simultaneously available all together, not through looking one by one. In this way, the new type of quantum computer appeared to be a kind of analog computer keeping and instantly processing information by and on sets of objects possessing an infinite number of degrees of freedom. As a practical implementation, the multithread GPUs with the CUDA language functionality allow the creation of software simulating that kind of field processing numbers of space/time discrete points only restricted by the GPU threads capacity.

Specific problems that can get huge speedups with the suggested analog quantum computers are mainly related to the continuous media calculations and blockchain schemes [8] .

Thus, the innovative scheme of quantum computing with its availability in the near future, incomparable low cost value and scalable simulation software platform is becoming a tremendous challenge for companies like those mentioned above.

The corresponding anticipated objective is to create simulating software running on a GPU, preferably programmed in the Nvidia CUDA language. In that way, we will get a quantum computer as an analog computer simulating instantly parallelizable processing, and transformation of states identified by points on the three-dimensional spheres, that replace the Hilbert space formalism. The amount of simultaneously available space/time points of the observable measured values is then only restricted by the overall available GPU number of threads.

The simulating software functionality will follow the requirements of further developing quantum computing in all areas where its paradigm will give tremendous effectiveness, first of all in speed increasing by orders, from running years to seconds: material science, finances, drug development, cryptography, weather forecasting, to mention just some immediately seen ones.

NOTES

1It is universally possible due to the hedgehog theorem.

2Remember, in the current theory scalars are real ones. “Complex” scalars have no sense.

3Good to remember that the two basic solutions F+ and F differ only by the sign of I3IS, which is caused by orientation of IS that in its turn defines if the triple $\left\{\stackrel{^}{E},\stackrel{^}{H},±{I}_{3}{I}_{S}\right\}$ is right-hand screw or left-hand screw oriented.

4The two received solutions are similar in their form to the Majorana operators, though with a bivector instead of formal imaginary unit. Thus, not surprising that the following computational scheme represents more general software simulation which gets around the problem of physical implementation of Majorana fermions.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

 [1] Aaronson, S. (2008) The Limits of Quantum Computers. Scientific American, 298, 62-69. https://doi.org/10.1038/scientificamerican0308-62 [2] Schatten, K. (2021) A Physical Origin for Quantum Entanglement and Probabilistic Behaviors. Journal of Modern Physics, 12, 50-58. https://doi.org/10.4236/jmp.2021.121005 [3] Soiguine, A. (2020) Scattering of Geometric Algebra Wave Functions and Collapse in Measurements. Journal of Applied Matrhematics and Physics, 8, 1838-1844. https://doi.org/10.4236/jamp.2020.89138 [4] Soiguine, A. (2022) The 3-Sphere Instead of Hilbert Space. Journal of Applied Mathematics and Physics, 10, 2733-2742. https://doi.org/10.4236/jamp.2022.109183 [5] Soiguine, A. (2023) Two Shaky Pillars of Quantum Computing. Journal of Applied Mathematics and Physics, 11, 448-456. https://doi.org/10.4236/jamp.2023.112027 [6] Soiguine, A. (2020) The Geometric Algebra Lift of Qubits and Beyond. Lambert Academic Publishing, Mauritius. [7] Soiguine, A.M. (1996) Complex Conjugation—Relative to What? In: Abłamowicz, R., Parra, J.M. and Lounesto, P., Eds., Clifford Algebras with Numeric and Symbolic Computations, Boston, 284-294. https://doi.org/10.1007/978-1-4615-8157-4_19 [8] Soiguine, A. (2023) Quantum Computer as Analog Computer on GPU. Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 23, 1-12.