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Quantum Computingvia Entanglement in Geometric Algebra Approach

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Journal of Applied Mathematics and Physics 2024年第2期12卷 445-457页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo USA

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.

关键词： Geometric Algebra Wave Functions Entanglement Maxwell Equations Three-Dimensional Sphere

The 3-Sphere Instead of Hilbert Space

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Journal of Applied Mathematics and Physics 2022年第9期10卷 2733-2742页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo USA

The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from changing of complex numbers by geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of weird quantum mechanical features, for example primitively considering atoms as a kind of solar system. The three-sphere S3 becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The S3 points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators.

关键词： Geometric Algebra States Observables Measurements

Two Shaky Pillars of Quantum Computing

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Journal of Applied Mathematics and Physics 2023年第2期11卷 448-456页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo USA

Superposition and entanglement are two theoretical pillars quantum computing rests upon. In the g-qubit theory quantum wave functions are identified by points on the surface of three-dimensional sphere S3. That gives different, more physically feasible explanation of what superposition and entanglement are. The core of quantum computing scheme should be in manipulation and transferring of wave functions on S3 as operators acting on observables and formulated in terms of geometrical algebra. In this way quantum computer will be a kind of analog computer keeping and processing information by sets of objects possessing infinite number of degrees of freedom, contrary to the two value bits or two-dimensional Hilbert space elements, qubits.

关键词： Geometric Algebra Wave Functions Observables Measurements

Quantum Computer on Nvidia GPU

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Journal of Applied Mathematics and Physics 2023年第8期11卷 2195-2204页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo USA

Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, unambiguous definition of states, observables, measurements, Maxwell equations solution in those terms, bring into reality a kind of physical fields 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, thus eliminating the concept of cause and effect, and perceiving of one-directional time. In the suggested theory all measured observable values get available all together, not through looking one by one. In this way 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 practical implementation, the multithread GPUs bearing the CUDA language functionality allow to simultaneously calculate observable measurement values at a number of space/time discrete points only restricted by the GPU threads capacity.

关键词： Geometric Algebra Quantum Mechanics Wave Functions Maxwell Equations GPU CUDA

Instantly Propagating States

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Journal of Applied Mathematics and Physics 2021年第3期9卷 468-475页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo USA

The current work considers the sprefield wave functions received as special g-qubit solutions of Maxwell equations in the terms of geometric algebra. I will call such g-qubits spreons or sprefields. The purpose of this article is to analyze behavior of such wave functions in scattering and measurements. It is shown that sprefields are defined through the whole three-dimensional space at all values of the time parameter. They instantly change all their values when get scattered, that is subjected to Clifford translation. In “measurements”, when a sprefield acts on a static geometric algebra element through the Hopf fibration, sprefield collapses and new geometric algebra non static, rotating element is thereby created.

关键词： Wave Functions Geometric Algebra Measurements Scattering Entanglement

Scattering of Geometric Algebra Wave Functions and Collapse in Measurements

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Journal of Applied Mathematics and Physics 2020年第9期8卷 1838-1844页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo CA USA

The research considers wavelike objects that are elements of even subalgebra of geometric algebra in three dimensions. The used formalism particularly eliminates long existing confusion about the reasons behind the appearance of the imaginary unit in quantum mechanics and introduces clear definition of wave functions. When a wave function acts through the Hopf fibration on a localized geometric algebra element, that is executing a measurement, the result can be named as “collapse” of the wave function.

关键词： Wave Functions Geometric Algebra Measurements Scattering Entanglement Dualism

Polarizations as States and Their Evolution in Geometric Algebra Terms with Variable Complex Plane

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Journal of Applied Mathematics and Physics 2018年第4期6卷 704-714页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo CA USA

Recently suggested scheme?[1] of quantum computing uses g-qubit states as circular polarizations from the solution of Maxwell equations in terms of geometric algebra, along with clear definition of a complex plane as bivector in three dimensions. Here all the details of receiving the solution, and its polarization transformations are analyzed. The results can particularly be applied to the problems of quantum computing and quantum cryptography. The suggested formalism replaces conventional quantum mechanics states as objects constructed in complex vector Hilbert space framework by geometrically feasible framework of multivectors.

关键词： Quantum Mechanics Quantum Computing Geometric Algebra Maxwell Equations

Instantaneous Spreading of the g-Qubit Fields

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Journal of Applied Mathematics and Physics 2019年第3期7卷 591-602页

作者： alexander soiguine soiguine Quantum Computing Aliso Viejo CA USA

The Geometric Algebra formalism opens the door to developing a theory deeper than conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, unambiguous definition of states, observables, measurements, Maxwell equations solution in that terms, bring into reality a kind of physical fields, states in the suggested theory, 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, thus eliminating the concept of cause and effect and perceiving of one-directional time.

关键词： Quantum States Clifford Translations Geometric Algebra