In this paper, methods for performing the Hadamard transform are analyzed to organize quantuminspired computations. The Hadamard transform is illustrated on the Bloch sphere and on the unit circle, both as a direct transformation and as a result of successive rotations of a vector. In addition, the effect of the Hadamard gate on the smoothly changing state of the qubit was investigated and the corresponding graph is presented. The paper proposes a qubit model in the form of a modified unit circle, which combines operations only with real numbers in the standard unit circle and symmetry of the Bloch sphere. This makes it possible to abandon the analysis of the vector phase when analyzing the behavior of the qubit model, that is, to abandon the use of complex numbers (amplitudes) when simulating. The model is oriented towards implementation in FPGA.
In this paper, methods for performing the Hadamard transform are analyzed to organize quantuminspired computations. The Hadamard transform is illustrated on the Bloch sphere and on the unit circle, both as a direct transformation and as a result of successive rotations of a vector. In addition, the effect of the Hadamard gate on the smoothly changing state of the qubit was investigated and the corresponding graph is presented. The paper proposes a qubit model in the form of a modified unit circle, which combines operations only with real numbers in the standard unit circle and symmetry of the Bloch sphere. This makes it possible to abandon the analysis of the vector phase when analyzing the behavior of the qubit model, that is, to abandon the use of complex numbers (amplitudes) when simulating. The model is oriented towards implementation in FPGA.
This work investigates the transformation and behavior of quantum states using Bloch sphere visualizations to provide an intuitive understanding of quantum phenomena, incorporating concepts from both quantum computing and nano physics.By applying quantum gates such as Hadamard, Pau-li-X, Pauli-Y, and Pauli-Z, the evolution of qubits is demonstrated.The study leverages principles of quantum superposition, entanglement, and coherence, which are pivotal in nanoscale systems, to analyze qubit dynamics.Additionally, phase noise, a critical factor in nanoscale quantum devices, is introduced to simulate real-world quantum errors, and fidelity is calculated to measure the robustness of quantum states under such noise.The approach addresses noise resilience in nanoscale quantum devices by combining phase noise analysis, fidelity measurement, and quantum entanglement.The novelty of the approach is emphasized through comparisons with the existing studies on quantum state tomography and machine learning-based quantum optimization.Results will have direct applications in quantum key distribution (QKD), quantum error correction, and secure quantum communication.
The Bloch sphere is a generalisation of the complex number z with |z|2 = 1 being represented in the complex plane as a point on the unit circle. The goal of the research is to create a simulation that can be used to visualise a Bloch sphere of a single quantum bit, also known as a Qbit. QISKIT (developed by IBM) is an open-source lab for education in the realm of quantum computing, and is used to test and validate this simulator. This study made use of both quantitative and qualitative methods of investigation.
Quantum computing leverages fundamental principles of quantum mechanics to perform computations far exceeding classical architectures. At the core of single-qubit quantum state manipulation is the Bloch sphere, a geometric representation mapping the state space of two-level quantum systems onto a three-dimensional unit sphere. This paper presents a comprehensive theoretical framework of qubits, starting from single-qubit geometric state vectors and expanding to rigorous multi-qubit mathematical formulations. Furthermore, it analyzes single-qubit quantum gates as unitary rotations on the Bloch sphere, evaluates multi-qubit system scaling, and explicitly proves state normalization conditions using Born's rule across multi-dimensional Hilbert spaces.
Abstract In quantum circuits, qubits and the quantum gates acting on them have traditionally been analysed using matrix algebra and Dirac notation. While powerful, these can be unintuitive for conceptual understanding and rapid problem solving. In this work, a new schematic representation method is developed that visualizes the effects of quantum gates on qubits without relying on complex mathematical operations. In the new notation, quantum bits (qubits) are represented using black (|0〉) and white (|1〉) circles. When a quantum gate is applied to a qubit, the circle representing the qubit is visually modified. For example, a Hadamard gate transforms a solid black or white circle into a half-black and half-white circle representing superposition. The work shows how this method simplifies the visualization of quantum algorithms, entanglement, and multi- qubit operations. The work proposes how this method can act as a visual aid to support the understanding of quantum algorithms, entanglement, and multi-qubit operations. Thus, the initial conceptual analysis of basic quantum circuits can be facilitated by simple schematic representations, reducing the immediate cognitive load of complex mathematical operations for beginners.
Abstract Quantum computing utilizes the concepts of quantum mechanics, including superposition and entanglement, to execute calculations that are impractical for traditional computers. Central to this approach is the qubit, a quantum equivalent of the traditional bit, capable of being in a superposition of states. Nonetheless, the conceptual essence of quantum states renders them hard to visualize and comprehend, particularly for students and professionals who are unfamiliar with the domain. To tackle this issue, we created an interactive qubit visualization tool using Streamlit, enabling real-time exploration of qubit behavior under quantum operations.The app enables users to control qubit states directly by modifying the real and imaginary components of probability amplitudes, guaranteeing the normalization of the state vector. It also incorporates a set of essential quantum gates, including Hadamard, Pauli-X, Pauli-Y, and Pauli-Z, as well as continuous rotation gates (Rx, Ry, Rz). In contrast to distinct button-driven functions, the rotation gates are implemented dynamically via angle sliders, enabling users to witness seamless, real-time updates to the qubit state. This provides an intuitive understanding of qubit evolution as it revolves around the Bloch sphere axes.The tool displays the outcomes in both numerical and graphical formats. Users are able to observe the quantum state vector, as well as the probabilities of obtaining |0⟩ and |1⟩, and visual depictions through probability bar charts and Bloch sphere visualizations. By providing a live, interactive setting, this platform connects the divide between mathematical theory and conceptual grasp of quantum mechanics. It functions as both a learning tool for students and a prototyping support for researchers, rendering quantum computation more approachable and captivating.