The Geometric Algebra Lift of Qubits and Beyond
Abstract
The book follows the Geometric Algebra formalism generalizing complex number theory. This mathematical structure provides, along with intuitively clear geometrical interpretation, much deeper description of physical phenomena superficially described by conventional quantum mechanics.
... is also solution of (1.2). The item in the second parenthesis is a weighted linear combination of two states (wave functions, g-qubits [4], [5]) with the same phase in the same plane but the opposite sense of orientation. The states are strictly coupled, entangled if you prefer, because the bivector plane should be the same for both, no matter what happens with that plane. ...
... In our torsion mechanics [7] states, wave functions, are identified by points on three-sphere 3 .A state 4 there is, see [5], Sec.2.5: ...
... Suppose a fluid is given in a three-dimensional region by continuously differential vector field ⃗( ), the fluid velocity. At every point we have a value ⃗( )that can be identified by = 0 + 1 1 + 2 2 + 3 3 ≡ ( ) ( ) with geometrically known entities [2]- [5]. The ⃗( ) is the infinitesimal volume density of the net vector circulation, that is magnitude and spatial orientation of the field around the point. ...
The geometric algebra lift of conventional quantum mechanics qubits is the gamechanging
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. It 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
practical implementation, the multithread GPUs with the CUDA language functionality allow
creating of software simulating that kind of fields processing numbers of space/time discrete
points only restricted by the GPU threads capacity.
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see, [3], [4], [7], [6], [8]. ...
... In the current formalism scalars can only be real numbers. "Complex" scalars make no sense anymore, see, for example,[4],[8]. ...
... Some other highly impressive perspectives of the approach comprise, particularly, explanations of the double-slit experiment, collapse of wave functions [9], possibility to modify blockchain information back and forth in time, see Subsection 6.3 of [8]. All that supports feasibility of the suggested approach to replace formalism of conventional quantum theory. ...
Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum theory (Mathematics Subject Classification, item 81). 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 planetary system, very familiar from macroscopic experience but recklessly used in a physically very different situation. In the current work the three-sphere becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The states as 3 points evolve, governed by updated Schrodinger equation, and act as operators on observables in measurements. ALEXANDER SOIGUINE 30
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see, [3], [4], [7], [6], [8]. ...
... The product of two exponents is again an exponent, because generally | 1 2 | = | 1 || 2 | and | 1 2 | = | 1 || 2 | = 1, see Sec.1.3 of [8]. ...
... Some other highly impressive perspectives of the approach comprise, particularly, explanations of the double-slit experiment, collapse of wave functions [9], possibility to modify blockchain information back and forth in time, see Sec.6.3 of [8]. All that supports feasibility of the suggested approach to replace formalism of conventional quantum theory. ...
... Wave functions as elements of 3 + are naturally mapped onto unit sphere 3 [2], [3], [4]. ...
... Wave function can always be conveniently written as exponent, see [2], Sec.2.5,: ...
... 1 2 = ( 1 + 1 1 )( 2 + 2 2 ) = 1 2 + 1 2 1 + 2 1 2 + 1 2 1 2 It is not commutative due to the not commutative product of bivectors 1 2 . Indeed, taking vectors to which 1 and 2 are dual: 1 = − 3 1 , 2 = − 3 2 , we have, see [2], sec.1.1: ...
Quantum computing rests upon two theoretical pillars: entanglement and superposition. But some physicists say that this is a very shaky foundation and quantum computing success will have to be based on a different theoretical foundation. The g-qubit theory supports this point of view. Current article is the second one of the two and about the entanglement. It gives different, more physically feasible, not mysterious, explanation of what the entanglement is. The suggested formalism demonstrates that the core of future quantum computing should not be in entanglement which only formally follows in conventional quantum mechanics from representation of the many particle states as tensor products of individual states. The core of quantum computing scheme should be in manipulation and transferring of wave functions on 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.
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see, [1], [2], [4], [5]. ...
... The product of two exponents is again an exponent, because generally | 1 2 | = | 1 || 2 | and | 1 2 | = | 1 || 2 | = 1, see Sec.1.3 of [5]. ...
... The result of measurement after multiple transformations reads: (proton.) 5 Rotation by the double of the exponential is known from rotational rules in three-dimensional geometric algebra, see, for example [3]. ...
The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of some weird quantum mechanical features, particularly, the results of double-slit experiments where particles create diffraction patterns inherent to a wave, or modeling atoms as a kind of solar system. Abstract-The Geometric Algebra formalism opens the door to developing a theory replacing conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions, followed by unambiguous definition of states, observables, measurements, bring into reality clear explanations of some weird quantum mechanical features, particularly, the results of double-slit experiments where particles create diffraction patterns inherent to a wave, or modeling atoms as a kind of solar system.
... Usage of even subalgebra 3 G + of geometric algebra 3 G [1] [2] [3] stems from generalization of complex numbers [3] [4]. The sprefield wave functions (states) received as special 3 G + solutions of Maxwell equations [5] [6]. ...
