Analysis of Electron Spin States in Magnetic Fields and Hardware Implementation of Quantum Computers

This paper systematically reviews the theoretical progression from blackbody radiation to the foundations of quantum computing. Beginning with Planck’s energy quantization and the representation of electromagnetic wave electric field components, the wave-particle duality of photons is analogized to introduce the de Broglie relation between momentum and wave vector for matter waves, leading to the establishment of the matter wave function and the Schrödinger equation as its governing partial differential equation. The consistency between the rigorous solution for a one-dimensional infinitely deep rectangular potential well and the conclusions for a one-dimensional hydrogen atom is demonstrated. Subsequently, within the Dirac notation formalism, the wave function is expressed, and the correspondence between spatial differentiation and momentum is derived, formalizing the momentum operator (often referred to as first quantization). To lay the physical groundwork for manipulating two-qubit states in quantum computing, the potential energy function of an electron’s orbital magnetic moment in an external magnetic field is introduced, starting from the torque and work on a current-carrying rectangular coil. This is extended to the magnetic moment and potential energy of a spinning electron, where the orientation of the magnetic field determines the form of the potential and accordingly selects the three components of the Pauli operators. The properties of quantum gates corresponding to the Pauli operators are then examined for single-qubit states. A two-qubit state, constructed via the coupling of two arbitrary single-qubit states represented in a two-level system or electron spin, is shown to be uniquely expressible as a superposition of four eigen two-qubit states. The paper further investigates joint manipulations of two-qubit states, including evolution under X and Z gates conditioned on high- and low-level states, the representation of control-flow circuits (with the low-level as control), and the formulation of measurement and CNOT gates. Throughout this work, essential mathematical tools—including integration, matrix operations, complex Euler’s formula, and solutions to simple second-order ordinary differential equations—are incorporated to provide a coherent theoretical framework for state representation and gate operations in quantum computing.