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arXiv:1210.7552

The Significance to Quantum Computing of the Classical Harmonic Nature of Energy Eigenstates

Steven Kenneth Kauffmann

Published Oct 20, 2012• 1 citation
arXivHTMLPDF

Abstract

Since a pure quantum system is incapable of faithfully simulating the solutions of the Schroedinger equation that actually pertains to itself, it is proposed that quantum computing technology (as opposed to cryptographic technology) not be based on pure quantum systems such as qubits but instead on physical systems which by their nature faithfully simulate the solutions of Schroedinger equations. Every Schroedinger equation is within a unitary transformation of being a set of mutually independent classical simple harmonic oscillator equations. Thus classical simple harmonic oscillators, or "chobits", are the mathematically fundamental building blocks for all Schroedinger equations. In addition, classical harmonic oscillators are, as a practical matter, far easier to deal with than any pure quantum system -- e.g., their phases and absolute amplitudes are readily physically accessible, they have little predilection for environmental decoherence, and they abound as cavity electromagnetic standing-wave modes. We study in mathematical detail the use of chobits to compute discrete quantum Fourier transforms, including gates, chobit counts, and chobit operation counts. The results suggest that thirty chobits and under a thousand chobit phase operations could generate discrete quantum Fourier transforms of a billion terms. Chobits can be technologically realized as semiconductor dynatron-type electronic oscillator circuits, which ought to be amenable to very considerable miniaturization.

DOI
10.48550/arxiv.1210.7552

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Since a pure quantum system is incapable of faithfully simulating the solutions of the Schroedinger equation that actually pertains to itself, it is proposed that quantum computing technology (as opposed to cryptographic technology) not be based on pure quantum systems such as qubits but instead on physical systems which by their nature faithfully simulate the solutions of Schroedinger equations. Every Schroedinger equation is within a unitary transformation of being a set of mutually independent classical simple harmonic oscillator equations. Thus classical simple harmonic oscillators, or "chobits", are the mathematically fundamental building blocks for all Schroedinger equations. In addition, classical harmonic oscillators are, as a practical matter, far easier to deal with than any pure quantum system -- e.g., their phases and absolute amplitudes are readily physically accessible, they have little predilection for environmental decoherence, and they abound as cavity electromagnetic standing-wave modes. We study in mathematical detail the use of chobits to compute discrete quantum Fourier transforms, including gates, chobit counts, and chobit operation counts. The results suggest that thirty chobits and under a thousand chobit phase operations could generate discrete quantum Fourier transforms of a billion terms. Chobits can be technologically realized as semiconductor dynatron-type electronic oscillator circuits, which ought to be amenable to very considerable miniaturization.
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The time-dependent Schroedinger equation with time-independent Hamiltonian matrix is a homogeneous linear oscillatory system in canonical form. We investigate whether any classical system that itself is linear, homogeneous, oscillatory and conservative is guaranteed to linearly map into a Schroedinger equation. Such oscillatory classical systems can be analyzed into their normal modes, which are mutually independent, uncoupled simple harmonic oscillators, and the equation of motion of such a system linearly maps into a Schroedinger equation whose Hamiltonian matrix is diagonal, with h times the individual simple harmonic oscillator frequencies as its diagonal entries. Therefore if the coupling-strength matrix of such an oscillatory system is presented in symmetric, positive-definite form, the Hamiltonian matrix of the Schroedinger equation it maps into is h times the square root of that coupling-strength matrix. We obtain a general expression for mapping this type of oscillatory classical equation of motion into a Schroedinger equation, and apply it to the real-valued classical Klein-Gordon equation and the source-free Maxwell equations, which results in relativistic Hamiltonian operators that are strictly compatible with the correspondence principle. Once such an oscillatory classical system has been mapped into a Schroedinger equation, it is automatically in canonical form, making second quantization of that Schroedinger equation a technically simple as well as a physically very interpretable way to quantize the original classical system.
Quantum computing leverages quantum-mechanical phenomena such as superposition, entanglement and interference to solve classes of problems that are intractable on classical machines.This paper unifies three complementary perspectives: (i) the central role of complex matrices in modelling quantum states, gates and error-correction protocols; (ii) the pervasive use of trigonometric functions in qubit representation, phase manipulation and the quantum Fourier transform (QFT); and (iii) a systematic comparison of classical and quantum computing paradigms with illustrative circuits and real-world applications.By integrating these dimensions, the study provides a consolidated reference that elucidates the mathematical foundations, circuit constructions, algorithmic advantages and practical challenges that define contemporary quantum information processing.
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The Wave-particle duality and the Schrodinger cat paradox pose important notions on quantum mechanical systems, in which a quantum system can exist in two physically opposite or distinct states. This motivates scientists from the days of Louis de Broglie to confidently look at matter as having a dual nature, which is a physical particle that can also be visualized as a wave entity. This paper seeks to provide an elegant method of solving the time-independent Schrodinger Equation (TISE) for a quantum simple harmonic oscillator (QSHO). A fully algebraic method is presented, the paper explains in detail how simple algebra, coupled with basic quantum mechanical operator commutation algebra can be employed to solve the TISE without resorting to complicated mathematical methods of solving second-order differential equations. The construction of the Hamiltonian is explained and an analogy between the classical simple harmonic oscillator (CSHO) and the QSHO is drawn to develop the solution method. The action of creation and annihilation operators on wave functions is exploited to generate states with higher and lower energies respectively and to generate a Hamiltonian that can act on a system and generate required quantum states. The aim is to generate the ground state of the QSHO and its associated energy value. From this state, other higher energy states can be generated by the action of the creation operator. The use of algebraic methods has been seen to be versatile as it does not explicitly rely on any particular basis or coordinate system. The method allows operators to be manipulated in different bases for example the momentum operator can be given as P ̂ x=-iℏ (∂/∂x) in a spatial coordinate basis or as a pure number p ̂ =p in the momentum basis.