... Usage of even subalgebra 3 G + of geometric algebra 3 G [1] [2] [3] stems from generalization of complex numbers [3] [4]. The sprefield wave functions (states) received as special 3 G + solutions of Maxwell equations [5] [6]. ...
... Usage of even subalgebra 3 G + of geometric algebra 3 G [1] [2] [3] stems from generalization of complex numbers [3] [4]. The sprefield wave functions (states) received as special 3 G + solutions of Maxwell equations [5] [6]. ...
... 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 processing information by and on sets of objects possessing an infinite number of degrees of freedom [3]. 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. ...
... = volumeSize; copyParams.kind = cudaMemcpyDeviceToDevice; checkCudaErrors ( cudaMemcpy3D(©Params) ); while ( 3 , , 0 , 0 , , 1 , 2 , 3 , , , ) also get instantly changed for all values of time of measurement, even if the Clifford translation was applied later than the measurement. That is an obvious demonstration that the suggested theory allows indefinite event casual order. ...
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 [1]. 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 [2]. 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 processing information by and on sets of objects possessing an infinite number of degrees of freedom [3]. 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.
... A theory that is an alternative to conventional quantum mechanics has been under development for a while, see [1] [2] [4] [5]. ...
... It is not commutative due to the not commutative product of bivectors In the current formalism scalars can only be real numbers. "Complex" scalars make no sense anymore, see, for example [2] [5]. Journal of Applied Mathematics and Physics ...
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 3 becomes the playground of the torsion kind states eliminating abstract Hilbert space vectors. The 3 points evolve, governed by updated Schrodinger equation, and act in measurements on observable as operators .
... In the same context, Soiguine [57] demonstrates how the the double split experiment results can be resolved with diffraction patterns inherent to wave diffraction. In his work, he exploits the GA formalism along with generalization of complex numbers and subsequent lift of the two-dimensional Hilbert space-valued qubits to geometrically feasible elements of an even GA subalgebra. ...
This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics.
... Wave function can always be conveniently written as exponent, see [7], Sec. 2. 5, Multiplication of an exponent by another exponent is often called Clifford translation. Using the term translation follows from the fact that Clifford translation does not change distances between the exponents it acts upon if we identify exponents as points on unit sphere 3 : ...
... The 3 elements of the form 3 = + , where is some unit bivector arbitrary placed in three-dimensional space, comprise so called even subalgebra of algebra 3 . This subalgebra is denoted by 3 + (Soiguine, 2015(Soiguine, , 2020. Elements of 3 + can be depict as in Figure 1. ...
The Geometric Algebra formalism opens the door to developing a theory upgrading conventional quantum mechanics. Generalizations, stemming from implementation of complex numbers as geometrically feasible objects in three dimensions; unambiguous definition of states, observables, measurements bring into reality clear explanations of conventional weird quantum mechanical features, particularly the results of double split experiments where particles create diffraction patterns inherent to wave diffraction. This weirdness of the double split experiment is milestone of all further difficulties in interpretation of quantum mechanics.
The superiority of hypothetical quantum computers is not due to faster cal-culations but due to different schemes of calculations running on specialhardware. The core of quantum computing follows the way a state of a quan-tum system is defined when basic things interact with each other. In conven-tional approach it is implemented through tensor product of qubits. In thegeometric algebra formalism simultaneous availability of all the results fornon-measured observables is based on the definition of states as points onthree-dimensional sphere.
(PDF) Parallelizable Calculation of Observables Values on Analog Quantum Computer. Available from: https://www.researchgate.net/publication/382341953_Parallelizable_Calculation_of_Observables_Values_on_Analog_Quantum_Computer#fullTextFileContent [accessed Jul 19 2024].
Geometric Algebra formalism opens the door to developing a theory deeperthan conventional quantum mechanics. Generalizations, stemming from im-plementation of complex numbers as geometrically feasible objects in threedimensions, unambiguous definition of states, observables, measurements,Maxwell equations solution in those terms, bring into reality a kind of physicalfields spreading through the whole three-dimensional space and values of thetime parameter. The fields can be modified instantly in all points of space andtime values, thus eliminating the concept of cause and effect, and perceivingof one-directional time. In the suggested theory all measured observable val-ues get available all together, not through looking one by one. In this wayquantum computer appeared to be a kind of analog computer keeping andinstantly processing information by and on sets of objects possessing an in-finite number of degrees of freedom. As practical implementation, the mul-tithread GPUs bearing the CUDA language functionality allow to simulta-neously calculate observable measurement values at a number of space/timediscrete points only restricted by the GPU threads capacity.
(PDF) Quantum Computer on Nvidia GPU. Available from: https://www.researchgate.net/publication/373089489_Quantum_Computer_on_Nvidia_GPU [accessed Oct 16 2023].
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